14.36 Problem number 175

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e \log \left (f x^m\right )\right )} \, dx \]

Optimal antiderivative \[ -\frac {b n}{2 e m \,x^{2}}-\frac {b \,{\mathrm e}^{\frac {2 d}{e m}} n \left (f \,x^{m}\right )^{\frac {2}{m}} \expIntegral \left (-\frac {2 \left (d +e \ln \left (f \,x^{m}\right )\right )}{e m}\right ) \left (d +e \ln \left (f \,x^{m}\right )\right )}{e^{2} m^{2} x^{2}}+\frac {{\mathrm e}^{\frac {2 d}{e m}} \left (f \,x^{m}\right )^{\frac {2}{m}} \expIntegral \left (-\frac {2 \left (d +e \ln \left (f \,x^{m}\right )\right )}{e m}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{e m \,x^{2}} \]

command

int((a+b*ln(c*x^n))/x^3/(d+e*ln(f*x^m)),x,method=_RETURNVERBOSE)

Maple 2022.1 output

method result size
risch \(\text {Expression too large to display}\) \(2341\)

Maple 2021.1 output

\[ \int \frac {b \ln \left (c \,x^{n}\right )+a}{\left (e \ln \left (f \,x^{m}\right )+d \right ) x^{3}}\, dx \]________________________________________________________________________________________