Optimal. Leaf size=216 \[ a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \text {PolyLog}(2,-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {PolyLog}(2,-c x)+\frac {1}{2} b d \text {PolyLog}(2,c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {PolyLog}(2,c x)-\frac {1}{2} a e \text {PolyLog}\left (2,c^2 x^2\right )-b e \log (1-c x) \text {PolyLog}(2,1-c x)+b e \log (1+c x) \text {PolyLog}(2,1+c x)+b e \text {PolyLog}(3,1-c x)-b e \text {PolyLog}(3,1+c x) \]
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Rubi [A]
time = 0.19, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6226, 6031,
6224, 2438, 6222, 2443, 2481, 2421, 6724} \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(-c x) \left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right )-\frac {1}{2} b e \text {Li}_2(c x) \left (-\log \left (1-c^2 x^2\right )+\log (1-c x)+\log (c x+1)\right )-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)+b e \text {Li}_3(1-c x)-b e \text {Li}_3(c x+1)-b e \text {Li}_2(1-c x) \log (1-c x)+b e \text {Li}_2(c x+1) \log (c x+1)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(c x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2438
Rule 2443
Rule 2481
Rule 6031
Rule 6222
Rule 6224
Rule 6226
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx &=d \int \frac {a+b \tanh ^{-1}(c x)}{x} \, dx+e \int \frac {\left (a+b \tanh ^{-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(-c x)+\frac {1}{2} b d \text {Li}_2(c x)+(a e) \int \frac {\log \left (1-c^2 x^2\right )}{x} \, dx+(b e) \int \frac {\tanh ^{-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(-c x)+\frac {1}{2} b d \text {Li}_2(c x)-\frac {1}{2} a e \text {Li}_2\left (c^2 x^2\right )-\frac {1}{2} (b e) \int \frac {\log ^2(1-c x)}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log ^2(1+c x)}{x} \, dx+\left (b e \left (-\log (1-c x)-\log (1+c x)+\log \left (1-c^2 x^2\right )\right )\right ) \int \frac {\tanh ^{-1}(c x)}{x} \, dx\\ &=a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(c x)-\frac {1}{2} a e \text {Li}_2\left (c^2 x^2\right )-(b c e) \int \frac {\log (c x) \log (1-c x)}{1-c x} \, dx-(b c e) \int \frac {\log (-c x) \log (1+c x)}{1+c x} \, dx\\ &=a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(c x)-\frac {1}{2} a e \text {Li}_2\left (c^2 x^2\right )+(b e) \text {Subst}\left (\int \frac {\log (x) \log \left (c \left (\frac {1}{c}-\frac {x}{c}\right )\right )}{x} \, dx,x,1-c x\right )-(b e) \text {Subst}\left (\int \frac {\log (x) \log \left (-c \left (-\frac {1}{c}+\frac {x}{c}\right )\right )}{x} \, dx,x,1+c x\right )\\ &=a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(c x)-\frac {1}{2} a e \text {Li}_2\left (c^2 x^2\right )-b e \log (1-c x) \text {Li}_2(1-c x)+b e \log (1+c x) \text {Li}_2(1+c x)+(b e) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-c x\right )-(b e) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+c x\right )\\ &=a d \log (x)-\frac {1}{2} b e \log (c x) \log ^2(1-c x)+\frac {1}{2} b e \log (-c x) \log ^2(1+c x)-\frac {1}{2} b d \text {Li}_2(-c x)+\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(-c x)+\frac {1}{2} b d \text {Li}_2(c x)-\frac {1}{2} b e \left (\log (1-c x)+\log (1+c x)-\log \left (1-c^2 x^2\right )\right ) \text {Li}_2(c x)-\frac {1}{2} a e \text {Li}_2\left (c^2 x^2\right )-b e \log (1-c x) \text {Li}_2(1-c x)+b e \log (1+c x) \text {Li}_2(1+c x)+b e \text {Li}_3(1-c x)-b e \text {Li}_3(1+c x)\\ \end {align*}
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Mathematica [F]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b \tanh ^{-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 = 40.17, size = 1638, normalized size = 7.58
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1638\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 156, normalized size = 0.72 \begin {gather*} -\frac {1}{2} \, {\left (\log \left (c x\right ) \log \left (-c x + 1\right )^{2} + 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 - 2 \, a e\right )} {\left (\log \left (c x\right ) \log \left (-c x + 1\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} + \frac {1}{2} \, {\left (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 {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 {atanh}{\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 {atanh}\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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