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sursumCorda

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Nearly fails to evaluate 'int(1/(z^2 - s...

sursumCorda 1284 Maple 2023
integration ode dsolve
May 02 2023
1 2

The issue arises from solving the following ODEs in Maple (where a is a suitable real parameter): 

ode__1 := a*(diff(y(x), x) + 1)^2 + (y(x) - x)^2*diff(y(x), x) = 0: # dsolve(ode__1);
ode__4 := a*(x*diff(y(x), x) + y(x))^2 - (y(x) + x)^2*diff(y(x), x) = 0: # dsolve(ode__4);

However, dsolve cannot give fully simplified solutions, so I have to compute these unevaluated integrals (i.e., expr1) manually: (For the sake of completeness, I list some related ODEs below.) 
 

> 

restart;
> 

ode__1 := a*(diff(y(x), x)+1)^2+(y(x)-x)^2*(diff(y(x), x)) = 0
ode__4 := a*(x*(diff(y(x), x))+y(x))^2-(y(x)+x)^2*(diff(y(x), x)) = 0
> 

dsolve(ode__1, y(x), explicit)
> 

expr__1 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int(1/(z^2+(z^4+4*a*z^2)^(1/2)+4*a), z), Int(-1/(z^2-(z^4+4*a*z^2)^(1/2)+4*a), z)]

 (1)
> 

value(expr__1)

[(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))-4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2))), -(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))+(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))+(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2)))]

 (2)
> 

verify(diff([-z/(z^2+sqrt(z^2*(z^2+4*a))), z/(z^2-sqrt(z^2*(z^2+4*a)))], z), `~`[op](1, expr__1), simplify)

true

 (3)
> 

dsolve(ode__4, y(x), explicit)
> 

expr__4 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int((z^2-4*a*z+(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)+2*z+1)/(z*(-4*a*z+z^2+2*z+1)), z), Int(-(z^2-4*a*z+2*z+1-((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2))/(z*(-4*a*z+z^2+2*z+1)), z)]

 (4)
> 

value(expr__4)

[(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))+ln(z), ((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))-ln(z)]

 (5)
> 

verify(diff([2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))+ln(z), 2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))-ln(z)], z), `~`[op](1, expr__4), simplify)

true

 (6)
> 

NULL


 

Download senseless_results_of_int.mw
 

> 

restart;
> 

ode__1 := a*(diff(y(x), x)+1)^2+(y(x)-x)^2*(diff(y(x), x)) = 0
ode__4 := a*(x*(diff(y(x), x))+y(x))^2-(y(x)+x)^2*(diff(y(x), x)) = 0
> 

dsolve(ode__1, y(x), explicit)
> 

expr__1 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int(1/(z^2+(z^4+4*a*z^2)^(1/2)+4*a), z), Int(-1/(z^2-(z^4+4*a*z^2)^(1/2)+4*a), z)]

 (1)
> 

value(expr__1)

[(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))-4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2))), -(1/2)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))+(1/2)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(1/2))+(z^4+4*a*z^2)^(1/2)*((1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)+2*(-a)^(1/2)*ln(z+((z-2*(-a)^(1/2))^2+4*(-a)^(1/2)*(z-2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)*(((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)-2*(-a)^(1/2)*ln(z+((z+2*(-a)^(1/2))^2-4*(-a)^(1/2)*(z+2*(-a)^(1/2)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))+(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)-2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^2-4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2)))-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)^2*(((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*ln(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)-2*(z^4+4*a*z^2)*a*ln((8*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)*((z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))^2+4*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)*(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1))/((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)+4*(z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2))/(z+2*(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)/(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*((z^4+4*a*z^2)*a/(z^2*(z^2+4*a)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)))^(1/2)))/((-(-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)+(-a)^(1/2))*((-a)^(1/2)*(z^4+4*a*z^2)/(z^2*(z^2+4*a))+(a*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))-1)*((z^4+4*a*z^2)^(1/2)/(z*(z^2+4*a)^(1/2))+1))^(1/2)-(-a)^(1/2))*(-(z^4+4*a*z^2)/(z^2*(z^2+4*a))+1)))/(z*(z^2+4*a)^(1/2))+4*a*(-(1/8)*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)*z^2*(z^2+4*a)*arctanh((1/2)*z*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1)/(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))/((z^4+4*a*z^2)*a*(a*((z^4+4*a*z^2)/(z^2*(z^2+4*a))-1))^(1/2))-(1/8)*z^2*(z^2+4*a)*arctan((1/2)*z/a^(1/2))/((z^4+4*a*z^2)*a^(3/2)))]

