Optimal. Leaf size=75 \[ -\frac{3 x}{32 a \left (1-a^2 x^2\right )}-\frac{x}{16 a \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \tanh ^{-1}(a x)}{32 a^2} \]
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Rubi [A] time = 0.0469405, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5994, 199, 206} \[ -\frac{3 x}{32 a \left (1-a^2 x^2\right )}-\frac{x}{16 a \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \tanh ^{-1}(a x)}{32 a^2} \]
Antiderivative was successfully verified.
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Rule 5994
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \frac{x \tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=\frac{\tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{\int \frac{1}{\left (1-a^2 x^2\right )^3} \, dx}{4 a}\\ &=-\frac{x}{16 a \left (1-a^2 x^2\right )^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \int \frac{1}{\left (1-a^2 x^2\right )^2} \, dx}{16 a}\\ &=-\frac{x}{16 a \left (1-a^2 x^2\right )^2}-\frac{3 x}{32 a \left (1-a^2 x^2\right )}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}-\frac{3 \int \frac{1}{1-a^2 x^2} \, dx}{32 a}\\ &=-\frac{x}{16 a \left (1-a^2 x^2\right )^2}-\frac{3 x}{32 a \left (1-a^2 x^2\right )}-\frac{3 \tanh ^{-1}(a x)}{32 a^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left (1-a^2 x^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0566228, size = 88, normalized size = 1.17 \[ \frac{3 x}{32 a \left (a^2 x^2-1\right )}-\frac{x}{16 a \left (a^2 x^2-1\right )^2}+\frac{\tanh ^{-1}(a x)}{4 a^2 \left (a^2 x^2-1\right )^2}+\frac{3 \log (1-a x)}{64 a^2}-\frac{3 \log (a x+1)}{64 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 92, normalized size = 1.2 \begin{align*}{\frac{{\it Artanh} \left ( ax \right ) }{4\,{a}^{2} \left ({a}^{2}{x}^{2}-1 \right ) ^{2}}}-{\frac{1}{64\,{a}^{2} \left ( ax-1 \right ) ^{2}}}+{\frac{3}{64\,{a}^{2} \left ( ax-1 \right ) }}+{\frac{3\,\ln \left ( ax-1 \right ) }{64\,{a}^{2}}}+{\frac{1}{64\,{a}^{2} \left ( ax+1 \right ) ^{2}}}+{\frac{3}{64\,{a}^{2} \left ( ax+1 \right ) }}-{\frac{3\,\ln \left ( ax+1 \right ) }{64\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.968843, size = 111, normalized size = 1.48 \begin{align*} \frac{\frac{2 \,{\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} x^{4} - 2 \, a^{2} x^{2} + 1} - \frac{3 \, \log \left (a x + 1\right )}{a} + \frac{3 \, \log \left (a x - 1\right )}{a}}{64 \, a} + \frac{\operatorname{artanh}\left (a x\right )}{4 \,{\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99264, size = 150, normalized size = 2. \begin{align*} \frac{6 \, a^{3} x^{3} - 10 \, a x -{\left (3 \, a^{4} x^{4} - 6 \, a^{2} x^{2} - 5\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )}{64 \,{\left (a^{6} x^{4} - 2 \, a^{4} x^{2} + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.02812, size = 158, normalized size = 2.11 \begin{align*} \begin{cases} - \frac{3 a^{4} x^{4} \operatorname{atanh}{\left (a x \right )}}{32 a^{6} x^{4} - 64 a^{4} x^{2} + 32 a^{2}} + \frac{3 a^{3} x^{3}}{32 a^{6} x^{4} - 64 a^{4} x^{2} + 32 a^{2}} + \frac{6 a^{2} x^{2} \operatorname{atanh}{\left (a x \right )}}{32 a^{6} x^{4} - 64 a^{4} x^{2} + 32 a^{2}} - \frac{5 a x}{32 a^{6} x^{4} - 64 a^{4} x^{2} + 32 a^{2}} + \frac{5 \operatorname{atanh}{\left (a x \right )}}{32 a^{6} x^{4} - 64 a^{4} x^{2} + 32 a^{2}} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18839, size = 113, normalized size = 1.51 \begin{align*} -\frac{3 \, \log \left ({\left | a x + 1 \right |}\right )}{64 \, a^{2}} + \frac{3 \, \log \left ({\left | a x - 1 \right |}\right )}{64 \, a^{2}} + \frac{3 \, a^{2} x^{3} - 5 \, x}{32 \,{\left (a^{2} x^{2} - 1\right )}^{2} a} + \frac{\log \left (-\frac{a x + 1}{a x - 1}\right )}{8 \,{\left (a^{2} x^{2} - 1\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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