• Only two independent variables. !!!!
• No "Method of lines"
• No "Finite element method"
• No "extended finite element method"
• No "spectral finite element method"
• No "meshfree finite element method"
• No "discontinuous Galerkin finite element method"
• No "Element-Free Galerkin Method"
• No "Interpolating Element-Free Galerkin Method"
• No "Finite volume method"
• No "Differential Quadrature Method"
• No tanh-sech method 
• No extended tanh method
• No hyperbolic function method ,
• No Jacobi elliptic function expansion method
• No F-expansion method
• No First Integral method
• No the sine-cosine method.

Mine is:

@vv 

@Kitonum 

@Carl Love 

plots[odeplot](de, [x, diff(y(x),x,x,x)], x=0..1);

The plot is incorrect !

@Kitonum 

@Carl Love 

should be:

J:= Int(I*sqrt((R*exp(I*theta)+1)/(R*exp(I*theta)-a)), theta= 0..Pi) 


not:

J:= Int(I*sqrt((R*exp(I*theta)+1)/(R*exp(I*theta)+a)), theta= 0..Pi): 

Its not the same.

@Carl Love 

RootOf function should handle a branch cuts.

Maple can't this integral to solve:

test.mw

Mariusz Iwaniuk

Thank you for your answer.

@Markiyan Hirnyk 

Thank you for your answers.
@Kitonum 

@Carl Love 

@Markiyan Hirnyk 

 SEE HERE: http://dlmf.nist.gov/31

M_Iwaniuk

@Markiyan Hirnyk 

@Markiyan Hirnyk 

How do You find  integration limits?