\[ \int \frac {x^3}{\log ^3\left (c \left (a+b x^2\right )\right )} \, dx \]
Optimal antiderivative \[ \frac {\expIntegral \left (2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{b^{2} c^{2}}-\frac {a \logarithmicIntegral \left (c \left (b \,x^{2}+a \right )\right )}{4 b^{2} c}-\frac {x^{2} \left (b \,x^{2}+a \right )}{4 b \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}-\frac {a \left (b \,x^{2}+a \right )}{4 b^{2} \ln \left (c \left (b \,x^{2}+a \right )\right )}-\frac {x^{2} \left (b \,x^{2}+a \right )}{2 b \ln \left (c \left (b \,x^{2}+a \right )\right )} \]
command
int(x^3/ln(c*(b*x^2+a))^3,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {\left (b \,x^{2}+a \right ) \left (2 \ln \left (c \left (b \,x^{2}+a \right )\right ) b \,x^{2}+b \,x^{2}+\ln \left (c \left (b \,x^{2}+a \right )\right ) a \right )}{4 b^{2} \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}+\frac {a \expIntegral \left (1, -\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{4 c \,b^{2}}-\frac {\expIntegral \left (1, -2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{c^{2} b^{2}}\) | \(105\) |
| default | \(\frac {-\frac {c^{2} \left (b \,x^{2}+a \right )^{2}}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}-\frac {c^{2} \left (b \,x^{2}+a \right )^{2}}{\ln \left (c \left (b \,x^{2}+a \right )\right )}-2 \expIntegral \left (1, -2 \ln \left (c \left (b \,x^{2}+a \right )\right )\right )-c a \left (-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )^{2}}-\frac {c \left (b \,x^{2}+a \right )}{2 \ln \left (c \left (b \,x^{2}+a \right )\right )}-\frac {\expIntegral \left (1, -\ln \left (c \left (b \,x^{2}+a \right )\right )\right )}{2}\right )}{2 c^{2} b^{2}}\) | \(143\) |
Maple 2021.1 output
\[ \int \frac {x^{3}}{\ln \left (\left (b \,x^{2}+a \right ) c \right )^{3}}\, dx \]________________________________________________________________________________________