How to integrate a function with an unknown constant

Question

How do I integrate $(e^x + e^{-x} -5)^p$ with respect to $x$ where $p>0$?

I have tried integrating by substitution with $u=e^x$ and $u= e^x + e^{-x} -5$ but I can’t make progress because of the unknown $p$

Is this the right approach?

Answer 1

Trying to make the problem more general, consider $$I_p=\int(e^x + e^{-x} +2k)^p\,dx=2^p\int(\cosh(x)+k)^p\,dx$$ So, using the binomial expansion, the integrand is a polynomial in $\cosh(x)$ $$(\cosh(x)+k)^p=\sum_{m=0}^p \binom{m}{p}\cosh^m(x)\, k^{p-m}$$

and the integrals $$J_{2m}=\int \cosh^{2m}(x)\,dx \qquad \text{and} \qquad J_{2m+1}=\int \cosh^{2m+1}(x)\,dx$$ are known (have a look at equations $2.413.3$ and $2.413.4$ in the "Table of Integrals, Series, and Products" (seventh edition) by I.S. Gradshteyn and I.M. Ryzhik.

As a general result, you should have something like $$\frac{I_p}{2^p}=a_p x+ \sum_{n=1}^p b_{p,n} \sinh(n x)$$

I do not see any pattern in the results but they could be expressed in terms of the gaussian hypergeometric function. Using a CAS $$J_{p}=\int \cosh^{p}(x)\,dx=$$ $$\frac{\sqrt{-\sinh ^2(x)} \,\text{csch}(x)\, \cosh ^{p+1}(x)}{p+1}\,\, _2F_1\left(\frac{1}{2},\frac{p+1}{2};\frac{p+3}{2};\cosh ^2(x)\right)$$