\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^2} \, dx \]
Optimal antiderivative \[ -\frac {b n \ln \left (d f \,x^{2}+1\right )}{x}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d f \,x^{2}+1\right )}{x}+2 b n \arctan \left (x \sqrt {d}\, \sqrt {f}\right ) \sqrt {d}\, \sqrt {f}+2 \arctan \left (x \sqrt {d}\, \sqrt {f}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {d}\, \sqrt {f}-i b n \polylog \left (2, -i x \sqrt {d}\, \sqrt {f}\right ) \sqrt {d}\, \sqrt {f}+i b n \polylog \left (2, i x \sqrt {d}\, \sqrt {f}\right ) \sqrt {d}\, \sqrt {f} \]
command
int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^2))/x^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {b \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right )}{x}-\frac {2 b d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{\sqrt {d f}}+\frac {2 b d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{\sqrt {d f}}-\frac {b n \ln \left (d f \,x^{2}+1\right )}{x}+\frac {2 b n d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )+b n \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )-b n \sqrt {-d f}\, \dilog \left (1+x \sqrt {-d f}\right )+b n \sqrt {-d f}\, \dilog \left (1-x \sqrt {-d f}\right )-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d f \,x^{2}+1\right )}{2 x}+\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d f \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 x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) d f \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} d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (d f \,x^{2}+1\right )}{2 x}+\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 ) \ln \left (d f \,x^{2}+1\right )}{2 x}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d f \,x^{2}+1\right )}{2 x}-\frac {b \ln \left (c \right ) \ln \left (d f \,x^{2}+1\right )}{x}+\frac {2 b \ln \left (c \right ) d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}-\frac {\ln \left (d f \,x^{2}+1\right ) a}{x}+\frac {2 a d f \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{\sqrt {d f}}\) | \(547\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x^{2}}\, dx \]________________________________________________________________________________________