 (2)
> 

verify(diff([-z/(z^2+sqrt(z^2*(z^2+4*a))), z/(z^2-sqrt(z^2*(z^2+4*a)))], z), `~`[op](1, expr__1), simplify)

true

 (3)
> 

dsolve(ode__4, y(x), explicit)
> 

expr__4 := convert(useInt(eval(selectfun([%], Intat), _Z = z)), list)

[Int((z^2-4*a*z+(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)+2*z+1)/(z*(-4*a*z+z^2+2*z+1)), z), Int(-(z^2-4*a*z+2*z+1-((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2))/(z*(-4*a*z+z^2+2*z+1)), z)]

 (4)
> 

value(expr__4)

[(-4*a*z^3+z^4-8*a*z^2+4*z^3-4*a*z+6*z^2+4*z+1)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))+ln(z), ((-4*a*z+z^2+2*z+1)*(z+1)^2)^(1/2)*(ln(z-2*a+1+(-4*a*z+z^2+2*z+1)^(1/2))+arctanh((2*a*z-z-1)/(-4*a*z+z^2+2*z+1)^(1/2)))/((z+1)*(-4*a*z+z^2+2*z+1)^(1/2))-ln(z)]

 (5)
> 

verify(diff([2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))+ln(z), 2*arctanh(sqrt((z+1)^2*(z*(z-2*(2*a-1))+1))/(z^2-1))-ln(z)], z), `~`[op](1, expr__4), simplify)

true

 (6)
> 

NULL


 

Download senseless_results_of_int.mw

 

As you can see, the lengthy output of is nearly meaningless! (And if you want to simplify it, Maple will simply return: Error, (in simplify/recurse) indeterminate expression of the form 0/0.) So, how do I get the simplified results in Maple?
The integrals are: 

expr__1 := [Int(1/(z^2 + sqrt(z^4 + 4*a*z^2) + 4*a), z), Int(-1/(z^2 - sqrt(z^4 + 4*a*z^2) + 4*a), z)]: # (value(expr__1));
expr__4 := [Int((z^2 - 4*a*z + sqrt(-4*a*z^3 + z^4 - 8*a*z^2 + 4*z^3 - 4*a*z + 6*z^2 + 4*z + 1) + 2*z + 1)/(z*(-4*a*z + z^2 + 2*z + 1)), z), Int(-(z^2 - 4*a*z + 2*z + 1 - sqrt((-4*a*z + z^2 + 2*z + 1)*(z + 1)^2))/(z*(-4*a*z + z^2 + 2*z + 1)), z)]: # (value(expr__4)):

Note. By the way, Mma can solve the original ODEs directly and explicitly: 

In[1]:= DSolve[a*(y'[x]+1)^2+(y[x]-x)^2*y'[x]==0,y[x],x,IncludeSingularSolutions->Automatic]

                                   2                3                    2
                  a - x C[1] - C[1]             16 a  - 4 a x C[1] - C[1]
Out[1]= {{y[x] -> ------------------}, {y[x] -> --------------------------}}
                       x + C[1]                     4 a (4 a x + C[1])

In[2]:= DSolve[a*(x*y'[x]+y[x])^2-(y[x]+x)^2*y'[x]==0,y[x],x,IncludeSingularSolutions->Automatic]

                     2 a C[1]       2 a C[1]     2  2 a C[1]
                  a E         (-(a E        ) + a  E         + x)
Out[2]= {{y[x] -> -----------------------------------------------}, 
                                     2 a C[1]
                                  a E         - x
 
                2 a C[1]    2 a C[1]
               E         (-E         + 2 a x)
>    {y[x] -> --------------------------------}}
                    2 a C[1]              2
              2 a (E         - 2 a x + 2 a  x)

Unfortunately, Maple fails to do so.

How do I obtain the solutions to this OD...

sursumCorda 1284 Maple 2023
ode dsolve differential-equations
April 30 2023
2 3

The ODE is: 

eqn := y(x)*(2*x*diff(y(x), x) + y(x)*(diff(y(x), x)^2 - 1)) = -1: # How about another ODE 'lhs(eqn) = +1' ?

