Optimal. Leaf size=44 \[ \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )+\frac{1}{3} \sqrt{-x^4+x^2+2} x+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.0401104, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {1091, 1180, 524, 424, 419} \[ \frac{1}{3} \sqrt{-x^4+x^2+2} x+F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 1091
Rule 1180
Rule 524
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \sqrt{2+x^2-x^4} \, dx &=\frac{1}{3} x \sqrt{2+x^2-x^4}+\frac{1}{3} \int \frac{4+x^2}{\sqrt{2+x^2-x^4}} \, dx\\ &=\frac{1}{3} x \sqrt{2+x^2-x^4}+\frac{2}{3} \int \frac{4+x^2}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{3} x \sqrt{2+x^2-x^4}+\frac{1}{3} \int \frac{\sqrt{2+2 x^2}}{\sqrt{4-2 x^2}} \, dx+2 \int \frac{1}{\sqrt{4-2 x^2} \sqrt{2+2 x^2}} \, dx\\ &=\frac{1}{3} x \sqrt{2+x^2-x^4}+\frac{1}{3} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )\\ \end{align*}
Mathematica [C] time = 0.0450932, size = 90, normalized size = 2.05 \[ \frac{-3 i \sqrt{-2 x^4+2 x^2+4} \text{EllipticF}\left (i \sinh ^{-1}(x),-\frac{1}{2}\right )-x^5+x^3+i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+2 x}{3 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 125, normalized size = 2.8 \begin{align*}{\frac{x}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{2\,\sqrt{2}}{3}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{\sqrt{2}}{6}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{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 \sqrt{-x^{4} + x^{2} + 2}\,{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 (\sqrt{-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 \sqrt{- x^{4} + 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 \sqrt{-x^{4} + x^{2} + 2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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