Optimal. Leaf size=40 \[ -8 x+\frac {4 \sqrt {-2 x^2-x^4} \text {ArcSin}\left (1+x^2\right )}{x}+x \text {ArcSin}\left (1+x^2\right )^2 \]
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Rubi [A]
time = 0.00, 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 = {4898, 8}
\begin {gather*} x \text {ArcSin}\left (x^2+1\right )^2+\frac {4 \sqrt {-x^4-2 x^2} \text {ArcSin}\left (x^2+1\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 = 40, 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.18, size = 39, normalized size = 0.98 \begin {gather*} 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 {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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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