Optimal. Leaf size=92 \[ -\sqrt {\frac {1}{2} \left (3+\sqrt {21}\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )+\sqrt {3+2 \sqrt {21}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1194, 538, 435,
430} \begin {gather*} \sqrt {3+2 \sqrt {21}} F\left (\text {ArcSin}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )-\sqrt {\frac {1}{2} \left (3+\sqrt {21}\right )} E\left (\text {ArcSin}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\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 {3-3 x^2-x^4}} \, dx &=2 \int \frac {3-x^2}{\sqrt {-3+\sqrt {21}-2 x^2} \sqrt {3+\sqrt {21}+2 x^2}} \, dx\\ &=\left (9+\sqrt {21}\right ) \int \frac {1}{\sqrt {-3+\sqrt {21}-2 x^2} \sqrt {3+\sqrt {21}+2 x^2}} \, dx-\int \frac {\sqrt {3+\sqrt {21}+2 x^2}}{\sqrt {-3+\sqrt {21}-2 x^2}} \, dx\\ &=-\sqrt {\frac {1}{2} \left (3+\sqrt {21}\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )+\sqrt {3+2 \sqrt {21}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.10, size = 107, normalized size = 1.16 \begin {gather*} -\frac {i \left (\left (-3+\sqrt {21}\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|-\frac {5}{2}-\frac {\sqrt {21}}{2}\right )-\left (-9+\sqrt {21}\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|-\frac {5}{2}-\frac {\sqrt {21}}{2}\right )\right )}{\sqrt {2 \left (-3+\sqrt {21}\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. 203 vs. \(2 (74 ) = 148\).
time = 0.08, size = 204, normalized size = 2.22
method | result | size |
default | \(\frac {36 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {18+6 \sqrt {21}}}{6}, \frac {i \sqrt {7}}{2}-\frac {i \sqrt {3}}{2}\right )-\EllipticE \left (\frac {x \sqrt {18+6 \sqrt {21}}}{6}, \frac {i \sqrt {7}}{2}-\frac {i \sqrt {3}}{2}\right )\right )}{\sqrt {18+6 \sqrt {21}}\, \sqrt {-x^{4}-3 x^{2}+3}\, \left (-3+\sqrt {21}\right )}+\frac {18 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {18+6 \sqrt {21}}}{6}, \frac {i \sqrt {7}}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {18+6 \sqrt {21}}\, \sqrt {-x^{4}-3 x^{2}+3}}\) | \(204\) |
elliptic | \(\frac {36 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {18+6 \sqrt {21}}}{6}, \frac {i \sqrt {7}}{2}-\frac {i \sqrt {3}}{2}\right )-\EllipticE \left (\frac {x \sqrt {18+6 \sqrt {21}}}{6}, \frac {i \sqrt {7}}{2}-\frac {i \sqrt {3}}{2}\right )\right )}{\sqrt {18+6 \sqrt {21}}\, \sqrt {-x^{4}-3 x^{2}+3}\, \left (-3+\sqrt {21}\right )}+\frac {18 \sqrt {1-\left (\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}}\, \sqrt {1-\left (\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}}\, \EllipticF \left (\frac {x \sqrt {18+6 \sqrt {21}}}{6}, \frac {i \sqrt {7}}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {18+6 \sqrt {21}}\, \sqrt {-x^{4}-3 x^{2}+3}}\) | \(204\) |
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 = 18, normalized size = 0.20 \begin {gather*} \frac {\sqrt {-x^{4} - 3 \, 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} - 3 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} - 3 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^4-3\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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