\[ \int \log \left (b \left (F^{e (c+d x)}\right )^n+\pi \right ) \, dx \]
Optimal antiderivative \[ x \ln \left (\pi \right )-\frac {\polylog \left (2, -\frac {b \left (F^{e \left (d x +c \right )}\right )^{n}}{\pi }\right )}{d e n \ln \left (F \right )} \]
command
integrate(log(b*(F^(e*(d*x+c)))^n+pi),x, algorithm="maxima")
Maxima 5.46 SBCL 2.0.1.debian via sagemath 9.6 output
\[ x \log \left (\pi + F^{{\left (d x + c\right )} n e} b\right ) - \frac {{\left (d n x e \log \left (\frac {F^{d n x e} F^{c n e} b}{\pi } + 1\right ) \log \left (F\right ) + {\rm Li}_2\left (-\frac {F^{d n x e} F^{c n e} b}{\pi }\right )\right )} e^{\left (-1\right )}}{d n \log \left (F\right )} \]
Maxima 5.44 via sagemath 9.3 output
\[ -\frac {1}{2} \, d e n x^{2} \log \left (F\right ) + \pi d e n \int \frac {x}{\pi + {\left (F^{d e x}\right )}^{n} {\left (F^{c e}\right )}^{n} b}\,{d x} \log \left (F\right ) + x \log \left (\pi + {\left (F^{d e x}\right )}^{n} {\left (F^{c e}\right )}^{n} b\right ) \]________________________________________________________________________________________