\[ \int x^5 \left (a+b \text {ArcTan}\left (c x^2\right )\right )^2 \, dx \]
Optimal antiderivative \[ \frac {b^{2} x^{2}}{6 c^{2}}-\frac {b^{2} \arctan \left (c \,x^{2}\right )}{6 c^{3}}-\frac {b \,x^{4} \left (a +b \arctan \left (c \,x^{2}\right )\right )}{6 c}-\frac {i \left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{6 c^{3}}+\frac {x^{6} \left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{6}-\frac {b \left (a +b \arctan \left (c \,x^{2}\right )\right ) \ln \left (\frac {2}{i c \,x^{2}+1}\right )}{3 c^{3}}-\frac {i b^{2} \polylog \left (2, 1-\frac {2}{i c \,x^{2}+1}\right )}{6 c^{3}} \]
command
int(x^5*(a+b*arctan(c*x^2))^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(\frac {x^{6} a^{2}}{6}+\frac {b a \ln \left (c^{2} x^{4}+1\right )}{6 c^{3}}+\frac {i b^{2} \ln \left (-i c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {b^{2} \ln \left (i c \,x^{2}+1\right ) \ln \left (-i c \,x^{2}+1\right ) x^{6}}{12}-\frac {a b \,x^{4}}{6 c}+\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c \,x^{2}}{2}\right ) \ln \left (-i c \,x^{2}+1\right )}{6 c^{3}}-\frac {i b^{2} \ln \left (\frac {1}{2}+\frac {i c \,x^{2}}{2}\right ) \ln \left (\frac {1}{2}-\frac {i c \,x^{2}}{2}\right )}{6 c^{3}}-\frac {i b a \,x^{6} \ln \left (i c \,x^{2}+1\right )}{6}-\frac {i b^{2} x^{4} \ln \left (-i c \,x^{2}+1\right )}{12 c}-\frac {i b^{2} \ln \left (i c \,x^{2}+1\right ) \ln \left (-i c \,x^{2}+1\right )}{12 c^{3}}-\frac {b^{2} \arctan \left (c \,x^{2}\right )}{6 c^{3}}+\frac {b^{2} x^{2}}{6 c^{2}}-\frac {i b^{2} \ln \left (i c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {i b^{2} x^{4} \ln \left (i c \,x^{2}+1\right )}{12 c}-\frac {17 i b^{2}}{108 c^{3}}-\frac {b^{2} x^{6} \ln \left (i c \,x^{2}+1\right )^{2}}{24}+\frac {i a b \,x^{6} \ln \left (-i c \,x^{2}+1\right )}{6}-\frac {b^{2} x^{6} \ln \left (-i c \,x^{2}+1\right )^{2}}{24}-\frac {i b^{2} \dilog \left (\frac {1}{2}-\frac {i c \,x^{2}}{2}\right )}{6 c^{3}}\) | \(367\) |
Maple 2021.1 output
\[ \int x^{5} \left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}\, dx \]________________________________________________________________________________________