Optimal. Leaf size=40 \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac {4 \sqrt {-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4814, 8} \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac {4 \sqrt {-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]
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 = 40, normalized size = 1.00 \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac {4 \sqrt {-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 39, normalized size = 0.98 \[ x \arctan \left (\frac {\sqrt {-x^{4} - 2 \, x^{2}} {\left (x^{2} + 1\right )}}{x^{4} + 2 \, x^{2}}\right )^{2} - 8 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \arcsin \left (x^{2} + 1\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, 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(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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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