Optimal. Leaf size=65 \[ \frac {\sqrt {-1+x^2} \sqrt {2+3 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {-1+x^2}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-2-x^2+3 x^4}} \]
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Rubi [A]
time = 0.00, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1111}
\begin {gather*} \frac {\sqrt {x^2-1} \sqrt {3 x^2+2} F\left (\text {ArcSin}\left (\frac {\sqrt {\frac {5}{2}} x}{\sqrt {x^2-1}}\right )|\frac {2}{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 {\frac {5}{2}} x}{\sqrt {-1+x^2}}\right )|\frac {2}{5}\right )}{\sqrt {5} \sqrt {-2-x^2+3 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.03, size = 60, normalized size = 0.92 \begin {gather*} -\frac {i \sqrt {1-x^2} \sqrt {2+3 x^2} F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {-6-3 x^2+9 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.04, size = 53, normalized size = 0.82
method | result | size |
default | \(-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {3 x^{4}-x^{2}-2}}\) | \(53\) |
elliptic | \(-\frac {i \sqrt {6}\, \sqrt {6 x^{2}+4}\, \sqrt {-x^{2}+1}\, \EllipticF \left (\frac {i x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )}{6 \sqrt {3 x^{4}-x^{2}-2}}\) | \(53\) |
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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