Optimal. Leaf size=63 \[ \frac {\sqrt {-3+x^2} \sqrt {1+2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {-3+x^2}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {-3-5 x^2+2 x^4}} \]
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Rubi [A]
time = 0.00, antiderivative size = 63, 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^2-3} \sqrt {2 x^2+1} F\left (\text {ArcSin}\left (\frac {\sqrt {7} x}{\sqrt {x^2-3}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {2 x^4-5 x^2-3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 1111
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-3-5 x^2+2 x^4}} \, dx &=\frac {\sqrt {-3+x^2} \sqrt {1+2 x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {7} x}{\sqrt {-3+x^2}}\right )|\frac {1}{7}\right )}{\sqrt {7} \sqrt {-3-5 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.03, size = 65, normalized size = 1.03 \begin {gather*} -\frac {i \sqrt {1-\frac {x^2}{3}} \sqrt {1+2 x^2} F\left (i \sinh ^{-1}\left (\sqrt {2} x\right )|-\frac {1}{6}\right )}{\sqrt {2} \sqrt {-3-5 x^2+2 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 = 53, normalized size = 0.84
method | result | size |
default | \(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {-3 x^{2}+9}\, \EllipticF \left (i \sqrt {2}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {2 x^{4}-5 x^{2}-3}}\) | \(53\) |
elliptic | \(-\frac {i \sqrt {2}\, \sqrt {2 x^{2}+1}\, \sqrt {-3 x^{2}+9}\, \EllipticF \left (i \sqrt {2}\, x , \frac {i \sqrt {6}}{6}\right )}{6 \sqrt {2 x^{4}-5 x^{2}-3}}\) | \(53\) |
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 {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 = -0.02 \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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