Optimal. Leaf size=10 \[ F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1109, 430}
\begin {gather*} F\left (\left .\text {ArcSin}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2+5 x^2-3 x^4}} \, dx &=\left (2 \sqrt {3}\right ) \int \frac {1}{\sqrt {12-6 x^2} \sqrt {2+6 x^2}} \, dx\\ &=F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-6\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.02, size = 65, normalized size = 6.50 \begin {gather*} -\frac {i \sqrt {1-\frac {x^2}{2}} \sqrt {1+3 x^2} F\left (i \sinh ^{-1}\left (\sqrt {3} x\right )|-\frac {1}{6}\right )}{\sqrt {3} \sqrt {2+5 x^2-3 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 50 vs. \(2 (13 ) = 26\).
time = 0.04, size = 51, normalized size = 5.10
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {3 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {6}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}+2}}\) | \(51\) |
elliptic | \(\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {3 x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {6}\right )}{2 \sqrt {-3 x^{4}+5 x^{2}+2}}\) | \(51\) |
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 [A]
time = 0.07, size = 8, normalized size = 0.80 \begin {gather*} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x, -6\right ) \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} + 5 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 = -0.10 \begin {gather*} \int \frac {1}{\sqrt {-3\,x^4+5\,x^2+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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