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 = 0.0062437, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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 = 0.0132387, size = 40, normalized size = 1. \[ x \sin ^{-1}\left (x^2+1\right )^2+\frac{4 \sqrt{-x^4-2 x^2} \sin ^{-1}\left (x^2+1\right )}{x}-8 x \]
Antiderivative was successfully verified.
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Maple [F] time = 0.052, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.26228, size = 84, normalized size = 2.1 \begin{align*} x \arctan \left (\frac{\sqrt{-x^{4} - 2 \, x^{2}}{\left (x^{2} + 1\right )}}{x^{4} + 2 \, x^{2}}\right )^{2} - 8 \, 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \arcsin \left (x^{2} + 1\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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