Optimal. Leaf size=10 \[ \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right ),-2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {419} \[ F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-x^2} \sqrt {1+x^2}} \, dx &=F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 19, normalized size = 1.90 \[ -\frac {i \operatorname {EllipticF}\left (i \sinh ^{-1}(x),-\frac {1}{2}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 2}}{x^{4} - x^{2} - 2}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {-x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 14, normalized size = 1.40 \[ \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{2} + 1} \sqrt {-x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.10 \[ \int \frac {1}{\sqrt {x^2+1}\,\sqrt {2-x^2}} \,d x \]
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 \[ \int \frac {1}{\sqrt {2 - x^{2}} \sqrt {x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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