Maple can solve it, but I find that (to get all four solutions) I have to execute the dsolve command twice: 
 

> 

restart;
> 

eqn := y(x)*(2*x*(diff(y(x), x))+y(x)*((diff(y(x), x))^2-1)) = -1
> 

dsolve(eqn, {y(x)}, 'parametric', 'singsol' = all)

y(x) = (c__1^2+2*c__1*x+1)^(1/2), y(x) = -(c__1^2+2*c__1*x+1)^(1/2)

 (1)
> 

dsolve(eqn, {y(x)}, 'singsol' = all)

y(x) = (-x^2+1)^(1/2), y(x) = -(-x^2+1)^(1/2), Int(-((_a^2+y(x)^2-1)^(1/2)*_a*y(x)^2-_a^2*y(x)^2-y(x)^4+2*_a^2+3*y(x)^2-2)/((y(x)^2+2*_a-2)*(-y(x)^2+2*_a+2)*(_a^2+y(x)^2-1)), _a = _b .. x)+Intat(-_f/(2*(_f^2+x^2-1)^(1/2)*x^2+(_f^2+x^2-1)^(1/2)*_f^2+2*x^3+2*x*_f^2-2*(_f^2+x^2-1)^(1/2)-2*x)-(Int(-(_a*_f^3/(_a^2+_f^2-1)^(1/2)+2*(_a^2+_f^2-1)^(1/2)*_a*_f-2*_a^2*_f-4*_f^3+6*_f)/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1))+2*((_a^2+_f^2-1)^(1/2)*_a*_f^2-_a^2*_f^2-_f^4+2*_a^2+3*_f^2-2)*_f/((_f^2+2*_a-2)^2*(-_f^2+2*_a+2)*(_a^2+_f^2-1))-2*((_a^2+_f^2-1)^(1/2)*_a*_f^2-_a^2*_f^2-_f^4+2*_a^2+3*_f^2-2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)^2*(_a^2+_f^2-1))+2*((_a^2+_f^2-1)^(1/2)*_a*_f^2-_a^2*_f^2-_f^4+2*_a^2+3*_f^2-2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1)^2), _a = _b .. x)), _f = y(x))+c__1 = 0, Int((_a^2*y(x)^2-2*_a^2+y(x)^4-3*y(x)^2+(_a^2+y(x)^2-1)^(1/2)*_a*y(x)^2+2)/((y(x)^2+2*_a-2)*(-y(x)^2+2*_a+2)*(_a^2+y(x)^2-1)), _a = _b .. x)+Intat(_f/(2*(_f^2+x^2-1)^(1/2)*x^2+(_f^2+x^2-1)^(1/2)*_f^2-2*x^3-2*x*_f^2-2*(_f^2+x^2-1)^(1/2)+2*x)-(Int((2*_a^2*_f+4*_f^3-6*_f+_a*_f^3/(_a^2+_f^2-1)^(1/2)+2*(_a^2+_f^2-1)^(1/2)*_a*_f)/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1))-2*(_a^2*_f^2-2*_a^2+_f^4-3*_f^2+(_a^2+_f^2-1)^(1/2)*_a*_f^2+2)*_f/((_f^2+2*_a-2)^2*(-_f^2+2*_a+2)*(_a^2+_f^2-1))+2*(_a^2*_f^2-2*_a^2+_f^4-3*_f^2+(_a^2+_f^2-1)^(1/2)*_a*_f^2+2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)^2*(_a^2+_f^2-1))-2*(_a^2*_f^2-2*_a^2+_f^4-3*_f^2+(_a^2+_f^2-1)^(1/2)*_a*_f^2+2)*_f/((_f^2+2*_a-2)*(-_f^2+2*_a+2)*(_a^2+_f^2-1)^2), _a = _b .. x)), _f = y(x))+c__1 = 0

 (2)
> 

NULL


 

Download dsolve_twice.mw

However, in MATLAB®, the complete solutions can be found just in one go: 

>> dsolve('y*(2*x*Dy + y*(Dy^2 - 1)) = -1', 'x') % require the Symbolic Math Toolbox™
ans =
                         1
                        -1
 -(-(x - 1)*(x + 1))^(1/2)
  (-(x - 1)*(x + 1))^(1/2)
 (C1^2 + 2*x*C1 + 1)^(1/2)
-(C1^2 + 2*x*C1 + 1)^(1/2)

Does anyone know why?

