Optimal. Leaf size=94 \[ -\frac{3}{16 a \left (1-a^2 x^2\right )}-\frac{1}{16 a \left (1-a^2 x^2\right )^2}+\frac{3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a} \]
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Rubi [A] time = 0.0448151, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5960, 5956, 261} \[ -\frac{3}{16 a \left (1-a^2 x^2\right )}-\frac{1}{16 a \left (1-a^2 x^2\right )^2}+\frac{3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac{3 \tanh ^{-1}(a x)^2}{16 a} \]
Antiderivative was successfully verified.
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Rule 5960
Rule 5956
Rule 261
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx &=-\frac{1}{16 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac{3}{4} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{1}{16 a \left (1-a^2 x^2\right )^2}+\frac{x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac{3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)^2}{16 a}-\frac{1}{8} (3 a) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx\\ &=-\frac{1}{16 a \left (1-a^2 x^2\right )^2}-\frac{3}{16 a \left (1-a^2 x^2\right )}+\frac{x \tanh ^{-1}(a x)}{4 \left (1-a^2 x^2\right )^2}+\frac{3 x \tanh ^{-1}(a x)}{8 \left (1-a^2 x^2\right )}+\frac{3 \tanh ^{-1}(a x)^2}{16 a}\\ \end{align*}
Mathematica [A] time = 0.0634113, size = 65, normalized size = 0.69 \[ \frac{3 a^2 x^2+3 \left (a^2 x^2-1\right )^2 \tanh ^{-1}(a x)^2+\left (10 a x-6 a^3 x^3\right ) \tanh ^{-1}(a x)-4}{16 a \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.061, size = 225, normalized size = 2.4 \begin{align*}{\frac{{\it Artanh} \left ( ax \right ) }{16\,a \left ( ax-1 \right ) ^{2}}}-{\frac{3\,{\it Artanh} \left ( ax \right ) }{16\,a \left ( ax-1 \right ) }}-{\frac{3\,{\it Artanh} \left ( ax \right ) \ln \left ( ax-1 \right ) }{16\,a}}-{\frac{{\it Artanh} \left ( ax \right ) }{16\,a \left ( ax+1 \right ) ^{2}}}-{\frac{3\,{\it Artanh} \left ( ax \right ) }{16\,a \left ( ax+1 \right ) }}+{\frac{3\,{\it Artanh} \left ( ax \right ) \ln \left ( ax+1 \right ) }{16\,a}}-{\frac{3\, \left ( \ln \left ( ax-1 \right ) \right ) ^{2}}{64\,a}}+{\frac{3\,\ln \left ( ax-1 \right ) }{32\,a}\ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }-{\frac{3\, \left ( \ln \left ( ax+1 \right ) \right ) ^{2}}{64\,a}}-{\frac{3}{32\,a}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{ax}{2}} \right ) }+{\frac{3\,\ln \left ( ax+1 \right ) }{32\,a}\ln \left ( -{\frac{ax}{2}}+{\frac{1}{2}} \right ) }-{\frac{1}{64\,a \left ( ax-1 \right ) ^{2}}}+{\frac{7}{64\,a \left ( ax-1 \right ) }}-{\frac{1}{64\,a \left ( ax+1 \right ) ^{2}}}-{\frac{7}{64\,a \left ( ax+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.979753, size = 246, normalized size = 2.62 \begin{align*} -\frac{1}{16} \,{\left (\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}\right )} \operatorname{artanh}\left (a x\right ) + \frac{{\left (12 \, a^{2} x^{2} - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right )^{2} + 6 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) \log \left (a x - 1\right ) - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )^{2} - 16\right )} a}{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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Fricas [A] time = 2.02996, size = 213, normalized size = 2.27 \begin{align*} \frac{12 \, a^{2} x^{2} + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} - 4 \,{\left (3 \, a^{3} x^{3} - 5 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 16}{64 \,{\left (a^{5} x^{4} - 2 \, a^{3} x^{2} + a\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}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, 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 )}{{\left (a^{2} x^{2} - 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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