Optimal. Leaf size=25 \[ -E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )+4 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1194, 21, 434,
435, 430} \begin {gather*} 4 F\left (\left .\text {ArcSin}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )-E\left (\left .\text {ArcSin}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 430
Rule 434
Rule 435
Rule 1194
Rubi steps
\begin {align*} \int \frac {3-x^2}{\sqrt {3+2 x^2-x^4}} \, dx &=2 \int \frac {3-x^2}{\sqrt {6-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=\int \frac {\sqrt {6-2 x^2}}{\sqrt {2+2 x^2}} \, dx\\ &=8 \int \frac {1}{\sqrt {6-2 x^2} \sqrt {2+2 x^2}} \, dx-\int \frac {\sqrt {2+2 x^2}}{\sqrt {6-2 x^2}} \, dx\\ &=-E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )+4 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {3}}\right )\right |-3\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.05, size = 19, normalized size = 0.76 \begin {gather*} -i \sqrt {3} E\left (i \sinh ^{-1}(x)|-\frac {1}{3}\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. 112 vs. \(2 (31 ) = 62\).
time = 0.03, size = 113, normalized size = 4.52
method | result | size |
default | \(\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (\frac {x \sqrt {3}}{3}, i \sqrt {3}\right )-\EllipticE \left (\frac {x \sqrt {3}}{3}, i \sqrt {3}\right )\right )}{3 \sqrt {-x^{4}+2 x^{2}+3}}+\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {3}}{3}, i \sqrt {3}\right )}{\sqrt {-x^{4}+2 x^{2}+3}}\) | \(113\) |
elliptic | \(\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {x^{2}+1}\, \left (\EllipticF \left (\frac {x \sqrt {3}}{3}, i \sqrt {3}\right )-\EllipticE \left (\frac {x \sqrt {3}}{3}, i \sqrt {3}\right )\right )}{3 \sqrt {-x^{4}+2 x^{2}+3}}+\frac {\sqrt {3}\, \sqrt {-3 x^{2}+9}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {x \sqrt {3}}{3}, i \sqrt {3}\right )}{\sqrt {-x^{4}+2 x^{2}+3}}\) | \(113\) |
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 = 0.07, size = 18, normalized size = 0.72 \begin {gather*} \frac {\sqrt {-x^{4} + 2 \, 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} + 2 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} + 2 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 = -0.04 \begin {gather*} \int -\frac {x^2-3}{\sqrt {-x^4+2\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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