Is it possible to upgrade some external ...

sursumCorda 1284 Maple 2023
external
April 28 2023
0 0

https://www.maplesoft.com/support/help/Maple/view.aspx?path=copyright lists some external packages used by Maple, but it appears that certain libraries are of outdated (albeit not obsolete) versions. For example, Maple 2023 uses FLINT 2.6.3 (released in 2020), but the newest stable version of FLINT is 2.9.0. Also, Maple 2023 uses Z3 4.5.0 (released in 2016), but the newest stable version of Z3 is 4.12.1. In addition, Maple 2023 uses GCC 10.2.0 (released in 2020), but the newest stable version of GCC is 13.1. Since they are distributed under free licenses, I can download the most recent (or even nightly) release's source code, but how can I replace the old components that Maple uses by the latest ones by myself?

`evala/Minpoly` and PolynomialTools:-Min...

sursumCorda 1284 Maple 2023
polynomial minpoly
April 21 2023
2 2

Here are three algebraic numbers: (In fact, they are solutions to some equation. See the attachment below.)

bSol := {RootOf(1216*_Z^4 + (264*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^8 + 408*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^7 - 1580*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^6 - 6832*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^5 + 3508*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^4 + 9944*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^3 + 9948*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^2 - 10752*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266) + 5204)*_Z^3 + (891*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^8 + 1652*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^7 - 4748*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^6 - 24076*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^5 + 5354*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^4 + 35356*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^3 + 29668*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^2 - 196*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266) + 3971)*_Z^2 + (506*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^8 + 980*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^7 - 2264*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^6 - 12420*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^5 + 3676*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^4 + 11596*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^3 + 33800*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^2 - 7772*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266) + 1210)*_Z - 473*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^8 - 720*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^7 + 2560*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^6 + 10960*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^5 - 8034*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^4 - 13840*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^3 - 9304*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266)^2 + 1104*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 0.2246 .. 0.2266) - 1133, index = real[2]), RootOf(1216*_Z^4 + (264*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^8 + 408*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^7 - 1580*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^6 - 6832*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^5 + 3508*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^4 + 9944*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^3 + 9948*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^2 - 10752*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68) + 5204)*_Z^3 + (891*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^8 + 1652*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^7 - 4748*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^6 - 24076*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^5 + 5354*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^4 + 35356*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^3 + 29668*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^2 - 196*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68) + 3971)*_Z^2 + (506*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^8 + 980*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^7 - 2264*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^6 - 12420*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^5 + 3676*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^4 + 11596*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^3 + 33800*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^2 - 7772*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68) + 1210)*_Z - 473*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^8 - 720*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^7 + 2560*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^6 + 10960*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^5 - 8034*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^4 - 13840*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^3 - 9304*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68)^2 + 1104*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 1.671 .. 1.68) - 1133, index = real[2]), RootOf(1216*_Z^4 + (264*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^8 + 408*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^7 - 1580*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^6 - 6832*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^5 + 3508*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^4 + 9944*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^3 + 9948*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^2 - 10752*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657) + 5204)*_Z^3 + (891*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^8 + 1652*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^7 - 4748*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^6 - 24076*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^5 + 5354*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^4 + 35356*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^3 + 29668*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^2 - 196*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657) + 3971)*_Z^2 + (506*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^8 + 980*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^7 - 2264*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^6 - 12420*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^5 + 3676*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^4 + 11596*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^3 + 33800*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^2 - 7772*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657) + 1210)*_Z - 473*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^8 - 720*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^7 + 2560*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^6 + 10960*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^5 - 8034*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^4 - 13840*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^3 - 9304*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657)^2 + 1104*RootOf(11*_Z^9 + 17*_Z^8 - 64*_Z^7 - 280*_Z^6 + 142*_Z^5 + 370*_Z^4 + 376*_Z^3 - 96*_Z^2 + 47*_Z - 11, 2.648 .. 2.657) - 1133, index = real[2])}:

One may check that 11_X9－47_X8＋96_X7－376_X6－370_X5－142_X4＋280_X3＋64_X2－17_X－11 is an “annihilating” polynomial of each of them (using another computer algebra system); accordingly, the degree of the minimal polynomial cannot be greater than 9. However, Maple's output indicates that the minimal polynomial is of degree 36! 

