Issues with integral

[parisse]

I just checked with a TI 89 and with a TI CX Nspire CAS, the integral is computed in approx mode, not in exact mode. On the Prime in CAS exact mode, the CAS tries to compute the integral exactly, and unfortunately this raises large computations that are hard to interrupt.
If you are interested in an approx value inside the CAS, just replace one boundary by an approx value, e.g. 0 by 0.0 and you will get almost instant the approx answer.
∫(1/(sqrt(cos(x))+sqrt(sin(x)))^4,x,0.0,π/2)
For an exact answer, you must help the CAS, like with roadrunner trick.