Optimal. Leaf size=96 \[ -\sqrt {\frac {1}{2} \left (-1+\sqrt {13}\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right )+\sqrt {7+2 \sqrt {13}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1194, 538, 435,
430} \begin {gather*} \sqrt {7+2 \sqrt {13}} F\left (\text {ArcSin}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right )-\sqrt {\frac {1}{2} \left (\sqrt {13}-1\right )} E\left (\text {ArcSin}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 430
Rule 435
Rule 538
Rule 1194
Rubi steps
\begin {align*} \int \frac {3-x^2}{\sqrt {3+x^2-x^4}} \, dx &=2 \int \frac {3-x^2}{\sqrt {1+\sqrt {13}-2 x^2} \sqrt {-1+\sqrt {13}+2 x^2}} \, dx\\ &=\left (5+\sqrt {13}\right ) \int \frac {1}{\sqrt {1+\sqrt {13}-2 x^2} \sqrt {-1+\sqrt {13}+2 x^2}} \, dx-\int \frac {\sqrt {-1+\sqrt {13}+2 x^2}}{\sqrt {1+\sqrt {13}-2 x^2}} \, dx\\ &=-\sqrt {\frac {1}{2} \left (-1+\sqrt {13}\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right )+\sqrt {7+2 \sqrt {13}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7-\sqrt {13}\right )\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.13, size = 103, normalized size = 1.07 \begin {gather*} -\frac {i \left (\left (1+\sqrt {13}\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7+\sqrt {13}\right )\right )-\left (-5+\sqrt {13}\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {13}}} x\right )|\frac {1}{6} \left (-7+\sqrt {13}\right )\right )\right )}{\sqrt {2 \left (1+\sqrt {13}\right )}} \end {gather*}
Warning: Unable to verify antiderivative.
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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. 199 vs. \(2 (74 ) = 148\).
time = 0.08, size = 200, normalized size = 2.08
method | result | size |
default | \(\frac {36 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}\, \left (1+\sqrt {13}\right )}+\frac {18 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}}\) | \(200\) |
elliptic | \(\frac {36 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )-\EllipticE \left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}\, \left (1+\sqrt {13}\right )}+\frac {18 \sqrt {1-\left (-\frac {1}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}}\, \sqrt {1-\left (-\frac {1}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {-6+6 \sqrt {13}}}{6}, \frac {i \sqrt {3}}{6}+\frac {i \sqrt {39}}{6}\right )}{\sqrt {-6+6 \sqrt {13}}\, \sqrt {-x^{4}+x^{2}+3}}\) | \(200\) |
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 = 0.10, size = 16, normalized size = 0.17 \begin {gather*} \frac {\sqrt {-x^{4} + x^{2} + 3}}{x} \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 {x^{2}}{\sqrt {- x^{4} + x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} + x^{2} + 3}}\right )\, 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 = -0.01 \begin {gather*} -\int \frac {x^2-3}{\sqrt {-x^4+x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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