Optimal. Leaf size=381 \[ -\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)-b e \log \left (\frac {c+\frac {1}{x}}{c}\right ) \text {PolyLog}\left (2,\frac {c+\frac {1}{x}}{c}\right )+b e \log \left (1-\frac {1}{c x}\right ) \text {PolyLog}\left (2,1-\frac {1}{c x}\right )+\frac {1}{2} b d \text {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \text {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \text {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} b e \log \left (-c^2 x^2\right ) \text {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \text {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \text {PolyLog}\left (2,c^2 x^2\right )+b e \text {PolyLog}\left (3,\frac {c+\frac {1}{x}}{c}\right )-b e \text {PolyLog}\left (3,1-\frac {1}{c x}\right )+b e \text {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \text {PolyLog}\left (3,\frac {1}{c x}\right ) \]
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Rubi [A]
time = 0.31, antiderivative size = 381, normalized size of antiderivative = 1.00, 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 = {6227, 6032,
6225, 2438, 6223, 2504, 2443, 2481, 2421, 6724, 6217} \begin {gather*} -\frac {1}{2} a e \text {Li}_2\left (c^2 x^2\right )+a d \log (x)+\frac {1}{2} b e \text {Li}_2\left (-\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )-\frac {1}{2} b e \text {Li}_2\left (-\frac {1}{c x}\right ) \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 )-\frac {1}{2} b e \text {Li}_2\left (\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )+\frac {1}{2} b e \text {Li}_2\left (\frac {1}{c x}\right ) \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 )+\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 )+b e \text {Li}_3\left (\frac {c+\frac {1}{x}}{c}\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 )+b e \text {Li}_2\left (1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )-b e \text {Li}_2\left (\frac {c+\frac {1}{x}}{c}\right ) \log \left (\frac {c+\frac {1}{x}}{c}\right )+\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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2438
Rule 2443
Rule 2481
Rule 2504
Rule 6032
Rule 6217
Rule 6223
Rule 6225
Rule 6227
Rule 6724
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) \text {Subst}\left (\int \frac {\log ^2\left (1-\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} (b e) \text {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) \text {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) \text {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) \text {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) \text {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) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-\frac {1}{c x}\right )+(b e) \text {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*}
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Mathematica [F]
time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 3.80, size = 864, normalized size = 2.27
method | result | size |
risch | \(\text {Expression too large to display}\) | \(864\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.37, size = 173, normalized size = 0.45 \begin {gather*} 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 (b d + {\left (i \, \pi b - 2 \, a\right )} 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 (b d + {\left (i \, \pi b + 2 \, a\right )} e\right )} {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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