Optimal. Leaf size=48 \[ \frac{\sqrt{2} \left (x^2+2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
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Rubi [A] time = 0.0290589, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {1456, 411} \[ \frac{\sqrt{2} \left (x^2+2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Rule 1456
Rule 411
Rubi steps
\begin{align*} \int \frac{2+x^2}{\left (1+x^2\right ) \sqrt{2+3 x^2+x^4}} \, dx &=\frac{\left (\sqrt{1+x^2} \sqrt{2+x^2}\right ) \int \frac{\sqrt{2+x^2}}{\left (1+x^2\right )^{3/2}} \, dx}{\sqrt{2+3 x^2+x^4}}\\ &=\frac{\sqrt{2} \left (2+x^2\right ) E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{\sqrt{\frac{2+x^2}{1+x^2}} \sqrt{2+3 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.135588, size = 94, normalized size = 1.96 \[ \frac{-i \sqrt{x^2+1} \sqrt{x^2+2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right ),2\right )+x^3+i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+2 x}{\sqrt{x^4+3 x^2+2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.028, size = 81, normalized size = 1.7 \begin{align*}{ \left ({x}^{2}+2 \right ) x{\frac{1}{\sqrt{ \left ({x}^{2}+1 \right ) \left ({x}^{2}+2 \right ) }}}}-{{\frac{i}{2}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}x\sqrt{2},\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+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{x^{2} + 2}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (x^{2} + 1\right )}}\,{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^{4} + 3 \, x^{2} + 2}}{x^{4} + 2 \, x^{2} + 1}, 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{x^{2} + 2}{\sqrt{\left (x^{2} + 1\right ) \left (x^{2} + 2\right )} \left (x^{2} + 1\right )}\, 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{x^{2} + 2}{\sqrt{x^{4} + 3 \, x^{2} + 2}{\left (x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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