# Hot Integral - 15

**Calculus**Level 5

\[\displaystyle \int\limits_{0}^{\infty} t^{\mu-1}\log^2 t f^2(t) dt = \frac{\sqrt{\pi}\Gamma^a(\frac{\mu}{b})}{c^2\Gamma(\frac{\mu+d}{e})}\left\{g\psi^{(h)}(\frac{\mu}{b})^i - j\psi^{(k)}(\frac{\mu +l}{m})\psi^{(n)}(\frac{\mu}{b}) + \psi^{(o)}(\frac{\mu +l}{m})^p + q\psi^{(r)}(\frac{\mu}{s}) - \psi^{(u)}(\frac{\mu + v}{w}) \right\}\]

where

\(\displaystyle f(x)=\int\limits_{0}^{\infty} \cos(x\sinh t) dt\)

Calculate \(a+b+c+d+e+g+h+i+j+k+l+m+n+o+p+q+r+s+u+v+w\)

**Clarifications:**

\(a,b,c,d,e,g,h,i,j,k,l,m,n,o,p,q,r,s,u,v,w\) are all integers.

\(\psi^{(\lambda)}(\cdot)\) represents their usual meanings.(Polygamma)

\(\Gamma(\cdot)\) is Gamma function.

\(\mu\) is a real constant.

This is a part of Hot Integrals