Optimal. Leaf size=191 \[ -\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}-\frac {3}{2} a \text {Li}_3\left (\frac {2}{a x+1}-1\right )-3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {3}{8} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {3}{8} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.44, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6030, 5982, 5916, 5988, 5932, 5948, 6056, 6610, 5956, 5994, 261} \[ -\frac {3}{2} a \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {3}{8} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 261
Rule 5916
Rule 5932
Rule 5948
Rule 5956
Rule 5982
Rule 5988
Rule 5994
Rule 6030
Rule 6056
Rule 6610
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )} \, dx\\ &=\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{8} a \tanh ^{-1}(a x)^4+a^2 \int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3\right ) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2} \, dx\\ &=-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x \left (1-a^2 x^2\right )} \, dx+\frac {1}{2} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx-\frac {1}{4} \left (3 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-\left (6 a^2\right ) \int \frac {\tanh ^{-1}(a x) \log \left (2-\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )+\left (3 a^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+a x}\right )}{1-a^2 x^2} \, dx\\ &=-\frac {3 a}{8 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{4 \left (1-a^2 x^2\right )}+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {3}{8} a \tanh ^{-1}(a x)^4+3 a \tanh ^{-1}(a x)^2 \log \left (2-\frac {2}{1+a x}\right )-3 a \tanh ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+a x}\right )-\frac {3}{2} a \text {Li}_3\left (-1+\frac {2}{1+a x}\right )\\ \end {align*}
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Mathematica [C] time = 0.35, size = 144, normalized size = 0.75 \[ \frac {1}{16} a \left (48 \tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )-24 \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )+6 \tanh ^{-1}(a x)^4-\frac {16 \tanh ^{-1}(a x)^3}{a x}-16 \tanh ^{-1}(a x)^3+48 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )+4 \tanh ^{-1}(a x)^3 \sinh \left (2 \tanh ^{-1}(a x)\right )+6 \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )-6 \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )-3 \cosh \left (2 \tanh ^{-1}(a x)\right )+2 i \pi ^3\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (a x\right )^{3}}{a^{4} x^{6} - 2 \, a^{2} x^{4} + x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.03, size = 442, normalized size = 2.31 \[ \frac {3 a^{2} x}{32 \left (a x -1\right )}+\frac {3 a^{2} x}{32 \left (a x +1\right )}+\frac {3 a}{32 \left (a x -1\right )}-\frac {3 a}{32 \left (a x +1\right )}-\frac {\arctanh \left (a x \right )^{3}}{x}-\frac {3 a \arctanh \left (a x \right )}{16 \left (a x -1\right )}-\frac {3 a \arctanh \left (a x \right )}{16 \left (a x +1\right )}+6 a \arctanh \left (a x \right ) \polylog \left (2, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+6 a \arctanh \left (a x \right ) \polylog \left (2, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {3 a \arctanh \left (a x \right )^{2}}{16 \left (a x -1\right )}-\frac {3 a \arctanh \left (a x \right )^{2}}{16 \left (a x +1\right )}+3 a \arctanh \left (a x \right )^{2} \ln \left (1+\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )+3 a \arctanh \left (a x \right )^{2} \ln \left (1-\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {a \arctanh \left (a x \right )^{3}}{8 \left (a x -1\right )}-\frac {a \arctanh \left (a x \right )^{3}}{8 \left (a x +1\right )}-6 a \polylog \left (3, -\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-6 a \polylog \left (3, \frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}\right )-a \arctanh \left (a x \right )^{3}+\frac {3 a \arctanh \left (a x \right )^{4}}{8}+\frac {\arctanh \left (a x \right )^{3} x \,a^{2}}{8 a x +8}+\frac {3 \arctanh \left (a x \right )^{2} x \,a^{2}}{16 \left (a x +1\right )}-\frac {3 \arctanh \left (a x \right ) a^{2} x}{16 \left (a x -1\right )}+\frac {3 \arctanh \left (a x \right ) a^{2} x}{16 \left (a x +1\right )}-\frac {\arctanh \left (a x \right )^{3} x \,a^{2}}{8 \left (a x -1\right )}+\frac {3 \arctanh \left (a x \right )^{2} x \,a^{2}}{16 \left (a x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,{\left (a^2\,x^2-1\right )}^2} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{x^{2} \left (a x - 1\right )^{2} \left (a x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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