Optimal. Leaf size=44 \[ -8 x-\frac {4 \sqrt {2 x^2-x^4} \text {ArcSin}\left (1-x^2\right )}{x}+x \text {ArcSin}\left (1-x^2\right )^2 \]
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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 = {4898, 8}
\begin {gather*} x \text {ArcSin}\left (1-x^2\right )^2-\frac {4 \sqrt {2 x^2-x^4} \text {ArcSin}\left (1-x^2\right )}{x}-8 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4898
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 = 0.01, size = 44, normalized size = 1.00 \begin {gather*} -8 x-\frac {4 \sqrt {2 x^2-x^4} \text {ArcSin}\left (1-x^2\right )}{x}+x \text {ArcSin}\left (1-x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.25, size = 43, normalized size = 0.98 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \operatorname {asin}^{2}{\left (x^{2} - 1 \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 58, normalized size = 1.32 \begin {gather*} 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 \, \sqrt {2} - 2 \, {\left | x \right |}\right )}}{\mathrm {sgn}\left (x\right )} \end {gather*}
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 \begin {gather*} \int {\mathrm {asin}\left (x^2-1\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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