> 

restart;
> 

alias(`~`[`=`](alpha__ || (1 .. 3), ` $`, RootOf(11*_Z^9+17*_Z^8-64*_Z^7-280*_Z^6+142*_Z^5+370*_Z^4+376*_Z^3-96*_Z^2+47*_Z-11, .2246 .. .2266), RootOf(11*_Z^9+17*_Z^8-64*_Z^7-280*_Z^6+142*_Z^5+370*_Z^4+376*_Z^3-96*_Z^2+47*_Z-11, 1.671 .. 1.68), RootOf(11*_Z^9+17*_Z^8-64*_Z^7-280*_Z^6+142*_Z^5+370*_Z^4+376*_Z^3-96*_Z^2+47*_Z-11, 2.648 .. 2.657)))
> 

({PDETools:-Solve})({`~`[`>=`](a, b, ` $`, 0), a^5*b+4*a^4*b^2+4*a^3*b^3-7*a^4*b-6*a^2*b^3-7*a*b^4+b^5-6*a^3*b+12*a^2*b^2+4*b^4+4*a^3-6*a*b^2+4*b^3+4*a^2-7*a*b+a = 0, a <> b})
bSol := `~`[subs](%, b)
> 

evalf[2*Digits](`~`[eval](11*_X^9-47*_X^8+96*_X^7-376*_X^6-370*_X^5-142*_X^4+280*_X^3+64*_X^2-17*_X-11, `~`[`=`](_X, bSol)))

{RootOf(1216*_Z^4+(264*alpha__1^8+408*alpha__1^7-1580*alpha__1^6-6832*alpha__1^5+3508*alpha__1^4+9944*alpha__1^3+9948*alpha__1^2-10752*alpha__1+5204)*_Z^3+(891*alpha__1^8+1652*alpha__1^7-4748*alpha__1^6-24076*alpha__1^5+5354*alpha__1^4+35356*alpha__1^3+29668*alpha__1^2-196*alpha__1+3971)*_Z^2+(506*alpha__1^8+980*alpha__1^7-2264*alpha__1^6-12420*alpha__1^5+3676*alpha__1^4+11596*alpha__1^3+33800*alpha__1^2-7772*alpha__1+1210)*_Z-473*alpha__1^8-720*alpha__1^7+2560*alpha__1^6+10960*alpha__1^5-8034*alpha__1^4-13840*alpha__1^3-9304*alpha__1^2+1104*alpha__1-1133, index = real[2]), RootOf(1216*_Z^4+(264*alpha__2^8+408*alpha__2^7-1580*alpha__2^6-6832*alpha__2^5+3508*alpha__2^4+9944*alpha__2^3+9948*alpha__2^2-10752*alpha__2+5204)*_Z^3+(891*alpha__2^8+1652*alpha__2^7-4748*alpha__2^6-24076*alpha__2^5+5354*alpha__2^4+35356*alpha__2^3+29668*alpha__2^2-196*alpha__2+3971)*_Z^2+(506*alpha__2^8+980*alpha__2^7-2264*alpha__2^6-12420*alpha__2^5+3676*alpha__2^4+11596*alpha__2^3+33800*alpha__2^2-7772*alpha__2+1210)*_Z-473*alpha__2^8-720*alpha__2^7+2560*alpha__2^6+10960*alpha__2^5-8034*alpha__2^4-13840*alpha__2^3-9304*alpha__2^2+1104*alpha__2-1133, index = real[2]), RootOf(1216*_Z^4+(264*alpha__3^8+408*alpha__3^7-1580*alpha__3^6-6832*alpha__3^5+3508*alpha__3^4+9944*alpha__3^3+9948*alpha__3^2-10752*alpha__3+5204)*_Z^3+(891*alpha__3^8+1652*alpha__3^7-4748*alpha__3^6-24076*alpha__3^5+5354*alpha__3^4+35356*alpha__3^3+29668*alpha__3^2-196*alpha__3+3971)*_Z^2+(506*alpha__3^8+980*alpha__3^7-2264*alpha__3^6-12420*alpha__3^5+3676*alpha__3^4+11596*alpha__3^3+33800*alpha__3^2-7772*alpha__3+1210)*_Z-473*alpha__3^8-720*alpha__3^7+2560*alpha__3^6+10960*alpha__3^5-8034*alpha__3^4-13840*alpha__3^3-9304*alpha__3^2+1104*alpha__3-1133, index = real[2])}

