Optimal. Leaf size=281 \[ -\frac {93 a}{128 \left (1-a^2 x^2\right )}-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{2} a \text {Li}_3\left (\frac {2}{a x+1}-1\right )-3 a \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.69, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {6030, 5982, 5916, 5988, 5932, 5948, 6056, 6610, 5956, 5994, 261, 5964, 5960} \[ -\frac {3}{2} a \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {93 a}{128 \left (1-a^2 x^2\right )}-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 261
Rule 5916
Rule 5932
Rule 5948
Rule 5956
Rule 5960
Rule 5964
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 )^3} \, dx &=a^2 \int \frac {\tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x^2 \left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {1}{8} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx+\frac {1}{4} \left (3 a^2\right ) \int \frac {\tanh ^{-1}(a x)^3}{\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 {3 a}{128 \left (1-a^2 x^2\right )^2}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {7}{32} a \tanh ^{-1}(a x)^4+\frac {1}{32} \left (9 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx+a^2 \int \frac {\tanh ^{-1}(a x)^3}{1-a^2 x^2} \, dx-\frac {1}{8} \left (9 a^3\right ) \int \frac {x \tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^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}{128 \left (1-a^2 x^2\right )^2}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {9 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {9}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac {\tanh ^{-1}(a x)^3}{x}+\frac {a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} 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}{8} \left (9 a^2\right ) \int \frac {\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, 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 {1}{64} \left (9 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {9 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \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}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} a \tanh ^{-1}(a x)^4+(3 a) \int \frac {\tanh ^{-1}(a x)^2}{x (1+a x)} \, dx-\frac {1}{16} \left (9 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx-\frac {1}{4} \left (3 a^3\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac {3 a}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \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}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} 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}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \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}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} 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}{128 \left (1-a^2 x^2\right )^2}-\frac {93 a}{128 \left (1-a^2 x^2\right )}+\frac {3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac {93}{128} a \tanh ^{-1}(a x)^2-\frac {3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {21 a \tanh ^{-1}(a x)^2}{16 \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}{4 \left (1-a^2 x^2\right )^2}+\frac {7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac {15}{32} 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 = 0.70, size = 218, normalized size = 0.78 \[ -a \left (-\frac {a x \tanh ^{-1}(a x)^3}{1-a^2 x^2}-3 \tanh ^{-1}(a x) \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )+\frac {3}{2} \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )-\frac {15}{32} \tanh ^{-1}(a x)^4+\frac {\tanh ^{-1}(a x)^3}{a x}+\tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac {1}{32} \tanh ^{-1}(a x)^3 \sinh \left (4 \tanh ^{-1}(a x)\right )-\frac {3}{4} \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )-\frac {3}{256} \tanh ^{-1}(a x) \sinh \left (4 \tanh ^{-1}(a x)\right )+\frac {3}{4} \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac {3}{128} \tanh ^{-1}(a x)^2 \cosh \left (4 \tanh ^{-1}(a x)\right )+\frac {3}{8} \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac {3 \cosh \left (4 \tanh ^{-1}(a x)\right )}{1024}-\frac {i \pi ^3}{8}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{3}}{a^{6} x^{8} - 3 \, a^{4} x^{6} + 3 \, a^{2} x^{4} - x^{2}}, x\right ) \]
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 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.04, size = 842, normalized size = 3.00 \[ \text {result too large to display} \]
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 = -0.00 \[ -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x^2\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
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 \[ - \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{8} - 3 a^{4} x^{6} + 3 a^{2} x^{4} - x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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