Optimal. Leaf size=89 \[ -\frac{2}{3 a \sqrt{1-a^2 x^2}}-\frac{1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0513439, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {5960, 5958} \[ -\frac{2}{3 a \sqrt{1-a^2 x^2}}-\frac{1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)}{3 \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5960
Rule 5958
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=-\frac{1}{9 a \left (1-a^2 x^2\right )^{3/2}}+\frac{x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2}{3} \int \frac{\tanh ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac{1}{9 a \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{3 a \sqrt{1-a^2 x^2}}+\frac{x \tanh ^{-1}(a x)}{3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 x \tanh ^{-1}(a x)}{3 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0512738, size = 49, normalized size = 0.55 \[ -\frac{-6 a^2 x^2+\left (6 a^3 x^3-9 a x\right ) \tanh ^{-1}(a x)+7}{9 a \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.208, size = 59, normalized size = 0.7 \begin{align*} -{\frac{6\,{a}^{3}{x}^{3}{\it Artanh} \left ( ax \right ) -6\,{a}^{2}{x}^{2}-9\,ax{\it Artanh} \left ( ax \right ) +7}{9\,a \left ({a}^{2}{x}^{2}-1 \right ) ^{2}}\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 = 0.962393, size = 100, normalized size = 1.12 \begin{align*} -\frac{1}{9} \, a{\left (\frac{6}{\sqrt{-a^{2} x^{2} + 1} a^{2}} + \frac{1}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} a^{2}}\right )} + \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-a^{2} x^{2} + 1}} + \frac{x}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\right )} \operatorname{artanh}\left (a x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45211, size = 161, normalized size = 1.81 \begin{align*} \frac{{\left (12 \, a^{2} x^{2} - 3 \,{\left (2 \, a^{3} x^{3} - 3 \, a x\right )} \log \left (-\frac{a x + 1}{a x - 1}\right ) - 14\right )} \sqrt{-a^{2} x^{2} + 1}}{18 \,{\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 )}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32977, size = 122, normalized size = 1.37 \begin{align*} -\frac{{\left (2 \, a^{2} x^{2} - 3\right )} \sqrt{-a^{2} x^{2} + 1} x \log \left (-\frac{a x + 1}{a x - 1}\right )}{6 \,{\left (a^{2} x^{2} - 1\right )}^{2}} - \frac{6 \, a^{2} x^{2} - 7}{9 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{-a^{2} x^{2} + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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