Optimal. Leaf size=193 \[ -\frac {3 a x}{8 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3}{4} \text {Li}_4\left (\frac {2}{a x+1}-1\right )-\frac {3}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2-\frac {3}{2} \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {1}{4} \tanh ^{-1}(a x)^3-\frac {3}{8} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \]
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Rubi [A] time = 0.39, antiderivative size = 193, 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 = {6030, 5988, 5932, 5948, 6056, 6060, 6610, 5994, 5956, 199, 206} \[ -\frac {3}{4} \text {PolyLog}\left (4,\frac {2}{a x+1}-1\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-\frac {3 a x}{8 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {1}{4} \tanh ^{-1}(a x)^3-\frac {3}{8} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 5932
Rule 5948
Rule 5956
Rule 5988
Rule 5994
Rule 6030
Rule 6056
Rule 6060
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )} \, dx\\ &=\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {1}{2} (3 a) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x (1+a x)} \, dx\\ &=-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-(3 a) \int \frac {\tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx+\frac {1}{2} \left (3 a^2\right ) \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {1}{4} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx+(3 a) \int \frac {\tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a x}{8 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )-\frac {1}{8} (3 a) \int \frac {1}{1-a^2 x^2} \, dx+\frac {1}{2} (3 a) \int \frac {\text {Li}_3\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a x}{8 \left (1-a^2 x^2\right )}-\frac {3}{8} \tanh ^{-1}(a x)+\frac {3 \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {1}{4} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4+\tanh ^{-1}(a x)^3 \log \left (2-\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {Li}_3\left (-1+\frac {2}{1+a x}\right )-\frac {3}{4} \text {Li}_4\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.20, size = 135, normalized size = 0.70 \[ \frac {1}{64} \left (96 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )-96 \tanh ^{-1}(a x) \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )+48 \text {Li}_4\left (e^{2 \tanh ^{-1}(a x)}\right )-16 \tanh ^{-1}(a x)^4+64 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-24 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )-12 \sinh \left (2 \tanh ^{-1}(a x)\right )+16 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+24 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )^{3}}{a^{4} x^{5} - 2 \, a^{2} x^{3} + x}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.74, size = 1387, normalized size = 7.19 \[ \text {result too large to display} \]
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 \[ \frac {{\left (a^{2} x^{2} - 1\right )} \log \left (-a x + 1\right )^{4} + 4 \, {\left ({\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) + 1\right )} \log \left (-a x + 1\right )^{3}}{64 \, {\left (a^{2} x^{2} - 1\right )}} - \frac {1}{8} \, \int -\frac {2 \, \log \left (a x + 1\right )^{3} - 6 \, \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) - 3 \, {\left (a^{2} x^{2} + a x + {\left (a^{4} x^{4} + a^{3} x^{3} - a^{2} x^{2} - a x - 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{2 \, {\left (a^{4} x^{5} - 2 \, a^{2} x^{3} + x\right )}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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