Optimal. Leaf size=30 \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),-2\right )}{\sqrt{2} \sqrt{x^2-1}} \]
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Rubi [A] time = 0.0146774, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {421, 419} \[ \frac{\sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2} \sqrt{x^2-1}} \]
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+4 x^2}} \, dx &=\frac{\sqrt{1-x^2} \int \frac{1}{\sqrt{1-x^2} \sqrt{2+4 x^2}} \, dx}{\sqrt{-1+x^2}}\\ &=\frac{\sqrt{1-x^2} F\left (\left .\sin ^{-1}(x)\right |-2\right )}{\sqrt{2} \sqrt{-1+x^2}}\\ \end{align*}
Mathematica [A] time = 0.0268143, size = 30, normalized size = 1. \[ \frac{\sqrt{1-x^2} \text{EllipticF}\left (\sin ^{-1}(x),-2\right )}{\sqrt{2} \sqrt{x^2-1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 34, normalized size = 1.1 \begin{align*}{-{\frac{i}{2}}{\it EllipticF} \left ( ix\sqrt{2},{\frac{i}{2}}\sqrt{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{4 \, 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{4 \, x^{2} + 2} \sqrt{x^{2} - 1}}{2 \,{\left (2 \, x^{4} - x^{2} - 1\right )}}, 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*} \frac{\sqrt{2} \int \frac{1}{\sqrt{x^{2} - 1} \sqrt{2 x^{2} + 1}}\, dx}{2} \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{4 \, 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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