Optimal. Leaf size=281 \[ -\frac{3}{2} a \text{PolyLog}\left (3,\frac{2}{a x+1}-1\right )-3 a \tanh ^{-1}(a x) \text{PolyLog}\left (2,\frac{2}{a x+1}-1\right )-\frac{93 a}{128 \left (1-a^2 x^2\right )}-\frac{3 a}{128 \left (1-a^2 x^2\right )^2}+\frac{7 a^2 x \tanh ^{-1}(a x)^3}{8 \left (1-a^2 x^2\right )}+\frac{a^2 x \tanh ^{-1}(a x)^3}{4 \left (1-a^2 x^2\right )^2}-\frac{21 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )}-\frac{3 a \tanh ^{-1}(a x)^2}{16 \left (1-a^2 x^2\right )^2}+\frac{93 a^2 x \tanh ^{-1}(a x)}{64 \left (1-a^2 x^2\right )}+\frac{3 a^2 x \tanh ^{-1}(a x)}{32 \left (1-a^2 x^2\right )^2}+\frac{15}{32} a \tanh ^{-1}(a x)^4+a \tanh ^{-1}(a x)^3-\frac{\tanh ^{-1}(a x)^3}{x}+\frac{93}{128} a \tanh ^{-1}(a x)^2+3 a \log \left (2-\frac{2}{a x+1}\right ) \tanh ^{-1}(a x)^2 \]
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Rubi [A] time = 0.690237, antiderivative size = 281, normalized size of antiderivative = 1., 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 6030
Rule 5982
Rule 5916
Rule 5988
Rule 5932
Rule 5948
Rule 6056
Rule 6610
Rule 5956
Rule 5994
Rule 261
Rule 5964
Rule 5960
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*}
Mathematica [C] time = 0.690842, size = 218, normalized size = 0.78 \[ -a \left (-3 \tanh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \tanh ^{-1}(a x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 \tanh ^{-1}(a x)}\right )-\frac{a x \tanh ^{-1}(a x)^3}{1-a^2 x^2}-\frac{15}{32} \tanh ^{-1}(a x)^4+\frac{\tanh ^{-1}(a x)^3}{a x}+\tanh ^{-1}(a x)^3-3 \tanh ^{-1}(a x)^2 \log \left (1-e^{2 \tanh ^{-1}(a x)}\right )-\frac{1}{32} \tanh ^{-1}(a x)^3 \sinh \left (4 \tanh ^{-1}(a x)\right )-\frac{3}{4} \tanh ^{-1}(a x) \sinh \left (2 \tanh ^{-1}(a x)\right )-\frac{3}{256} \tanh ^{-1}(a x) \sinh \left (4 \tanh ^{-1}(a x)\right )+\frac{3}{4} \tanh ^{-1}(a x)^2 \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac{3}{128} \tanh ^{-1}(a x)^2 \cosh \left (4 \tanh ^{-1}(a x)\right )+\frac{3}{8} \cosh \left (2 \tanh ^{-1}(a x)\right )+\frac{3 \cosh \left (4 \tanh ^{-1}(a x)\right )}{1024}-\frac{i \pi ^3}{8}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.714, size = 842, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\operatorname{artanh}\left (a x\right )^{3}}{{\left (a^{2} x^{2} - 1\right )}^{3} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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