Optimal. Leaf size=14 \[ -F\left (\left .\cos ^{-1}\left (\sqrt {\frac {2}{3}} x\right )\right |3\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, 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, 431}
\begin {gather*} -F\left (\left .\text {ArcCos}\left (\sqrt {\frac {2}{3}} x\right )\right |3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 431
Rule 1109
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3+5 x^2-2 x^4}} \, dx &=\left (2 \sqrt {2}\right ) \int \frac {1}{\sqrt {6-4 x^2} \sqrt {-4+4 x^2}} \, dx\\ &=-F\left (\left .\cos ^{-1}\left (\sqrt {\frac {2}{3}} x\right )\right |3\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(14)=28\).
time = 10.03, size = 53, normalized size = 3.79 \begin {gather*} \frac {\sqrt {3-2 x^2} \sqrt {1-x^2} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{3}} x\right )|\frac {3}{2}\right )}{\sqrt {-6+10 x^2-4 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 50, normalized size = 3.57
method | result | size |
default | \(\frac {\sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {6}}{3}, \frac {\sqrt {6}}{2}\right )}{6 \sqrt {-2 x^{4}+5 x^{2}-3}}\) | \(50\) |
elliptic | \(\frac {\sqrt {6}\, \sqrt {-6 x^{2}+9}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {6}}{3}, \frac {\sqrt {6}}{2}\right )}{6 \sqrt {-2 x^{4}+5 x^{2}-3}}\) | \(50\) |
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 {- 2 x^{4} + 5 x^{2} - 3}}\, 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 = -0.07 \begin {gather*} \int \frac {1}{\sqrt {-2\,x^4+5\,x^2-3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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