\[ \int x^{11} \left (a+b \tanh ^{-1}\left (c x^3\right )\right )^2 \, dx \]
Optimal antiderivative \[ \frac {a b \,x^{3}}{6 c^{3}}+\frac {b^{2} x^{6}}{36 c^{2}}+\frac {b^{2} x^{3} \arctanh \left (c \,x^{3}\right )}{6 c^{3}}+\frac {b \,x^{9} \left (a +b \arctanh \left (c \,x^{3}\right )\right )}{18 c}-\frac {\left (a +b \arctanh \left (c \,x^{3}\right )\right )^{2}}{12 c^{4}}+\frac {x^{12} \left (a +b \arctanh \left (c \,x^{3}\right )\right )^{2}}{12}+\frac {b^{2} \ln \left (-c^{2} x^{6}+1\right )}{9 c^{4}} \]
command
int(x^11*(a+b*arctanh(c*x^3))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {b^{2} \left (x^{12} c^{4}-1\right ) \ln \left (c \,x^{3}+1\right )^{2}}{48 c^{4}}+\frac {b \left (-3 x^{12} b \ln \left (-c \,x^{3}+1\right ) c^{4}+6 a \,c^{4} x^{12}+2 b \,c^{3} x^{9}+6 b c \,x^{3}+3 b \ln \left (-c \,x^{3}+1\right )\right ) \ln \left (c \,x^{3}+1\right )}{72 c^{4}}+\frac {b^{2} x^{12} \ln \left (-c \,x^{3}+1\right )^{2}}{48}-\frac {a b \,x^{12} \ln \left (-c \,x^{3}+1\right )}{12}+\frac {x^{12} a^{2}}{12}-\frac {b^{2} x^{9} \ln \left (-c \,x^{3}+1\right )}{36 c}+\frac {a b \,x^{9}}{18 c}+\frac {b^{2} x^{6}}{36 c^{2}}-\frac {b^{2} x^{3} \ln \left (-c \,x^{3}+1\right )}{12 c^{3}}+\frac {a b \,x^{3}}{6 c^{3}}-\frac {b^{2} \ln \left (-c \,x^{3}+1\right )^{2}}{48 c^{4}}+\frac {b \ln \left (-c \,x^{3}+1\right ) a}{12 c^{4}}+\frac {b^{2} \ln \left (-c \,x^{3}+1\right )}{9 c^{4}}-\frac {b \ln \left (-c \,x^{3}-1\right ) a}{12 c^{4}}+\frac {b^{2} \ln \left (-c \,x^{3}-1\right )}{9 c^{4}}\) | \(298\) |
Maple 2021.1 output
\[ \int x^{11} \left (a +b \arctanh \left (c \,x^{3}\right )\right )^{2}\, dx \]________________________________________________________________________________________