Optimal. Leaf size=187 \[ a d \log (x)+\frac {a e \log ^2\left (f x^m\right )}{2 m}-\frac {i b d \text {Li}_2\left (-\frac {i x^{-n}}{c}\right )}{2 n}+\frac {i b d \text {Li}_2\left (\frac {i x^{-n}}{c}\right )}{2 n}-\frac {i b e \text {Li}_2\left (-\frac {i x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}+\frac {i b e \text {Li}_2\left (\frac {i x^{-n}}{c}\right ) \log \left (f x^m\right )}{2 n}-\frac {i b e m \text {Li}_3\left (-\frac {i x^{-n}}{c}\right )}{2 n^2}+\frac {i b e m \text {Li}_3\left (\frac {i x^{-n}}{c}\right )}{2 n^2} \]
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Rubi [A] time = 0.61, antiderivative size = 187, normalized size of antiderivative = 1.00, 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 2374
Rule 2391
Rule 4849
Rule 5006
Rule 5008
Rule 5032
Rule 6589
Rule 6742
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*}
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Mathematica [C] time = 0.36, size = 132, normalized size = 0.71 \[ -\frac {b c x^n \left (d+e \log \left (f x^m\right )\right ) \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right )}{n}+\frac {b c e m x^n \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2},\frac {3}{2};-c^2 x^{2 n}\right )}{n^2}-\frac {1}{2} \log (x) \left (e m \log (x)-2 \left (d+e \log \left (f x^m\right )\right )\right ) \left (a+b \tan ^{-1}\left (c x^n\right )+b \cot ^{-1}\left (c x^n\right )\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.82, size = 246, normalized size = 1.32 \[ \frac {2 \, a e m n^{2} \log \relax (x)^{2} - 2 i \, b e m {\rm polylog}\left (3, i \, c x^{n}\right ) + 2 i \, b e m {\rm polylog}\left (3, -i \, c x^{n}\right ) + 2 \, {\left (b e m n^{2} \log \relax (x)^{2} + 2 \, {\left (b e n^{2} \log \relax (f) + b d n^{2}\right )} \log \relax (x)\right )} \operatorname {arccot}\left (c x^{n}\right ) + {\left (2 i \, b e m n \log \relax (x) + 2 i \, b e n \log \relax (f) + 2 i \, b d n\right )} {\rm Li}_2\left (i \, c x^{n}\right ) + {\left (-2 i \, b e m n \log \relax (x) - 2 i \, b e n \log \relax (f) - 2 i \, b d n\right )} {\rm Li}_2\left (-i \, c x^{n}\right ) + {\left (-i \, b e m n^{2} \log \relax (x)^{2} + {\left (-2 i \, b e n^{2} \log \relax (f) - 2 i \, b d n^{2}\right )} \log \relax (x)\right )} \log \left (i \, c x^{n} + 1\right ) + {\left (i \, b e m n^{2} \log \relax (x)^{2} + {\left (2 i \, b e n^{2} \log \relax (f) + 2 i \, b d n^{2}\right )} \log \relax (x)\right )} \log \left (-i \, c x^{n} + 1\right ) + 4 \, {\left (a e n^{2} \log \relax (f) + a d n^{2}\right )} \log \relax (x)}{4 \, n^{2}} \]
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 \[ \int \frac {{\left (b \operatorname {arccot}\left (c x^{n}\right ) + a\right )} {\left (e \log \left (f x^{m}\right ) + d\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.57, size = 1058, normalized size = 5.66 \[ \text {result too large to display} \]
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 \[ \frac {a e \log \left (f x^{m}\right )^{2}}{2 \, m} + a d \log \relax (x) - \frac {1}{2} \, {\left (b e m \log \relax (x)^{2} - 2 \, b e \log \relax (x) \log \left (x^{m}\right ) - 2 \, {\left (b e \log \relax (f) + b d\right )} \log \relax (x)\right )} \arctan \left (\frac {1}{c x^{n}}\right ) + \int -\frac {b c e m n x^{n} \log \relax (x)^{2} - 2 \, b c e n x^{n} \log \relax (x) \log \left (x^{m}\right ) - 2 \, {\left (b c e \log \relax (f) + b c d\right )} n x^{n} \log \relax (x)}{2 \, {\left (c^{2} x x^{2 \, n} + x\right )}}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acot}\left (c\,x^n\right )\right )\,\left (d+e\,\ln \left (f\,x^m\right )\right )}{x} \,d x \]
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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