\[ \int x^5 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \]
Optimal antiderivative \[ \frac {b^{2} x^{2}}{6 c^{2}}-\frac {b^{2} \arctanh \left (c \,x^{2}\right )}{6 c^{3}}+\frac {b \,x^{4} \left (a +b \arctanh \left (c \,x^{2}\right )\right )}{6 c}+\frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{6 c^{3}}+\frac {x^{6} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{6}-\frac {b \left (a +b \arctanh \left (c \,x^{2}\right )\right ) \ln \left (\frac {2}{-c \,x^{2}+1}\right )}{3 c^{3}}-\frac {b^{2} \polylog \left (2, 1-\frac {2}{-c \,x^{2}+1}\right )}{6 c^{3}} \]
command
int(x^5*(a+b*arctanh(c*x^2))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {b^{2} x^{2}}{6 c^{2}}+\frac {b a \,x^{6} \ln \left (c \,x^{2}+1\right )}{6}+\frac {b a \ln \left (c \,x^{2}+1\right )}{6 c^{3}}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right ) \ln \left (c \,x^{2}+1\right ) x^{6}}{12}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right ) \ln \left (c \,x^{2}+1\right )}{12 c^{3}}+\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{2}}{2}\right ) \ln \left (c \,x^{2}+1\right )}{6 c^{3}}-\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{2}}{2}\right ) \ln \left (\frac {c \,x^{2}}{2}+\frac {1}{2}\right )}{6 c^{3}}+\frac {a b \,x^{4}}{6 c}-\frac {17 b^{2}}{108 c^{3}}-\frac {b^{2} x^{4} \ln \left (-c \,x^{2}+1\right )}{12 c}-\frac {a b \,x^{6} \ln \left (-c \,x^{2}+1\right )}{6}+\frac {a b \ln \left (c \,x^{2}-1\right )}{6 c^{3}}+\frac {b^{2} x^{6} \ln \left (-c \,x^{2}+1\right )^{2}}{24}+\frac {11 b^{2} \ln \left (-c \,x^{2}+1\right )}{36 c^{3}}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {b^{2} x^{6} \ln \left (c \,x^{2}+1\right )^{2}}{24}-\frac {b^{2} \ln \left (c \,x^{2}+1\right )}{12 c^{3}}+\frac {b^{2} \ln \left (c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {x^{6} a^{2}}{6}-\frac {b^{2} \dilog \left (\frac {c \,x^{2}}{2}+\frac {1}{2}\right )}{6 c^{3}}-\frac {2 b^{2} \ln \left (c \,x^{2}-1\right )}{9 c^{3}}+\frac {b^{2} x^{4} \ln \left (c \,x^{2}+1\right )}{12 c}\) | \(380\) |
Maple 2021.1 output
\[ \int x^{5} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}\, dx \]________________________________________________________________________________________