18.1 Problem number 102

\[ \int \frac {x^3}{\log \left (c \left (a+b x^2\right )^p\right )} \, dx \]

Optimal antiderivative \[ -\frac {a \left (b \,x^{2}+a \right ) \expIntegral \left (\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{p}\right ) \left (c \left (b \,x^{2}+a \right )^{p}\right )^{-\frac {1}{p}}}{2 b^{2} p}+\frac {\left (b \,x^{2}+a \right )^{2} \expIntegral \left (\frac {2 \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{p}\right ) \left (c \left (b \,x^{2}+a \right )^{p}\right )^{-\frac {2}{p}}}{2 b^{2} p} \]

command

int(x^3/ln(c*(b*x^2+a)^p),x,method=_RETURNVERBOSE)

Maple 2022.1 output

method result size
risch \(-\frac {\left (b \,x^{2}+a \right )^{2} c^{-\frac {2}{p}} \left (\left (b \,x^{2}+a \right )^{p}\right )^{-\frac {2}{p}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )-\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )\right ) \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )\right )}{p}} \expIntegral \left (1, -2 \ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{p}\right )}{2 b^{2} p}+\frac {a \left (b \,x^{2}+a \right ) c^{-\frac {1}{p}} \left (\left (b \,x^{2}+a \right )^{p}\right )^{-\frac {1}{p}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \left (\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right )-\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )\right ) \left (\mathrm {csgn}\left (i c \right )-\mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )\right )}{2 p}} \expIntegral \left (1, -\ln \left (b \,x^{2}+a \right )-\frac {i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \mathrm {csgn}\left (i c \right )-i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{3}+i \pi \mathrm {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (c \right )+2 \ln \left (\left (b \,x^{2}+a \right )^{p}\right )-2 p \ln \left (b \,x^{2}+a \right )}{2 p}\right )}{2 b^{2} p}\) \(547\)

Maple 2021.1 output

\[ \int \frac {x^{3}}{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}\, dx \]________________________________________________________________________________________