Optimal. Leaf size=187 \[ -\frac{i b d \text{PolyLog}\left (2,-\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b d \text{PolyLog}\left (2,\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e \log \left (f x^m\right ) \text{PolyLog}\left (2,-\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b e \log \left (f x^m\right ) \text{PolyLog}\left (2,\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e m \text{PolyLog}\left (3,-\frac{i x^{-n}}{c}\right )}{2 n^2}+\frac{i b e m \text{PolyLog}\left (3,\frac{i x^{-n}}{c}\right )}{2 n^2}+a d \log (x)+\frac{a e \log ^2\left (f x^m\right )}{2 m} \]
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Rubi [A] time = 0.607913, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2301, 6742, 5032, 4849, 2391, 5008, 5006, 2374, 6589} \[ -\frac{i b d \text{PolyLog}\left (2,-\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b d \text{PolyLog}\left (2,\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e \log \left (f x^m\right ) \text{PolyLog}\left (2,-\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b e \log \left (f x^m\right ) \text{PolyLog}\left (2,\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e m \text{PolyLog}\left (3,-\frac{i x^{-n}}{c}\right )}{2 n^2}+\frac{i b e m \text{PolyLog}\left (3,\frac{i x^{-n}}{c}\right )}{2 n^2}+a d \log (x)+\frac{a e \log ^2\left (f x^m\right )}{2 m} \]
Antiderivative was successfully verified.
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Rule 2301
Rule 6742
Rule 5032
Rule 4849
Rule 2391
Rule 5008
Rule 5006
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \cot ^{-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 \cot ^{-1}\left (c x^n\right )\right )}{x}+\frac{e \left (a+b \cot ^{-1}\left (c x^n\right )\right ) \log \left (f x^m\right )}{x}\right ) \, dx\\ &=d \int \frac{a+b \cot ^{-1}\left (c x^n\right )}{x} \, dx+e \int \frac{\left (a+b \cot ^{-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{\cot ^{-1}\left (c x^n\right ) \log \left (f x^m\right )}{x} \, dx+\frac{d \operatorname{Subst}\left (\int \frac{a+b \cot ^{-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{1}{2} (i b e) \int \frac{\log \left (f x^m\right ) \log \left (1-\frac{i x^{-n}}{c}\right )}{x} \, dx-\frac{1}{2} (i b e) \int \frac{\log \left (f x^m\right ) \log \left (1+\frac{i x^{-n}}{c}\right )}{x} \, dx+\frac{(i b d) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i}{c x}\right )}{x} \, dx,x,x^n\right )}{2 n}-\frac{(i b d) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i}{c x}\right )}{x} \, dx,x,x^n\right )}{2 n}\\ &=a d \log (x)+\frac{a e \log ^2\left (f x^m\right )}{2 m}-\frac{i b d \text{Li}_2\left (-\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e \log \left (f x^m\right ) \text{Li}_2\left (-\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b d \text{Li}_2\left (\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b e \log \left (f x^m\right ) \text{Li}_2\left (\frac{i x^{-n}}{c}\right )}{2 n}+\frac{(i b e m) \int \frac{\text{Li}_2\left (-\frac{i x^{-n}}{c}\right )}{x} \, dx}{2 n}-\frac{(i b e m) \int \frac{\text{Li}_2\left (\frac{i x^{-n}}{c}\right )}{x} \, dx}{2 n}\\ &=a d \log (x)+\frac{a e \log ^2\left (f x^m\right )}{2 m}-\frac{i b d \text{Li}_2\left (-\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e \log \left (f x^m\right ) \text{Li}_2\left (-\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b d \text{Li}_2\left (\frac{i x^{-n}}{c}\right )}{2 n}+\frac{i b e \log \left (f x^m\right ) \text{Li}_2\left (\frac{i x^{-n}}{c}\right )}{2 n}-\frac{i b e m \text{Li}_3\left (-\frac{i x^{-n}}{c}\right )}{2 n^2}+\frac{i b e m \text{Li}_3\left (\frac{i x^{-n}}{c}\right )}{2 n^2}\\ \end{align*}
Mathematica [C] time = 0.335828, size = 132, normalized size = 0.71 \[ -\frac{b c x^n \left (d+e \log \left (f x^m\right )\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-c^2 x^{2 n}\right )}{n}+\frac{b c e m x^n \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2},\frac{3}{2}\right \},-c^2 x^{2 n}\right )}{n^2}-\frac{1}{2} \log (x) \left (e m \log (x)-2 \left (d+e \log \left (f x^m\right )\right )\right ) \left (a+b \tan ^{-1}\left (c x^n\right )+b \cot ^{-1}\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.371, size = 1058, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, b e m \arctan \left (1, c x^{n}\right ) \log \left (x\right )^{2} + b e \arctan \left (1, c x^{n}\right ) \log \left (x\right ) \log \left (x^{m}\right ) + \frac{a e \log \left (f x^{m}\right )^{2}}{2 \, m} + a d \log \left (x\right ) +{\left (b e \arctan \left (1, c x^{n}\right ) \log \left (f\right ) + b d \arctan \left (1, c x^{n}\right )\right )} \log \left (x\right ) + \int -\frac{b c e m n x^{n} \log \left (x\right )^{2} - 2 \, b c e n x^{n} \log \left (x\right ) \log \left (x^{m}\right ) - 2 \,{\left (b c e \log \left (f\right ) + b c d\right )} n x^{n} \log \left (x\right )}{2 \,{\left (c^{2} x x^{2 \, n} + x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.40855, size = 711, normalized size = 3.8 \begin{align*} \frac{2 \, a e m n^{2} \log \left (x\right )^{2} - 2 i \, b e m{\rm polylog}\left (3, i \, c x^{n}\right ) + 2 i \, b e m{\rm polylog}\left (3, -i \, c x^{n}\right ) + 2 \,{\left (b e m n^{2} \log \left (x\right )^{2} + 2 \,{\left (b e n^{2} \log \left (f\right ) + b d n^{2}\right )} \log \left (x\right )\right )} \operatorname{arccot}\left (c x^{n}\right ) +{\left (2 i \, b e m n \log \left (x\right ) + 2 i \, b e n \log \left (f\right ) + 2 i \, b d n\right )}{\rm Li}_2\left (i \, c x^{n}\right ) +{\left (-2 i \, b e m n \log \left (x\right ) - 2 i \, b e n \log \left (f\right ) - 2 i \, b d n\right )}{\rm Li}_2\left (-i \, c x^{n}\right ) +{\left (-i \, b e m n^{2} \log \left (x\right )^{2} +{\left (-2 i \, b e n^{2} \log \left (f\right ) - 2 i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (i \, c x^{n} + 1\right ) +{\left (i \, b e m n^{2} \log \left (x\right )^{2} +{\left (2 i \, b e n^{2} \log \left (f\right ) + 2 i \, b d n^{2}\right )} \log \left (x\right )\right )} \log \left (-i \, c x^{n} + 1\right ) + 4 \,{\left (a e n^{2} \log \left (f\right ) + a d n^{2}\right )} \log \left (x\right )}{4 \, n^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arccot}\left (c x^{n}\right ) + a\right )}{\left (e \log \left (f x^{m}\right ) + d\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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