@vv 

Good Job and great code,but only work for simple cases.

R := {(16*x^5-20*x^3+5*x)^2-(16*x^5-20*x^3+5*x)*(8*y^4-8*y^2+1)+(8*y^4-8*y^2+1)^2 <= -3/4}
Area(R)

With this code Maple give a 0 and Mathematica an error.

This is a very good question.

I putt in Google "region measure in MAPLE" and  I'm  not find anything conclusive.

In Maple  there no such built functions like in Mathematica to measure region.

Maybe in next  release in Maple 2017.4  we will see these function. Maybe !

@vv 

I thought there would be something simpler and shorter.

Thanks for answer.

I will test it.

Thanks.

frac functon replaced by another functions:

    frac(t) = t-floor(t) = 1/2-I*ln(-exp((2*I)*Pi*t))/(2*Pi) = 1/2-arctan(cot(Pi*t))/Pi

plot([frac(t), t-floor(t), 1/2-I*ln(-exp((2*I)*Pi*t))/(2*Pi), 1/2-arctan(cot(Pi*t))/Pi], t = 0 .. 3.2, legend = [typeset("Curve: ", frac(t)), typeset("Curve: ", t-floor(t)), typeset("Curve: ", 1/2-I*ln(-exp((2*I)*Pi*t))/(2*Pi)), typeset("Curve: ", 1/2-arctan(cot(Pi*t))/Pi)])

For integrals may this helps:

http://12000.org/my_notes/CAS_integration_tests/index.htm

http://12000.org/my_notes/ten_hard_integrals/index.htm

The DirectSearch package will solve your problems, as Mr. Markiyan Hirnyk suggested.

@ernilesh80 

Download summation..mw

@kuwait1 

Well, if alpha =3 in function RootOf you have a equation 5 degree:

sol := {x = b*RootOf(16*_Z^5*b^4-32*V*_Z^4*b^3+24*V^2*_Z^3*b^2-8*V^3*_Z^2*b+V^4*_Z-3*V^3*W*a)/a, y = RootOf(16*_Z^5*b^4-32*V*_Z^4*b^3+24*V^2*_Z^3*b^2-8*V^3*_Z^2*b+V^4*_Z-3*V^3*W*a)}

if alpha =4 in function RootOf you have a equation 6 degree:

sol := {x = b*RootOf(32*_Z^6*b^5-80*V*_Z^5*b^4+80*V^2*_Z^4*b^3-40*V^3*_Z^3*b^2+10*V^4*_Z^2*b-V^5*_Z+4*V^4*W*a)/a, y = RootOf(32*_Z^6*b^5-80*V*_Z^5*b^4+80*V^2*_Z^4*b^3-40*V^3*_Z^3*b^2+10*V^4*_Z^2*b-V^5*_Z+4*V^4*W*a)}

These equations can only be solved in some cases symbolicaly,but in our case this is not possible, can be done only numerically.

An Example see worksheet:

Download Example.mw

For more see:

https://en.wikipedia.org/wiki/Quintic_function

https://en.wikipedia.org/wiki/Sextic_equation

@Leinegold 

Can you clarify the question?

For backup yours work (worksheet) Use : Tools->Option..->General->Auto save-> x minute

:)

@dellair 

It seems a bug in Maple 2016.2 or earlier version.

Using: evalf(Int(evalf(fe11_1), p = 1 .. 9, method = _Dexp)) I'm speed up caluculation for 500 times.

See file attached.

evalfandintPerformance_ver2.mw

@dellair 

Try add first:

Digits := 16;

Checked with Maple 2016.2 on Windows 8.1.

Thank you for a extensive reply.

@tomleslie 

Edited 13.02.2017.

I'm corrrcted my answer.

Better way to solve is numerical,but Mape can't handle nonlinear -elliptic PDE.

I'm use a finite difference schemes,and works.

Here you can find the necessary information that can help you: http://mathematica.stackexchange.com/questions/137494/steady-transonic-gas-flow-nonlinear-pde-giving-initial-condition-and-boundary-va

PDE_-finite_case_2.mw

@Zeineb