\[ \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^2\right )\right )}{x^4} \, dx \]
Optimal antiderivative \[ -\frac {8 b d f n}{9 x}-\frac {2 b \,d^{\frac {3}{2}} f^{\frac {3}{2}} n \arctan \left (x \sqrt {d}\, \sqrt {f}\right )}{9}-\frac {2 d f \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 x}-\frac {2 d^{\frac {3}{2}} f^{\frac {3}{2}} \arctan \left (x \sqrt {d}\, \sqrt {f}\right ) \left (a +b \ln \left (c \,x^{n}\right )\right )}{3}-\frac {b n \ln \left (d f \,x^{2}+1\right )}{9 x^{3}}-\frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d f \,x^{2}+1\right )}{3 x^{3}}+\frac {i b \,d^{\frac {3}{2}} f^{\frac {3}{2}} n \polylog \left (2, -i x \sqrt {d}\, \sqrt {f}\right )}{3}-\frac {i b \,d^{\frac {3}{2}} f^{\frac {3}{2}} n \polylog \left (2, i x \sqrt {d}\, \sqrt {f}\right )}{3} \]
command
int((a+b*ln(c*x^n))*ln(d*(1/d+f*x^2))/x^4,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| risch | \(-\frac {\ln \left (d f \,x^{2}+1\right ) a}{3 x^{3}}+\frac {b n d f \sqrt {-d f}\, \dilog \left (1+x \sqrt {-d f}\right )}{3}-\frac {b n d f \sqrt {-d f}\, \dilog \left (1-x \sqrt {-d f}\right )}{3}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d f}{3 x}+\frac {2 b \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) n \ln \left (x \right )}{3 \sqrt {d f}}+\frac {b n d f \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1+x \sqrt {-d f}\right )}{3}-\frac {b n d f \sqrt {-d f}\, \ln \left (x \right ) \ln \left (1-x \sqrt {-d f}\right )}{3}+\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 )}{6 x^{3}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d f}{3 x}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d f}{3 x}-\frac {2 a \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 \sqrt {d f}}-\frac {2 a d f}{3 x}-\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 )}{6 x^{3}}-\frac {b \ln \left (c \right ) \ln \left (d f \,x^{2}+1\right )}{3 x^{3}}-\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 )}{6 x^{3}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 \sqrt {d f}}-\frac {2 b \ln \left (c \right ) d f}{3 x}-\frac {b \ln \left (d f \,x^{2}+1\right ) \ln \left (x^{n}\right )}{3 x^{3}}-\frac {2 b d f \ln \left (x^{n}\right )}{3 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 ) d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 \sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 \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}{3 x}-\frac {b n \ln \left (d f \,x^{2}+1\right )}{9 x^{3}}-\frac {2 b n \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{9 \sqrt {d f}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 \sqrt {d f}}+\frac {i b \pi \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (d f \,x^{2}+1\right )}{6 x^{3}}-\frac {2 b \ln \left (c \right ) d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right )}{3 \sqrt {d f}}-\frac {8 b d f n}{9 x}-\frac {2 b \,d^{2} f^{2} \arctan \left (\frac {x d f}{\sqrt {d f}}\right ) \ln \left (x^{n}\right )}{3 \sqrt {d f}}\) | \(734\) |
Maple 2021.1 output
\[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \,x^{2}+\frac {1}{d}\right ) d \right )}{x^{4}}\, dx \]________________________________________________________________________________________