Optimal. Leaf size=381 \[ -\frac{1}{2} a e \text{PolyLog}\left (2,c^2 x^2\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b d \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b d \text{PolyLog}\left (2,\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,\frac{c+\frac{1}{x}}{c}\right )-b e \text{PolyLog}\left (3,1-\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,-\frac{1}{c x}\right )-b e \text{PolyLog}\left (3,\frac{1}{c x}\right )+b e \log \left (1-\frac{1}{c x}\right ) \text{PolyLog}\left (2,1-\frac{1}{c x}\right )-b e \log \left (\frac{c+\frac{1}{x}}{c}\right ) \text{PolyLog}\left (2,\frac{c+\frac{1}{x}}{c}\right )+a d \log (x)+\frac{1}{2} b e \log \left (\frac{1}{c x}\right ) \log ^2\left (1-\frac{1}{c x}\right )-\frac{1}{2} b e \log ^2\left (\frac{1}{c x}+1\right ) \log \left (-\frac{1}{c x}\right ) \]
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Rubi [A] time = 0.439495, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {6080, 5913, 6078, 2391, 6076, 2454, 2396, 2433, 2374, 6589, 6070} \[ -\frac{1}{2} a e \text{PolyLog}\left (2,c^2 x^2\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac{1}{c x}\right )+\log \left (\frac{1}{c x}+1\right )\right ) \text{PolyLog}\left (2,\frac{1}{c x}\right )+\frac{1}{2} b d \text{PolyLog}\left (2,-\frac{1}{c x}\right )-\frac{1}{2} b d \text{PolyLog}\left (2,\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,\frac{c+\frac{1}{x}}{c}\right )-b e \text{PolyLog}\left (3,1-\frac{1}{c x}\right )+b e \text{PolyLog}\left (3,-\frac{1}{c x}\right )-b e \text{PolyLog}\left (3,\frac{1}{c x}\right )+b e \log \left (1-\frac{1}{c x}\right ) \text{PolyLog}\left (2,1-\frac{1}{c x}\right )-b e \log \left (\frac{c+\frac{1}{x}}{c}\right ) \text{PolyLog}\left (2,\frac{c+\frac{1}{x}}{c}\right )+a d \log (x)+\frac{1}{2} b e \log \left (\frac{1}{c x}\right ) \log ^2\left (1-\frac{1}{c x}\right )-\frac{1}{2} b e \log ^2\left (\frac{1}{c x}+1\right ) \log \left (-\frac{1}{c x}\right ) \]
Antiderivative was successfully verified.
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Rule 6080
Rule 5913
Rule 6078
Rule 2391
Rule 6076
Rule 2454
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rule 6070
Rubi steps
\begin{align*} \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx &=d \int \frac{a+b \coth ^{-1}(c x)}{x} \, dx+e \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )+(a e) \int \frac{\log \left (1-c^2 x^2\right )}{x} \, dx+(b e) \int \frac{\coth ^{-1}(c x) \log \left (1-c^2 x^2\right )}{x} \, dx\\ &=a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-\frac{1}{2} (b e) \int \frac{\log ^2\left (1-\frac{1}{c x}\right )}{x} \, dx+\frac{1}{2} (b e) \int \frac{\log ^2\left (1+\frac{1}{c x}\right )}{x} \, dx+(b e) \int \frac{\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x} \, dx+\left (b e \left (-\log \left (1-\frac{1}{c x}\right )-\log \left (1+\frac{1}{c x}\right )-\log \left (-c^2 x^2\right )+\log \left (1-c^2 x^2\right )\right )\right ) \int \frac{\coth ^{-1}(c x)}{x} \, dx\\ &=a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-\frac{1}{2} (b e) \int \frac{\log \left (1-\frac{1}{c x}\right ) \log \left (-c^2 x^2\right )}{x} \, dx+\frac{1}{2} (b e) \int \frac{\log \left (1+\frac{1}{c x}\right ) \log \left (-c^2 x^2\right )}{x} \, dx+\frac{1}{2} (b e) \operatorname{Subst}\left (\int \frac{\log ^2\left (1-\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )-\frac{1}{2} (b e) \operatorname{Subst}\left (\int \frac{\log ^2\left (1+\frac{x}{c}\right )}{x} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-(b e) \int \frac{\text{Li}_2\left (-\frac{1}{c x}\right )}{x} \, dx+(b e) \int \frac{\text{Li}_2\left (\frac{1}{c x}\right )}{x} \, dx+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (\frac{x}{c}\right ) \log \left (1-\frac{x}{c}\right )}{1-\frac{x}{c}} \, dx,x,\frac{1}{x}\right )}{c}+\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{x}{c}\right ) \log \left (1+\frac{x}{c}\right )}{1+\frac{x}{c}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )+b e \text{Li}_3\left (-\frac{1}{c x}\right )-b e \text{Li}_3\left (\frac{1}{c x}\right )-(b e) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (\frac{c-c x}{c}\right )}{x} \, dx,x,1-\frac{1}{c x}\right )+(b e) \operatorname{Subst}\left (\int \frac{\log (x) \log \left (-\frac{-c+c x}{c}\right )}{x} \, dx,x,1+\frac{1}{c x}\right )\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+b e \log \left (1-\frac{1}{c x}\right ) \text{Li}_2\left (1-\frac{1}{c x}\right )-b e \log \left (1+\frac{1}{c x}\right ) \text{Li}_2\left (1+\frac{1}{c x}\right )+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )+b e \text{Li}_3\left (-\frac{1}{c x}\right )-b e \text{Li}_3\left (\frac{1}{c x}\right )-(b e) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1-\frac{1}{c x}\right )+(b e) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,1+\frac{1}{c x}\right )\\ &=-\frac{1}{2} b e \log ^2\left (1+\frac{1}{c x}\right ) \log \left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log ^2\left (1-\frac{1}{c x}\right ) \log \left (\frac{1}{c x}\right )+a d \log (x)+b e \log \left (1-\frac{1}{c x}\right ) \text{Li}_2\left (1-\frac{1}{c x}\right )-b e \log \left (1+\frac{1}{c x}\right ) \text{Li}_2\left (1+\frac{1}{c x}\right )+\frac{1}{2} b d \text{Li}_2\left (-\frac{1}{c x}\right )+\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (-\frac{1}{c x}\right )-\frac{1}{2} b d \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} b e \log \left (-c^2 x^2\right ) \text{Li}_2\left (\frac{1}{c x}\right )+\frac{1}{2} b e \left (\log \left (1-\frac{1}{c x}\right )+\log \left (1+\frac{1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text{Li}_2\left (\frac{1}{c x}\right )-\frac{1}{2} a e \text{Li}_2\left (c^2 x^2\right )-b e \text{Li}_3\left (1-\frac{1}{c x}\right )+b e \text{Li}_3\left (1+\frac{1}{c x}\right )+b e \text{Li}_3\left (-\frac{1}{c x}\right )-b e \text{Li}_3\left (\frac{1}{c x}\right )\\ \end{align*}
Mathematica [F] time = 0.212522, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 1.104, size = 864, normalized size = 2.3 \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 [C] time = 1.77925, size = 225, normalized size = 0.59 \begin{align*} i \, \pi a e \log \left (x\right ) - \frac{1}{2} \,{\left (\log \left (c x - 1\right )^{2} \log \left (c x\right ) + 2 \,{\rm Li}_2\left (-c x + 1\right ) \log \left (c x - 1\right ) - 2 \,{\rm Li}_{3}(-c x + 1)\right )} b e + \frac{1}{2} \,{\left (\log \left (c x + 1\right )^{2} \log \left (-c x\right ) + 2 \,{\rm Li}_2\left (c x + 1\right ) \log \left (c x + 1\right ) - 2 \,{\rm Li}_{3}(c x + 1)\right )} b e + a d \log \left (x\right ) - \frac{1}{2} \,{\left (i \, \pi b e + b d - 2 \, a e\right )}{\left (\log \left (c x - 1\right ) \log \left (c x\right ) +{\rm Li}_2\left (-c x + 1\right )\right )} - \frac{1}{2} \,{\left (-i \, \pi b e - b d - 2 \, a e\right )}{\left (\log \left (c x + 1\right ) \log \left (-c x\right ) +{\rm Li}_2\left (c x + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b d \operatorname{arcoth}\left (c x\right ) + a d +{\left (b e \operatorname{arcoth}\left (c x\right ) + a e\right )} \log \left (-c^{2} x^{2} + 1\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{acoth}{\left (c x \right )}\right ) \left (d + e \log{\left (- c^{2} x^{2} + 1 \right )}\right )}{x}\, dx \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{arcoth}\left (c x\right ) + a\right )}{\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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