Optimal. Leaf size=92 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-2 x^4-2 x^2-3}} \]
[Out]
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Rubi [A] time = 0.0529185, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4+2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{12} \left (6-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{-2 x^4-2 x^2-3}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[-3 - 2*x^2 - 2*x^4],x]
[Out]
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Rubi in Sympy [A] time = 3.60811, size = 94, normalized size = 1.02 \[ \frac{6^{\frac{3}{4}} \sqrt{- \frac{- 2 x^{4} - 2 x^{2} - 3}{\left (\frac{\sqrt{6} x^{2}}{3} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{3} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | - \frac{\sqrt{6}}{12} + \frac{1}{2}\right )}{12 \sqrt{- 2 x^{4} - 2 x^{2} - 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(-2*x**4-2*x**2-3)**(1/2),x)
[Out]
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Mathematica [C] time = 0.141384, size = 144, normalized size = 1.57 \[ -\frac{i \sqrt{1-\frac{2 x^2}{-1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{-1+i \sqrt{5}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{-1-i \sqrt{5}}} x\right )|\frac{-1-i \sqrt{5}}{-1+i \sqrt{5}}\right )}{\sqrt{2} \sqrt{-\frac{1}{-1-i \sqrt{5}}} \sqrt{-2 x^4-2 x^2-3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[-3 - 2*x^2 - 2*x^4],x]
[Out]
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Maple [C] time = 0.066, size = 87, normalized size = 1. \[ 3\,{\frac{\sqrt{1- \left ( -1/3-i/3\sqrt{5} \right ){x}^{2}}\sqrt{1- \left ( -1/3+i/3\sqrt{5} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,\sqrt{-3-3\,i\sqrt{5}}x,1/3\,\sqrt{-6-3\,i\sqrt{5}} \right ) }{\sqrt{-3-3\,i\sqrt{5}}\sqrt{-2\,{x}^{4}-2\,{x}^{2}-3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(-2*x^4-2*x^2-3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} - 2 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-2*x^4 - 2*x^2 - 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{\sqrt{-2 \, x^{4} - 2 \, x^{2} - 3}}{2 \, x^{4} + 2 \, x^{2} + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-2*x^4 - 2*x^2 - 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- 2 x^{4} - 2 x^{2} - 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(-2*x**4-2*x**2-3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-2 \, x^{4} - 2 \, x^{2} - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-2*x^4 - 2*x^2 - 3),x, algorithm="giac")
[Out]