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 = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 8
Rule 4814
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*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 1.00 \[ x \sin ^{-1}\left (1-x^2\right )^2-\frac {4 \sqrt {2 x^2-x^4} \sin ^{-1}\left (1-x^2\right )}{x}-8 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 43, normalized size = 0.98 \[ \frac {x^{2} \arcsin \left (x^{2} - 1\right )^{2} - 8 \, x^{2} + 4 \, \sqrt {-x^{4} + 2 \, x^{2}} \arcsin \left (x^{2} - 1\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.76, size = 58, normalized size = 1.32 \[ x \arcsin \left (x^{2} - 1\right )^{2} + 2 \, {\left (\sqrt {2} \pi - 4 \, \sqrt {2}\right )} \mathrm {sgn}\relax (x) + \frac {4 \, {\left (\sqrt {-x^{2} + 2} \arcsin \left (x^{2} - 1\right ) + 2 \, \sqrt {2} - 2 \, {\left | x \right |}\right )}}{\mathrm {sgn}\relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \arcsin \left (x^{2}-1\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \arctan \left (x^{2} - 1, \sqrt {-x^{2} + 2} x\right )^{2} + 4 \, \int \frac {\sqrt {-x^{2} + 2} x \arctan \left (x^{2} - 1, \sqrt {-x^{2} + 2} x\right )}{x^{2} - 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {asin}\left (x^2-1\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asin}^{2}{\left (x^{2} - 1 \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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