Optimal. Leaf size=44 \[ x \sin ^{-1}\left (1-x^2\right )^2-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left (1-x^2\right )}{x}-8 x \]
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Rubi [A] time = 0.0071933, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4814, 8} \[ x \sin ^{-1}\left (1-x^2\right )^2-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left (1-x^2\right )}{x}-8 x \]
Antiderivative was successfully verified.
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Rule 4814
Rule 8
Rubi steps
\begin{align*} \int \sin ^{-1}\left (1-x^2\right )^2 \, dx &=-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left (1-x^2\right )}{x}+x \sin ^{-1}\left (1-x^2\right )^2-8 \int 1 \, dx\\ &=-8 x-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left (1-x^2\right )}{x}+x \sin ^{-1}\left (1-x^2\right )^2\\ \end{align*}
Mathematica [A] time = 0.0147286, size = 44, normalized size = 1. \[ x \sin ^{-1}\left (1-x^2\right )^2-\frac{4 \sqrt{2 x^2-x^4} \sin ^{-1}\left (1-x^2\right )}{x}-8 x \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int \left ( \arcsin \left ({x}^{2}-1 \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} x \arctan \left (x^{2} - 1, \sqrt{-x^{2} + 2} x\right )^{2} + 4 \, \int \frac{\sqrt{-x^{2} + 2} x \arctan \left (x^{2} - 1, \sqrt{-x^{2} + 2} x\right )}{x^{2} - 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99209, size = 100, normalized size = 2.27 \begin{align*} \frac{x^{2} \arcsin \left (x^{2} - 1\right )^{2} - 8 \, x^{2} + 4 \, \sqrt{-x^{4} + 2 \, x^{2}} \arcsin \left (x^{2} - 1\right )}{x} \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 \operatorname{asin}^{2}{\left (x^{2} - 1 \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21066, size = 77, normalized size = 1.75 \begin{align*} x \arcsin \left (x^{2} - 1\right )^{2} + 2 \,{\left (\sqrt{2} \pi - 4 \, \sqrt{2}\right )} \mathrm{sgn}\left (x\right ) + \frac{4 \,{\left (\sqrt{-x^{2} + 2} \arcsin \left (x^{2} - 1\right ) - 2 \, x + 2 \, \sqrt{2}\right )}}{\mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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