Optimal. Leaf size=44 \[ \frac {1}{3} \sqrt {-x^4+x^2+2} x+F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {1}{3} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 419
Rule 424
Rule 524
Rule 1091
Rule 1180
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*}
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Mathematica [C] time = 0.05, size = 90, normalized size = 2.05 \[ \frac {-x^5+x^3-3 i \sqrt {-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+i \sqrt {-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+2 x}{3 \sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{4} + x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-x^{4} + x^{2} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 125, normalized size = 2.84 \[ \frac {\sqrt {-x^{4}+x^{2}+2}\, x}{3}+\frac {2 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{3 \sqrt {-x^{4}+x^{2}+2}}-\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {-x^{4}+x^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-x^{4} + x^{2} + 2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {-x^4+x^2+2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- x^{4} + x^{2} + 2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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