Optimal. Leaf size=96 \[ \sqrt {9+2 \sqrt {21}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right )-\sqrt {\frac {1}{2} \left (\sqrt {21}-3\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right ) \]
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Rubi [A] time = 0.18, antiderivative size = 96, 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 = {1180, 524, 424, 419} \[ \sqrt {9+2 \sqrt {21}} F\left (\sin ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right )-\sqrt {\frac {1}{2} \left (\sqrt {21}-3\right )} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5-\sqrt {21}\right )\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1180
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 (3+\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 )+\frac {1}{2} \sqrt {36+8 \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] time = 0.17, size = 103, normalized size = 1.07 \[ -\frac {i \left (\left (3+\sqrt {21}\right ) E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )-\left (\sqrt {21}-3\right ) F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{-3+\sqrt {21}}} x\right )|\frac {1}{2} \left (-5+\sqrt {21}\right )\right )\right )}{\sqrt {2 \left (3+\sqrt {21}\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + 3 \, x^{2} + 3} {\left (x^{2} - 3\right )}}{x^{4} - 3 \, x^{2} - 3}, x\right ) \]
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 \[ \int -\frac {x^{2} - 3}{\sqrt {-x^{4} + 3 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 204, normalized size = 2.12 \[ \frac {18 \sqrt {-\left (-\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-18+6 \sqrt {21}}\, x}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )}{\sqrt {-18+6 \sqrt {21}}\, \sqrt {-x^{4}+3 x^{2}+3}}+\frac {36 \sqrt {-\left (-\frac {1}{2}+\frac {\sqrt {21}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {\sqrt {21}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-18+6 \sqrt {21}}\, x}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )+\EllipticF \left (\frac {\sqrt {-18+6 \sqrt {21}}\, x}{6}, \frac {i \sqrt {3}}{2}+\frac {i \sqrt {7}}{2}\right )\right )}{\sqrt {-18+6 \sqrt {21}}\, \sqrt {-x^{4}+3 x^{2}+3}\, \left (3+\sqrt {21}\right )} \]
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 \[ -\int \frac {x^{2} - 3}{\sqrt {-x^{4} + 3 \, x^{2} + 3}}\,{d x} \]
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 \[ \int -\frac {x^2-3}{\sqrt {-x^4+3\,x^2+3}} \,d x \]
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 \[ - \int \frac {x^{2}}{\sqrt {- x^{4} + 3 x^{2} + 3}}\, dx - \int \left (- \frac {3}{\sqrt {- x^{4} + 3 x^{2} + 3}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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