Optimal. Leaf size=49 \[ \frac{\sqrt{2 x^2+1} \text{EllipticF}\left (\tan ^{-1}(x),-1\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x^2+1}{x^2+1}}} \]
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Rubi [A] time = 0.0092925, 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} \[ \frac{\sqrt{2 x^2+1} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x^2+1}{x^2+1}}} \]
Antiderivative was successfully verified.
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Rule 418
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1+x^2} \sqrt{2+4 x^2}} \, dx &=\frac{\sqrt{1+2 x^2} F\left (\left .\tan ^{-1}(x)\right |-1\right )}{\sqrt{2} \sqrt{1+x^2} \sqrt{\frac{1+2 x^2}{1+x^2}}}\\ \end{align*}
Mathematica [C] time = 0.0277553, size = 17, normalized size = 0.35 \[ -\frac{i \text{EllipticF}\left (i \sinh ^{-1}(x),2\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 15, normalized size = 0.3 \begin{align*} -{\frac{i}{2}}{\it EllipticF} \left ( ix,\sqrt{2} \right ) \sqrt{2} \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} + 3 \, 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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