Optimal. Leaf size=227 \[ -\frac {3 x}{8 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)}{8 a^4}+\frac {3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^4}+\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {3 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,1-\frac {2}{1-a x}\right )}{2 a^4}+\frac {3 \tanh ^{-1}(a x) \text {PolyLog}\left (3,1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {3 \text {PolyLog}\left (4,1-\frac {2}{1-a x}\right )}{4 a^4} \]
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Rubi [A]
time = 0.29, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6175, 6131,
6055, 6095, 6205, 6209, 6745, 6141, 6103, 205, 212} \begin {gather*} -\frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{4 a^4}-\frac {3 \text {Li}_2\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^2}{2 a^4}+\frac {3 \text {Li}_3\left (1-\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)}{2 a^4}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\tanh ^{-1}(a x)^3}{4 a^4}-\frac {3 \tanh ^{-1}(a x)}{8 a^4}-\frac {\log \left (\frac {2}{1-a x}\right ) \tanh ^{-1}(a x)^3}{a^4}+\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac {3 x}{8 a^3 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 212
Rule 6055
Rule 6095
Rule 6103
Rule 6131
Rule 6141
Rule 6175
Rule 6205
Rule 6209
Rule 6745
Rubi steps
\begin {align*} \int \frac {x^3 \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx &=\frac {\int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx}{a^2}-\frac {\int \frac {x \tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx}{a^2}\\ &=\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\int \frac {\tanh ^{-1}(a x)^3}{1-a x} \, dx}{a^3}-\frac {3 \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^3}\\ &=-\frac {3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^4}+\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^4}+\frac {3 \int \frac {\tanh ^{-1}(a x)^2 \log \left (\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}+\frac {3 \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 a^2}\\ &=\frac {3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^4}+\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {3 \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx}{4 a^3}+\frac {3 \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{a^3}\\ &=-\frac {3 x}{8 a^3 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^4}+\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^4}+\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {3 \int \frac {1}{1-a^2 x^2} \, dx}{8 a^3}-\frac {3 \int \frac {\text {Li}_3\left (1-\frac {2}{1-a x}\right )}{1-a^2 x^2} \, dx}{2 a^3}\\ &=-\frac {3 x}{8 a^3 \left (1-a^2 x^2\right )}-\frac {3 \tanh ^{-1}(a x)}{8 a^4}+\frac {3 \tanh ^{-1}(a x)}{4 a^4 \left (1-a^2 x^2\right )}-\frac {3 x \tanh ^{-1}(a x)^2}{4 a^3 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{4 a^4}+\frac {\tanh ^{-1}(a x)^3}{2 a^4 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^4}{4 a^4}-\frac {\tanh ^{-1}(a x)^3 \log \left (\frac {2}{1-a x}\right )}{a^4}-\frac {3 \tanh ^{-1}(a x)^2 \text {Li}_2\left (1-\frac {2}{1-a x}\right )}{2 a^4}+\frac {3 \tanh ^{-1}(a x) \text {Li}_3\left (1-\frac {2}{1-a x}\right )}{2 a^4}-\frac {3 \text {Li}_4\left (1-\frac {2}{1-a x}\right )}{4 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 139, normalized size = 0.61 \begin {gather*} \frac {-4 \tanh ^{-1}(a x)^4+6 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+4 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )-16 \tanh ^{-1}(a x)^3 \log \left (1+e^{-2 \tanh ^{-1}(a x)}\right )+24 \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )+24 \tanh ^{-1}(a x) \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(a x)}\right )+12 \text {PolyLog}\left (4,-e^{-2 \tanh ^{-1}(a x)}\right )-3 \sinh \left (2 \tanh ^{-1}(a x)\right )-6 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )}{16 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 107.56, size = 806, normalized size = 3.55 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 \begin {gather*} \text {Failed to integrate} \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 {x^{3} \operatorname {atanh}^{3}{\left (a x \right )}}{\left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, 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 {x^3\,{\mathrm {atanh}\left (a\,x\right )}^3}{{\left (a^2\,x^2-1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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