Computation of parametric integral

Question

I am trying to compute the integral

Integrate[(g^(u^(g - 1)))/(1 + u^g), {u, 0, t}]

but as an answer I get my input expression. There must be something wrong with parameters but even if I give values to t or g mathematica'a answer is the same: it just gives me back my input expression...

My output is

How can I compute this integral?

Answer 1

It appears that this integral can be calculated symbolically only for special integer arguments g. One can use e.g. Assumptions in Integrate :

Integrate[(g^(u^(g - 1)))/(1 + u^g), {u, 0, t}, Assumptions -> g == # && t > 0] & /@ Range[4]

{Log[1 + t], 
 I 2^(-1 - I) (Cos[Log[4]] ExpIntegralEi[-I Log[2]] - ExpIntegralEi[I Log[2]] + 
 ExpIntegralEi[(I + t) Log[2]] - ExpIntegralEi[(-I + t) Log[2]] (Cos[Log[4]] + I Sin[Log[4]])
  + I ExpIntegralEi[-I Log[2]] Sin[Log[4]]), 
 Integrate[g^u^(-1 + g)/(1 + u^g), {u, 0, t}, Assumptions -> g == 3 && t > 0], 
 Integrate[g^u^(-1 + g)/(1 + u^g), {u, 0, t}, Assumptions -> g == 4 && t > 0]}

We can see that only for g == 1 or g == 2 we can get exact results. For g rational e.g. g == 3/2 we can't compute symbolically this integral either. For other values of g you could use Nintegrateto find numerical results.Integrate hasn't been changed in version 9, nethertheless some improvements of the core functionality has been described in Enhanced Core Algorithms. To find out more interesting remarks on Integrate I recommend to read Some Notes on Internal Implementation, especially this paragraph : Differentiation and Integration, where it says e.g. that the algorithms in Mathematica cover all of the indefinite integrals in standard reference books such as Gradshteyn-Ryzhik as well as that Integrate uses about 500 pages of Mathematica code and 600 pages of C code.