Optimal. Leaf size=62 \[ -\frac{1}{6 \left (1-x^2\right )}+\frac{1}{3} \log \left (1-x^2\right )+\frac{2 x \sin ^{-1}(x)}{3 \sqrt{1-x^2}}+\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0391695, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4655, 4651, 260, 261} \[ -\frac{1}{6 \left (1-x^2\right )}+\frac{1}{3} \log \left (1-x^2\right )+\frac{2 x \sin ^{-1}(x)}{3 \sqrt{1-x^2}}+\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4655
Rule 4651
Rule 260
Rule 261
Rubi steps
\begin{align*} \int \frac{\sin ^{-1}(x)}{\left (1-x^2\right )^{5/2}} \, dx &=\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}-\frac{1}{3} \int \frac{x}{\left (1-x^2\right )^2} \, dx+\frac{2}{3} \int \frac{\sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}} \, dx\\ &=-\frac{1}{6 \left (1-x^2\right )}+\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(x)}{3 \sqrt{1-x^2}}-\frac{2}{3} \int \frac{x}{1-x^2} \, dx\\ &=-\frac{1}{6 \left (1-x^2\right )}+\frac{x \sin ^{-1}(x)}{3 \left (1-x^2\right )^{3/2}}+\frac{2 x \sin ^{-1}(x)}{3 \sqrt{1-x^2}}+\frac{1}{3} \log \left (1-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0879138, size = 45, normalized size = 0.73 \[ \frac{1}{6} \left (\frac{1}{x^2-1}+2 \log \left (1-x^2\right )-\frac{2 x \left (2 x^2-3\right ) \sin ^{-1}(x)}{\left (1-x^2\right )^{3/2}}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 63, normalized size = 1. \begin{align*}{\frac{\arcsin \left ( x \right ) x}{3\, \left ({x}^{2}-1 \right ) ^{2}}\sqrt{-{x}^{2}+1}}+{\frac{1}{6\,{x}^{2}-6}}-{\frac{2\,\arcsin \left ( x \right ) x}{3\,{x}^{2}-3}\sqrt{-{x}^{2}+1}}+{\frac{\ln \left ( -{x}^{2}+1 \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41139, size = 65, normalized size = 1.05 \begin{align*} \frac{1}{3} \,{\left (\frac{2 \, x}{\sqrt{-x^{2} + 1}} + \frac{x}{{\left (-x^{2} + 1\right )}^{\frac{3}{2}}}\right )} \arcsin \left (x\right ) + \frac{1}{6 \,{\left (x^{2} - 1\right )}} + \frac{1}{3} \, \log \left (-3 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53875, size = 151, normalized size = 2.44 \begin{align*} -\frac{2 \,{\left (2 \, x^{3} - 3 \, x\right )} \sqrt{-x^{2} + 1} \arcsin \left (x\right ) - x^{2} - 2 \,{\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x^{2} - 1\right ) + 1}{6 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08977, size = 73, normalized size = 1.18 \begin{align*} -\frac{{\left (2 \, x^{2} - 3\right )} \sqrt{-x^{2} + 1} x \arcsin \left (x\right )}{3 \,{\left (x^{2} - 1\right )}^{2}} - \frac{2 \, x^{2} - 3}{6 \,{\left (x^{2} - 1\right )}} + \frac{1}{3} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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