  

{-0.7765721e-11, -0.40e-16, -0.2e-17}

 (1)
> 

`~`[`@`(evala, Minpoly)](bSol, _X)

{-17799961-(10941904462/121)*_X+(61823634144236824/14641)*_X^9-(31748793508955524/14641)*_X^8-(101389427707536/14641)*_X^7+(2187899683524768/14641)*_X^6+(660533278629392/14641)*_X^5-(35195970681077/1331)*_X^4+(4540912173250/1331)*_X^3-(226104907168/1331)*_X^2+_X^36+(562/11)*_X^35+(1306112/1331)*_X^34-(18882494/14641)*_X^33-(1885893201/14641)*_X^32-(8021957456/14641)*_X^31+(128807680096/14641)*_X^30+(601684442192/14641)*_X^29+(136952065956/14641)*_X^28-(7313279407608/14641)*_X^27-(20755313257248/14641)*_X^26-(72279502775080/14641)*_X^25-(235147325265588/14641)*_X^24+(407012808852624/14641)*_X^23-(2003920103008/1331)*_X^22-(2647129453154576/14641)*_X^21-(5329535956015778/14641)*_X^20-(11189597881735324/14641)*_X^19+(18014890583299168/14641)*_X^18-(25692630236542548/14641)*_X^17+(57603516516708946/14641)*_X^16-(875402744452912/121)*_X^15+(36990665431348512/14641)*_X^14+(67887070781490608/14641)*_X^13+(643327218250876/1331)*_X^12-(81888059180050616/14641)*_X^11+(306280599794336/14641)*_X^10}

 (2)
> 

`~`[PolynomialTools[MinimalPolynomial]](bSol, _X)

{14641*_X^36+748022*_X^35+14367232*_X^34-18882494*_X^33-1885893201*_X^32-8021957456*_X^31+128807680096*_X^30+601684442192*_X^29+136952065956*_X^28-7313279407608*_X^27-20755313257248*_X^26-72279502775080*_X^25-235147325265588*_X^24+407012808852624*_X^23-22043121133088*_X^22-2647129453154576*_X^21-5329535956015778*_X^20-11189597881735324*_X^19+18014890583299168*_X^18-25692630236542548*_X^17+57603516516708946*_X^16-105923732078802352*_X^15+36990665431348512*_X^14+67887070781490608*_X^13+7076599400759636*_X^12-81888059180050616*_X^11+306280599794336*_X^10+61823634144236824*_X^9-31748793508955524*_X^8-101389427707536*_X^7+2187899683524768*_X^6+660533278629392*_X^5-387155677491847*_X^4+49950033905750*_X^3-2487153978848*_X^2-1323970439902*_X-260609229001}

 (3)
> 

factor({{-260609229001-1323970439902*_X+407012808852624*_X^23-22043121133088*_X^22-2647129453154576*_X^21-5329535956015778*_X^20-11189597881735324*_X^19+18014890583299168*_X^18-25692630236542548*_X^17+57603516516708946*_X^16-105923732078802352*_X^15+36990665431348512*_X^14+67887070781490608*_X^13+7076599400759636*_X^12-81888059180050616*_X^11+306280599794336*_X^10+61823634144236824*_X^9-31748793508955524*_X^8-101389427707536*_X^7+2187899683524768*_X^6+660533278629392*_X^5-387155677491847*_X^4+49950033905750*_X^3-2487153978848*_X^2+14641*_X^36+748022*_X^35+14367232*_X^34-18882494*_X^33-1885893201*_X^32-8021957456*_X^31+128807680096*_X^30+601684442192*_X^29+136952065956*_X^28-7313279407608*_X^27-20755313257248*_X^26-72279502775080*_X^25-235147325265588*_X^24}[], {-17799961-(10941904462/121)*_X+(407012808852624/14641)*_X^23-(2003920103008/1331)*_X^22-(2647129453154576/14641)*_X^21-(5329535956015778/14641)*_X^20-(11189597881735324/14641)*_X^19+(18014890583299168/14641)*_X^18-(25692630236542548/14641)*_X^17+(57603516516708946/14641)*_X^16-(875402744452912/121)*_X^15+(36990665431348512/14641)*_X^14+(67887070781490608/14641)*_X^13+(643327218250876/1331)*_X^12-(81888059180050616/14641)*_X^11+(306280599794336/14641)*_X^10+(61823634144236824/14641)*_X^9-(31748793508955524/14641)*_X^8-(101389427707536/14641)*_X^7+(2187899683524768/14641)*_X^6+(660533278629392/14641)*_X^5-(35195970681077/1331)*_X^4+(4540912173250/1331)*_X^3-(226104907168/1331)*_X^2+_X^36+(562/11)*_X^35+(1306112/1331)*_X^34-(18882494/14641)*_X^33-(1885893201/14641)*_X^32-(8021957456/14641)*_X^31+(128807680096/14641)*_X^30+(601684442192/14641)*_X^29+(136952065956/14641)*_X^28-(7313279407608/14641)*_X^27-(20755313257248/14641)*_X^26-(72279502775080/14641)*_X^25-(235147325265588/14641)*_X^24}[]})

