Optimal. Leaf size=63 \[ \frac{2 x}{\sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0396381, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5962, 191} \[ \frac{2 x}{\sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 5962
Rule 191
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx &=-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+2 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=\frac{2 x}{\sqrt{1-a^2 x^2}}-\frac{2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0406033, size = 38, normalized size = 0.6 \[ \frac{2 a x+a x \tanh ^{-1}(a x)^2-2 \tanh ^{-1}(a x)}{a \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.225, size = 49, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( ax \right ) \right ) ^{2}ax+2\,ax-2\,{\it Artanh} \left ( ax \right ) }{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955922, size = 77, normalized size = 1.22 \begin{align*} \frac{x \operatorname{artanh}\left (a x\right )^{2}}{\sqrt{-a^{2} x^{2} + 1}} + \frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} - \frac{2 \, \operatorname{artanh}\left (a x\right )}{\sqrt{-a^{2} x^{2} + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.28507, size = 150, normalized size = 2.38 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x \log \left (-\frac{a x + 1}{a x - 1}\right )^{2} + 8 \, a x - 4 \, \log \left (-\frac{a x + 1}{a x - 1}\right )\right )}}{4 \,{\left (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}^{2}{\left (a x \right )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{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 )^{2}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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