Maple has this function Totient(n) built-in.This command was introduced in Maple 2016. Using showstat function we can preview the code.

with(NumberTheory):
showstat(Totient);
showstat(PrimeFactors);

NumberTheory:-Totient := proc(n::{posint, And(algebraic,Not({boolean, `in`, complexcons, extended_numeric}))}, $)
local prime_factors, p;
   1   if not type(n,'posint') then
   2       return ('procname')(n)
       elif isprime(n) then
   3       return n-1
       else
   4       prime_factors := NumberTheory:-PrimeFactors(n);
   5       return n*mul(p-1,`in`(p,prime_factors))/convert(prime_factors,'`*`')
       end if
end proc

NumberTheory:-PrimeFactors := proc(n::{integer, And(algebraic,Not({boolean, `in`, complexcons, extended_numeric}))}, $)
local i;
   1   if n = 0 then
   2       error "cannot represent all prime factors of %1", n
       elif type(n,{'negint', 'posint'}) then
   3       return {seq(i[1],`in`(i,ifactors(n)[2]))}
       else
   4       return ('procname')(n)
       end if
end proc

Another code you can find on this webpage: http://oeis.org/A000010 see:MAPLE -> # version 2.

Have fun !

If yours integral is:

int(sqrt(ln(x)/x^2-1/x), x)

then Maple can't find it. Possible it has No closed form.

See: https://en.wikipedia.org/wiki/Nonelementary_integral.

Mathematica, Rubi,Axiom,SymPy, Maxima  cannot do it, either.

Second one:

int((ln(x)^(a-1)/x^2-1/x)^(1/2),x)

the same case.

EDITED:

If You want approximate integral by series see worksheet.

Approximate_indefine_integral_by_Series.mw

convert(exp(x), Sum, dummy = n)

or:

convert(exp(x), FormalPowerSeries)

For Simple function (2 methods):

you can use inverse ztransform:

Sum(x^n*invztrans(eval(exp(x), x = 1/x), x, n), n = 0 .. infinity)#only works if invztrans can find transfrom.

or n-th derivative:

Sum(x^n*(eval(diff(exp(x), x$n), x = 0))/factorial(n), n = 0 .. infinity)#only works if n-th derivative can find.

Only another way to calculate Julian date.

Formula from:http://aa.usno.navy.mil/faq/docs/JD_Formula.php

Compute the JD corresponding to 2018 January 11, 18h30m30s UT.

Substituting Y = 2018, M = 1, D = 11,h = 18, m = 30, s = 30;

restart:
JD := proc (Y, M, D, h, m, s) local x; 
x := evalf(367*Y-trunc((7/4)*Y+(7/4)*trunc((1/12)*M+3/4))+trunc((275/9)*M)+D+1721013.5-
(1/2)*signum(100*Y+M-190002.5)+1/2+(1/24)*h+(1/1440)*m+(1/86400)*s); end proc:

JD(2018, 1, 11, 18, 30, 30);

# 2.458130271*10^6

It may not possible to find anti-derivatives(closed form).

https://math.stackexchange.com/questions/1469123/integral-of-ex2

https://math.stackexchange.com/questions/168860/functions-cannot-be-integrated-as-simple-functions?rq=1

Mathematica gives No solution.

An aproximation only can be done e.g by series:

Download Only_aproximation.mw

Use Maple built function intsolve with method Neumann or with this  paper https://arxiv.org/ftp/arxiv/papers/1309/1309.6311.pdf see another attached file (Fredholm_integral_2ver.)

Download _Fredholm_integral.mw

_Fredholm_integral_2ver.mw

Only for k=1,4,5,6,7 is real solution.

Download sol.mw

solve(15-(1/100)*m = 5+(1/600)*m, {m})
#{m = 6000/7}

solve(15-1/(100*m) = 5+1/(600*m), {m})
#{m = 7/6000}

Regards,Mariusz

Please read in Maple Help (Ctrl+F1 and put "How Do I,Solve an Ordinary Differential Equation?" in Search->Enter):

Scroll down to:

      -Solving an ODE Numerically

      - Taking Derivatives and Integrals of Numeric Solutions

      -  Why You Should Not Use Int or Diff on a Numeric Solution 

Corrected file:question-vers_2.mw

(Edited: 2 times) :P

Download Critical_Points.mw

sol := n > ceil(evalf(solve(product(exp(1/i), i = 1 .. n) = 100, n))); solve(sol, n);

# sol := 56 < n

# RealRange(Open(56), infinity)

Calculating point with newton method:

restart;
with(Student[NumericalAnalysis]):
with(plots):
f := x^3-x^2-x-1;
P := Newton(f, x = 2.0, tolerance = 10^(-2));
p1 := pointplot([[P, 0]], color = [blue], symbolsize = 20, symbol = circle, axes = normal):
p2 := plot(f, x = 0 .. 2.2):
display({p1, p2});

First iteration:   

evalf(eval(simplify(x-f/(diff(f, x))), x = 2));
# 1.857142857

Hi

I'm only changed "Equation" to "2*M" , "Solu" to "Solu[1]" and other things.

Regards Mariusz.

Help_v2.mw

 I'm not an expert on that topic.

With help from: https://www.maplesoft.com/applications/view.aspx?sid=4971  (See: A Numeric Approach).

restart;
f := proc (u) options operator, arrow; piecewise(0 <= u and u <= 1, 0, 1 < u and u <= 2, 1) end proc;
ode := diff(y(u), u$2) = 4*Pi^2*(f(u)-e)*y(u); bc := y(0) = 0, y(2) = 0, (D(y))(0) = 1;
Eigen1 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-3), approxsoln = [y(u) = exp(-u), e = 1]))(0)[4];
Eigen2 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-3), approxsoln = [y(u) = u, e = 3]))(0)[4];
Eigen3 := (dsolve({bc, ode}, numeric, range = 0 .. 2, maxmesh = 8192, abserr = 1.*10^(-3), approxsoln = [y(u) = u, e = 6]))(0)[4];

In Maple there is not much choice how to do it.

One of the possibilities is textplot.

with(plots):
p1 := pointplot([[3, 8], [-5, 16], [11, 32], [3, -8]], color = [black], symbol = solidbox, axes = none, symbolsize = 12): 
p2 := textplot({[-5, 16+2, "lampart"], [3, (-8)-2, "dog"], [3, 8+3, "cat"], [11, 32+4, "panthera"]}, axes = none):
p3 := implicitplot(y^2 = x^3-43*x+166, x = -40 .. 40, y = -40 .. 40, axes = normal, gridrefine = 2): 
display({p1, p2, p3})