Optimal. Leaf size=136 \[ a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}-\frac {b e \text {Li}_2\left (-c x^n\right ) \log \left (f x^m\right )}{2 n}+\frac {b e \text {Li}_2\left (c x^n\right ) \log \left (f x^m\right )}{2 n}+\frac {b e m \text {Li}_3\left (-c x^n\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (c x^n\right )}{2 n^2} \]
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Rubi [A] time = 0.54, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2301, 6742, 6095, 5912, 6071, 6069, 2374, 6589} \[ -\frac {b d \text {PolyLog}\left (2,-c x^n\right )}{2 n}+\frac {b d \text {PolyLog}\left (2,c x^n\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,-c x^n\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {PolyLog}\left (2,c x^n\right )}{2 n}+\frac {b e m \text {PolyLog}\left (3,-c x^n\right )}{2 n^2}-\frac {b e m \text {PolyLog}\left (3,c x^n\right )}{2 n^2}+a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 2374
Rule 5912
Rule 6069
Rule 6071
Rule 6095
Rule 6589
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \left (d+e \log \left (f x^m\right )\right )}{x} \, dx &=\int \left (\frac {d \left (a+b \tanh ^{-1}\left (c x^n\right )\right )}{x}+\frac {e \left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x}\right ) \, dx\\ &=d \int \frac {a+b \tanh ^{-1}\left (c x^n\right )}{x} \, dx+e \int \frac {\left (a+b \tanh ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x} \, dx\\ &=(a e) \int \frac {\log \left (f x^m\right )}{x} \, dx+(b e) \int \frac {\tanh ^{-1}\left (c x^n\right ) \log \left (f x^m\right )}{x} \, dx+\frac {d \operatorname {Subst}\left (\int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^n\right )}{n}\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}-\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1-c x^n\right )}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log \left (f x^m\right ) \log \left (1+c x^n\right )}{x} \, dx\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (c x^n\right )}{2 n}+\frac {(b e m) \int \frac {\text {Li}_2\left (-c x^n\right )}{x} \, dx}{2 n}-\frac {(b e m) \int \frac {\text {Li}_2\left (c x^n\right )}{x} \, dx}{2 n}\\ &=a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {b d \text {Li}_2\left (-c x^n\right )}{2 n}-\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (-c x^n\right )}{2 n}+\frac {b d \text {Li}_2\left (c x^n\right )}{2 n}+\frac {b e \log \left (f x^m\right ) \text {Li}_2\left (c x^n\right )}{2 n}+\frac {b e m \text {Li}_3\left (-c x^n\right )}{2 n^2}-\frac {b e m \text {Li}_3\left (c x^n\right )}{2 n^2}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 114, normalized size = 0.84 \[ \frac {b c x^n \left (d+e \log \left (f x^m\right )\right ) \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right )}{n}-\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};c^2 x^{2 n}\right )}{n^2}+\frac {1}{2} a \log (x) \left (2 d+2 e \log \left (f x^m\right )-e m \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.55, size = 327, normalized size = 2.40 \[ \frac {2 \, a e m n^{2} \log \relax (x)^{2} - 2 \, b e m {\rm polylog}\left (3, c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right )\right ) + 2 \, b e m {\rm polylog}\left (3, -c \cosh \left (n \log \relax (x)\right ) - c \sinh \left (n \log \relax (x)\right )\right ) + 2 \, {\left (b e m n \log \relax (x) + b e n \log \relax (f) + b d n\right )} {\rm Li}_2\left (c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right )\right ) - 2 \, {\left (b e m n \log \relax (x) + b e n \log \relax (f) + b d n\right )} {\rm Li}_2\left (-c \cosh \left (n \log \relax (x)\right ) - c \sinh \left (n \log \relax (x)\right )\right ) - {\left (b e m n^{2} \log \relax (x)^{2} + 2 \, {\left (b e n^{2} \log \relax (f) + b d n^{2}\right )} \log \relax (x)\right )} \log \left (c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right ) + 1\right ) + {\left (b e m n^{2} \log \relax (x)^{2} + 2 \, {\left (b e n^{2} \log \relax (f) + b d n^{2}\right )} \log \relax (x)\right )} \log \left (-c \cosh \left (n \log \relax (x)\right ) - c \sinh \left (n \log \relax (x)\right ) + 1\right ) + 4 \, {\left (a e n^{2} \log \relax (f) + a d n^{2}\right )} \log \relax (x) + {\left (b e m n^{2} \log \relax (x)^{2} + 2 \, {\left (b e n^{2} \log \relax (f) + b d n^{2}\right )} \log \relax (x)\right )} \log \left (-\frac {c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right ) + 1}{c \cosh \left (n \log \relax (x)\right ) + c \sinh \left (n \log \relax (x)\right ) - 1}\right )}{4 \, n^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x^{n}\right ) + a\right )} {\left (e \log \left (f x^{m}\right ) + d\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.40, size = 668, normalized size = 4.91 \[ -\frac {e b m \ln \relax (x ) \polylog \left (2, -c \,x^{n}\right )}{2 n}+\frac {e b \dilog \left (c \,x^{n}+1\right ) m \ln \relax (x )}{2 n}+\frac {e b m \ln \relax (x ) \polylog \left (2, c \,x^{n}\right )}{2 n}+\frac {e b \dilog \left (c \,x^{n}\right ) m \ln \relax (x )}{2 n}+\frac {\ln \left (x^{n}\right ) a d}{n}+\frac {e a \ln \left (x^{m}\right )^{2}}{2 m}+\frac {i \dilog \left (c \,x^{n}+1\right ) \pi b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )}{4 n}-\frac {i \dilog \left (1-c \,x^{n}\right ) \pi b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )}{4 n}-\frac {i \ln \left (x^{n}\right ) \pi a e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )}{2 n}+\frac {b e m \polylog \left (3, -c \,x^{n}\right )}{2 n^{2}}-\frac {b e m \polylog \left (3, c \,x^{n}\right )}{2 n^{2}}-\frac {e b \dilog \left (c \,x^{n}+1\right ) \ln \left (x^{m}\right )}{2 n}-\frac {e b \dilog \left (c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}+\frac {\dilog \left (1-c \,x^{n}\right ) \ln \relax (f ) b e}{2 n}-\frac {\dilog \left (c \,x^{n}+1\right ) \ln \relax (f ) b e}{2 n}+\frac {i \ln \left (x^{n}\right ) \pi a e \,\mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{2 n}+\frac {i \dilog \left (1-c \,x^{n}\right ) \pi b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}+\frac {e b \ln \left (1-c \,x^{n}\right ) \ln \left (c \,x^{n}\right ) m \ln \relax (x )}{2 n}+\frac {\ln \left (x^{n}\right ) \ln \relax (f ) a e}{n}-\frac {i \dilog \left (c \,x^{n}+1\right ) \pi b e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}+\frac {i \dilog \left (1-c \,x^{n}\right ) \pi b e \,\mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}-\frac {i \dilog \left (c \,x^{n}+1\right ) \pi b e \,\mathrm {csgn}\left (i x^{m}\right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{4 n}-\frac {e b \ln \left (1-c \,x^{n}\right ) \ln \left (c \,x^{n}\right ) \ln \left (x^{m}\right )}{2 n}-\frac {\dilog \left (c \,x^{n}+1\right ) b d}{2 n}+\frac {\dilog \left (1-c \,x^{n}\right ) b d}{2 n}+\frac {i \ln \left (x^{n}\right ) \pi a e \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{m}\right )^{2}}{2 n}-\frac {i \ln \left (x^{n}\right ) \pi a e \mathrm {csgn}\left (i f \,x^{m}\right )^{3}}{2 n}-\frac {i \dilog \left (1-c \,x^{n}\right ) \pi b e \mathrm {csgn}\left (i f \,x^{m}\right )^{3}}{4 n}+\frac {i \dilog \left (c \,x^{n}+1\right ) \pi b e \mathrm {csgn}\left (i f \,x^{m}\right )^{3}}{4 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a e \log \left (f x^{m}\right )^{2}}{2 \, m} + a d \log \relax (x) - \frac {1}{4} \, {\left (b e m \log \relax (x)^{2} - 2 \, b e \log \relax (x) \log \left (x^{m}\right ) - 2 \, {\left (e \log \relax (f) + d\right )} b \log \relax (x)\right )} \log \left (c x^{n} + 1\right ) + \frac {1}{4} \, {\left (b e m \log \relax (x)^{2} - 2 \, b e \log \relax (x) \log \left (x^{m}\right ) - 2 \, {\left (e \log \relax (f) + d\right )} b \log \relax (x)\right )} \log \left (-c x^{n} + 1\right ) + \int \frac {2 \, b c e n x^{n} \log \relax (x) \log \left (x^{m}\right ) - {\left (b c e m n \log \relax (x)^{2} - 2 \, {\left (e n \log \relax (f) + d n\right )} b c \log \relax (x)\right )} x^{n}}{2 \, {\left (c^{2} x x^{2 \, n} - x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {atanh}\left (c\,x^n\right )\right )\,\left (d+e\,\ln \left (f\,x^m\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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