Optimal. Leaf size=277 \[ -\frac {141 a x}{256 \left (1-a^2 x^2\right )}-\frac {3 a x}{128 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3}{4} \text {Li}_4\left (\frac {2}{a x+1}-1\right )-\frac {3}{2} \text {Li}_2\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)^2-\frac {3}{2} \text {Li}_3\left (\frac {2}{a x+1}-1\right ) \tanh ^{-1}(a x)+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {11}{32} \tanh ^{-1}(a x)^3-\frac {141}{256} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \]
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Rubi [A] time = 0.63, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6030, 5988, 5932, 5948, 6056, 6060, 6610, 5994, 5956, 199, 206, 5964} \[ -\frac {3}{4} \text {PolyLog}\left (4,\frac {2}{a x+1}-1\right )-\frac {3}{2} \tanh ^{-1}(a x)^2 \text {PolyLog}\left (2,\frac {2}{a x+1}-1\right )-\frac {3}{2} \tanh ^{-1}(a x) \text {PolyLog}\left (3,\frac {2}{a x+1}-1\right )-\frac {141 a x}{256 \left (1-a^2 x^2\right )}-\frac {3 a x}{128 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {11}{32} \tanh ^{-1}(a x)^3-\frac {141}{256} \tanh ^{-1}(a x)+\log \left (2-\frac {2}{a x+1}\right ) \tanh ^{-1}(a x)^3 \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 5932
Rule 5948
Rule 5956
Rule 5964
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 )^3} \, dx &=a^2 \int \frac {x \tanh ^{-1}(a x)^3}{\left (1-a^2 x^2\right )^3} \, dx+\int \frac {\tanh ^{-1}(a x)^3}{x \left (1-a^2 x^2\right )^2} \, dx\\ &=\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac {1}{4} (3 a) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, 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 {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\frac {\tanh ^{-1}(a x)^3}{2 \left (1-a^2 x^2\right )}+\frac {1}{4} \tanh ^{-1}(a x)^4-\frac {1}{32} (3 a) \int \frac {1}{\left (1-a^2 x^2\right )^3} \, dx-\frac {1}{16} (9 a) \int \frac {\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx-\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}{128 \left (1-a^2 x^2\right )^2}+\frac {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\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 {1}{128} (9 a) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx-(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}{16} \left (9 a^2\right ) \int \frac {x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^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 a x}{128 \left (1-a^2 x^2\right )^2}-\frac {9 a x}{256 \left (1-a^2 x^2\right )}+\frac {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\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}{256} (9 a) \int \frac {1}{1-a^2 x^2} \, dx-\frac {1}{32} (9 a) \int \frac {1}{\left (1-a^2 x^2\right )^2} \, dx-\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}{128 \left (1-a^2 x^2\right )^2}-\frac {141 a x}{256 \left (1-a^2 x^2\right )}-\frac {9}{256} \tanh ^{-1}(a x)+\frac {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\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}{64} (9 a) \int \frac {1}{1-a^2 x^2} \, dx-\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}{128 \left (1-a^2 x^2\right )^2}-\frac {141 a x}{256 \left (1-a^2 x^2\right )}-\frac {141}{256} \tanh ^{-1}(a x)+\frac {3 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac {33 \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )}-\frac {3 a x \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}-\frac {33 a x \tanh ^{-1}(a x)^2}{32 \left (1-a^2 x^2\right )}-\frac {11}{32} \tanh ^{-1}(a x)^3+\frac {\tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}+\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 = 0.30, size = 189, normalized size = 0.68 \[ \frac {1536 \tanh ^{-1}(a x)^2 \text {Li}_2\left (e^{2 \tanh ^{-1}(a x)}\right )-1536 \tanh ^{-1}(a x) \text {Li}_3\left (e^{2 \tanh ^{-1}(a x)}\right )+768 \text {Li}_4\left (e^{2 \tanh ^{-1}(a x)}\right )-256 \tanh ^{-1}(a x)^4+1024 \tanh ^{-1}(a x)^3 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-576 \tanh ^{-1}(a x)^2 \sinh \left (2 \tanh ^{-1}(a x)\right )-24 \tanh ^{-1}(a x)^2 \sinh \left (4 \tanh ^{-1}(a x)\right )-288 \sinh \left (2 \tanh ^{-1}(a x)\right )-3 \sinh \left (4 \tanh ^{-1}(a x)\right )+384 \tanh ^{-1}(a x)^3 \cosh \left (2 \tanh ^{-1}(a x)\right )+32 \tanh ^{-1}(a x)^3 \cosh \left (4 \tanh ^{-1}(a x)\right )+576 \tanh ^{-1}(a x) \cosh \left (2 \tanh ^{-1}(a x)\right )+12 \tanh ^{-1}(a x) \cosh \left (4 \tanh ^{-1}(a x)\right )+16 \pi ^4}{1024} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\operatorname {artanh}\left (a x\right )^{3}}{a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x}, x\right ) \]
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 \[ \int -\frac {\operatorname {artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.86, size = 1533, normalized size = 5.53 \[ \text {result too large to display} \]
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 \[ \frac {{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-a x + 1\right )^{4} + 2 \, {\left (2 \, a^{2} x^{2} + 2 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) - 3\right )} \log \left (-a x + 1\right )^{3}}{64 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )}} - \frac {1}{8} \, \int \frac {4 \, \log \left (a x + 1\right )^{3} - 12 \, \log \left (a x + 1\right )^{2} \log \left (-a x + 1\right ) + 3 \, {\left (2 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - 3 \, a x + 2 \, {\left (a^{6} x^{6} + a^{5} x^{5} - 2 \, a^{4} x^{4} - 2 \, a^{3} x^{3} + a^{2} x^{2} + a x + 2\right )} \log \left (a x + 1\right )\right )} \log \left (-a x + 1\right )^{2}}{4 \, {\left (a^{6} x^{7} - 3 \, a^{4} x^{5} + 3 \, a^{2} x^{3} - x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ -\int \frac {{\mathrm {atanh}\left (a\,x\right )}^3}{x\,{\left (a^2\,x^2-1\right )}^3} \,d x \]
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 \[ - \int \frac {\operatorname {atanh}^{3}{\left (a x \right )}}{a^{6} x^{7} - 3 a^{4} x^{5} + 3 a^{2} x^{3} - x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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