Optimal. Leaf size=31 \[ \frac{\sqrt{x^2+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{\sqrt{-x^2-1}} \]
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Rubi [A] time = 0.0156464, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {421, 419} \[ \frac{\sqrt{x^2+1} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{\sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
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Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-x^2} \sqrt{2-x^2}} \, dx &=\frac{\sqrt{1+x^2} \int \frac{1}{\sqrt{2-x^2} \sqrt{1+x^2}} \, dx}{\sqrt{-1-x^2}}\\ &=\frac{\sqrt{1+x^2} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{\sqrt{-1-x^2}}\\ \end{align*}
Mathematica [C] time = 0.0248291, size = 39, normalized size = 1.26 \[ -\frac{i \sqrt{x^2+1} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 34, normalized size = 1.1 \begin{align*}{{\frac{i}{2}}{\it EllipticF} \left ( ix,{\frac{i}{2}}\sqrt{2} \right ) \sqrt{2}\sqrt{-{x}^{2}-1}{\frac{1}{\sqrt{{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-x^{2} + 2} \sqrt{-x^{2} - 1}}{x^{4} - x^{2} - 2}, x\right ) \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 \frac{1}{\sqrt{2 - x^{2}} \sqrt{- x^{2} - 1}}\, 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 \frac{1}{\sqrt{-x^{2} + 2} \sqrt{-x^{2} - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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