Optimal. Leaf size=44 \[ \sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (-3+\sqrt{15}\right )} x\right )|-4-\sqrt{15}\right ) \]
[Out]
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Rubi [A] time = 0.264769, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \sqrt{\frac{1}{6} \left (3+\sqrt{15}\right )} F\left (\sin ^{-1}\left (\sqrt{\frac{1}{3} \left (-3+\sqrt{15}\right )} x\right )|-4-\sqrt{15}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[3 + 6*x^2 - 2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 19.6277, size = 68, normalized size = 1.55 \[ \frac{2 \sqrt{6} F\left (\operatorname{asin}{\left (\frac{\sqrt{3} x \sqrt{-3 + \sqrt{15}}}{3} \right )}\middle | -4 - \sqrt{15}\right )}{\sqrt{-6 + 2 \sqrt{15}} \sqrt{-3 + \sqrt{15}} \sqrt{6 + 2 \sqrt{15}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-2*x**4+6*x**2+3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0916383, size = 43, normalized size = 0.98 \[ -\frac{i F\left (i \sinh ^{-1}\left (\sqrt{1+\sqrt{\frac{5}{3}}} x\right )|-4+\sqrt{15}\right )}{\sqrt{3+\sqrt{15}}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/Sqrt[3 + 6*x^2 - 2*x^4],x]
[Out]
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Maple [B] time = 0.119, size = 84, normalized size = 1.9 \[ 3\,{\frac{\sqrt{1- \left ( -1+1/3\,\sqrt{15} \right ){x}^{2}}\sqrt{1- \left ( -1-1/3\,\sqrt{15} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-9+3\,\sqrt{15}},i/2\sqrt{6}+i/2\sqrt{10} \right ) }{\sqrt{-9+3\,\sqrt{15}}\sqrt{-2\,{x}^{4}+6\,{x}^{2}+3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-2*x^4+6*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-2*x^4 + 6*x^2 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-2*x^4 + 6*x^2 + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 2 x^{4} + 6 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-2*x**4+6*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} + 6 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-2*x^4 + 6*x^2 + 3),x, algorithm="giac")
[Out]