Optimal. Leaf size=63 \[ \frac {\sqrt {1+x^2} \sqrt {-2+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {5} x}{\sqrt {-2+3 x^2}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {-2+x^2+3 x^4}} \]
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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1111}
\begin {gather*} \frac {\sqrt {x^2+1} \sqrt {3 x^2-2} F\left (\text {ArcSin}\left (\frac {\sqrt {5} x}{\sqrt {3 x^2-2}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {3 x^4+x^2-2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1111
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2+x^2+3 x^4}} \, dx &=\frac {\sqrt {1+x^2} \sqrt {-2+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {5} x}{\sqrt {-2+3 x^2}}\right )|\frac {3}{5}\right )}{\sqrt {5} \sqrt {-2+x^2+3 x^4}}\\ \end {align*}
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Mathematica [A]
time = 10.03, size = 48, normalized size = 0.76 \begin {gather*} \frac {\sqrt {\left (\frac {2}{3}-x^2\right ) \left (1+x^2\right )} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {-2+x^2+3 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.03, size = 43, normalized size = 0.68
method | result | size |
default | \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \EllipticF \left (i x , \frac {i \sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+x^{2}-2}}\) | \(43\) |
elliptic | \(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-6 x^{2}+4}\, \EllipticF \left (i x , \frac {i \sqrt {6}}{2}\right )}{2 \sqrt {3 x^{4}+x^{2}-2}}\) | \(43\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {3 x^{4} + x^{2} - 2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {3\,x^4+x^2-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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