Optimal. Leaf size=85 \[ \frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \coth ^{-1}(a x)}{a}+x \coth ^{-1}(a x)^3+\frac {\coth ^{-1}(a x)^3}{a}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a} \]
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Rubi [A] time = 0.17, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5911, 5985, 5919, 5949, 6059, 6610} \[ \frac {3 \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a}-\frac {3 \coth ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{a}+x \coth ^{-1}(a x)^3+\frac {\coth ^{-1}(a x)^3}{a}-\frac {3 \log \left (\frac {2}{1-a x}\right ) \coth ^{-1}(a x)^2}{a} \]
Antiderivative was successfully verified.
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Rule 5911
Rule 5919
Rule 5949
Rule 5985
Rule 6059
Rule 6610
Rubi steps
\begin {align*} \int \coth ^{-1}(a x)^3 \, dx &=x \coth ^{-1}(a x)^3-(3 a) \int \frac {x \coth ^{-1}(a x)^2}{1-a^2 x^2} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-3 \int \frac {\coth ^{-1}(a x)^2}{1-a x} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}+6 \int \frac {\coth ^{-1}(a x) \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+3 \int \frac {\text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx\\ &=\frac {\coth ^{-1}(a x)^3}{a}+x \coth ^{-1}(a x)^3-\frac {3 \coth ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{a}-\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{a}+\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 79, normalized size = 0.93 \[ -\frac {3 \coth ^{-1}(a x) \text {Li}_2\left (e^{2 \coth ^{-1}(a x)}\right )}{a}+\frac {3 \text {Li}_3\left (e^{2 \coth ^{-1}(a x)}\right )}{2 a}+x \coth ^{-1}(a x)^3+\frac {\coth ^{-1}(a x)^3}{a}-\frac {3 \coth ^{-1}(a x)^2 \log \left (1-e^{2 \coth ^{-1}(a x)}\right )}{a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arcoth}\left (a x\right )^{3}, x\right ) \]
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 \[ \int \operatorname {arcoth}\left (a x\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 180, normalized size = 2.12 \[ x \mathrm {arccoth}\left (a x \right )^{3}+\frac {\mathrm {arccoth}\left (a x \right )^{3}}{a}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (1-\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {3 \mathrm {arccoth}\left (a x \right )^{2} \ln \left (1+\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}-\frac {6 \,\mathrm {arccoth}\left (a x \right ) \polylog \left (2, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {6 \polylog \left (3, \frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a}+\frac {6 \polylog \left (3, -\frac {1}{\sqrt {\frac {a x -1}{a x +1}}}\right )}{a} \]
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 \[ \frac {{\left (a x + 1\right )} \log \left (a x + 1\right )^{3} - 3 \, {\left (a x - 1\right )} \log \left (a x + 1\right )^{2} \log \left (a x - 1\right )}{8 \, a} + \frac {1}{8} \, \int -\frac {{\left (a x + 1\right )} \log \left (a x - 1\right )^{3} - 3 \, {\left ({\left (a x + 1\right )} \log \left (a x - 1\right )^{2} + 2 \, {\left (a x - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + 1\right )}{a x + 1}\,{d x} \]
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 \[ \int {\mathrm {acoth}\left (a\,x\right )}^3 \,d x \]
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 \[ \int \operatorname {acoth}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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