{(11*_X^9-47*_X^8+96*_X^7-376*_X^6-370*_X^5-142*_X^4+280*_X^3+64*_X^2-17*_X-11)*(83746429305*_X-163433814*_X^23-1409885474*_X^22+7323055726*_X^21+92878340298*_X^20+291711433585*_X^19-28358008525*_X^18-1146850616945*_X^17+2003142623069*_X^16+7054039060380*_X^15+10860482240404*_X^14+4410674835220*_X^13-23715924119108*_X^12+39935154074341*_X^11-76564178781009*_X^10+246946329497683*_X^9-303627746551159*_X^8+41661161235738*_X^7+181533634595246*_X^6-146573328877410*_X^5+44279227597786*_X^4-3813039868649*_X^3+234521505317*_X^2+1331*_X^27+73689*_X^26+1609349*_X^25+4562111*_X^24+23691748091), (1/14641)*(11*_X^9-47*_X^8+96*_X^7-376*_X^6-370*_X^5-142*_X^4+280*_X^3+64*_X^2-17*_X-11)*(83746429305*_X-163433814*_X^23-1409885474*_X^22+7323055726*_X^21+92878340298*_X^20+291711433585*_X^19-28358008525*_X^18-1146850616945*_X^17+2003142623069*_X^16+7054039060380*_X^15+10860482240404*_X^14+4410674835220*_X^13-23715924119108*_X^12+39935154074341*_X^11-76564178781009*_X^10+246946329497683*_X^9-303627746551159*_X^8+41661161235738*_X^7+181533634595246*_X^6-146573328877410*_X^5+44279227597786*_X^4-3813039868649*_X^3+234521505317*_X^2+1331*_X^27+73689*_X^26+1609349*_X^25+4562111*_X^24+23691748091)}

 (4)
> 

``


 

Download minpoly.mw

Isn't the results incorrect? 

Generally speaking, will eliminating the...

sursumCorda 1284 Maple 2023
April 20 2023
1 4

Let us begin with the official descriptions of loops. The Maple® documentation claims that: 

Note that the examples above don't necessarily illustrate the best way to perform these operations. Often a functional form like seq, map, add, or mul is far more efficient.

Mma's tech tutorial also claims that: 

If you have a big program full of If, Do, Return, etc., you're probably not doing things right. 
Often, however, you can make more elegant and efficient programs using the functional programming constructs ….

Also, MatLab's Techniques to Improve Performance and Measure and Improve GPU Performance claims that: 

You can achieve better performance by rewriting loops to make use of higher-dimensional operations. The performance of a wide variety of element-wise functions can be improved … instead of looping over the matrices.

Well, I'm confused. Why did the official help page say like this? Actually, I find that lots of users in this forum still (and usually) use traditional for-loops instead of something which fits in with the alleged functional programming ideas. Did I misconstrue those statements? 
(For instance, as for the functional operations, it's unfortunate that Maple's built-in map cannot operate on arbitrary expression trees of any depth; so I have to use the loops to apply some procedure indirectly, which is not so convenient. In my opinion, owing to such limitation, people have to, and then gradually tend to, use the loops.) 

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