\[ \int \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right ) \, dx \]
Optimal antiderivative \[ 4 b n x -2 x \left (a +b \ln \left (c \,x^{n}\right )\right )-b n x \ln \left (d f \,x^{2}+1\right )+x \left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d f \,x^{2}+1\right )-\frac {2 b n \arctan \left (x \sqrt {d}\, \sqrt {f}\right )}{\sqrt {d}\, \sqrt {f}}+\frac {2 \arctan \left (x \sqrt {d}\, \sqrt {f}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{\sqrt {d}\, \sqrt {f}}-\frac {i b n \polylog \left (2, -i x \sqrt {d}\, \sqrt {f}\right )}{\sqrt {d}\, \sqrt {f}}+\frac {i b n \polylog \left (2, i x \sqrt {d}\, \sqrt {f}\right )}{\sqrt {d}\, \sqrt {f}} \]
command
int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^2)),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(x \ln \left (d f \,x^{2}+1\right ) a -2 \ln \left (c \right ) b x +i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x +\frac {2 a \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}+4 b n x +\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x \ln \left (d f \,x^{2}+1\right )}{2}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) x \ln \left (d f \,x^{2}+1\right )}{2}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x \ln \left (d f \,x^{2}+1\right )}{2}-\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{d f}+\frac {b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{d f}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {b n \sqrt {-d f}\, \dilog \left (1+x \sqrt {-d f}\right )}{d f}+\frac {b n \sqrt {-d f}\, \dilog \left (1-x \sqrt {-d f}\right )}{d f}+b \ln \left (c \right ) x \ln \left (d f \,x^{2}+1\right )+\frac {2 b \ln \left (c \right ) \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}+b \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right ) x +\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{\sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-2 a x -\frac {2 b \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{\sqrt {d f}}-\frac {2 b n \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-2 b x \ln \left (x^{n}\right )+i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x -b n x \ln \left (d f \,x^{2}+1\right )-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x -\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} x \ln \left (d f \,x^{2}+1\right )}{2}-i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} x\) | \(643\) |
Maple 2021.1 output
\[ \int \left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )\, dx \]________________________________________________________________________________________