Optimal. Leaf size=49 \[ \frac{\sqrt{x^2+2} \text{EllipticF}\left (\tan ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
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Rubi [A] time = 0.0087737, antiderivative size = 49, normalized size of antiderivative = 1., 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 = {418} \[ \frac{\sqrt{x^2+2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-x^2-1} \sqrt{\frac{x^2+2}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-x^2} \sqrt{2+x^2}} \, dx &=\frac{\sqrt{2+x^2} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{2} \sqrt{-1-x^2} \sqrt{\frac{2+x^2}{1+x^2}}}\\ \end{align*}
Mathematica [C] time = 0.0351139, size = 53, normalized size = 1.08 \[ -\frac{i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}(x),\frac{1}{2}\right )}{\sqrt{2} \sqrt{-\left (x^2+1\right ) \left (x^2+2\right )}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.019, size = 33, normalized size = 0.7 \begin{align*}{{\frac{i}{2}}\sqrt{2}{\it EllipticF} \left ( ix,{\frac{\sqrt{2}}{2}} \right ) \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} + 3 \, 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{- x^{2} - 1} \sqrt{x^{2} + 2}}\, 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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