\[ \int x^7 \left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2 \, dx \]
Optimal antiderivative \[ \frac {a b \,x^{2}}{4 c^{3}}+\frac {b^{2} x^{4}}{24 c^{2}}+\frac {b^{2} x^{2} \arctanh \left (c \,x^{2}\right )}{4 c^{3}}+\frac {b \,x^{6} \left (a +b \arctanh \left (c \,x^{2}\right )\right )}{12 c}-\frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{8 c^{4}}+\frac {x^{8} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{8}+\frac {b^{2} \ln \left (-c^{2} x^{4}+1\right )}{6 c^{4}} \]
command
int(x^7*(a+b*arctanh(c*x^2))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {b^{2} \left (x^{8} c^{4}-1\right ) \ln \left (c \,x^{2}+1\right )^{2}}{32 c^{4}}+\frac {b \left (-3 x^{8} b \ln \left (-c \,x^{2}+1\right ) c^{4}+6 a \,c^{4} x^{8}+2 b \,c^{3} x^{6}+6 b c \,x^{2}+3 b \ln \left (-c \,x^{2}+1\right )\right ) \ln \left (c \,x^{2}+1\right )}{48 c^{4}}+\frac {b^{2} x^{8} \ln \left (-c \,x^{2}+1\right )^{2}}{32}-\frac {a b \,x^{8} \ln \left (-c \,x^{2}+1\right )}{8}+\frac {x^{8} a^{2}}{8}-\frac {b^{2} x^{6} \ln \left (-c \,x^{2}+1\right )}{24 c}+\frac {a b \,x^{6}}{12 c}+\frac {b^{2} x^{4}}{24 c^{2}}-\frac {b^{2} x^{2} \ln \left (-c \,x^{2}+1\right )}{8 c^{3}}+\frac {a b \,x^{2}}{4 c^{3}}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right )^{2}}{32 c^{4}}+\frac {b \ln \left (-c \,x^{2}+1\right ) a}{8 c^{4}}+\frac {b^{2} \ln \left (-c \,x^{2}+1\right )}{6 c^{4}}-\frac {b \ln \left (-c \,x^{2}-1\right ) a}{8 c^{4}}+\frac {b^{2} \ln \left (-c \,x^{2}-1\right )}{6 c^{4}}\) | \(298\) |
Maple 2021.1 output
\[ \int x^{7} \left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}\, dx \]________________________________________________________________________________________