Optimal. Leaf size=90 \[ \frac{\left (\sqrt{6} x^2+3\right ) \sqrt{\frac{2 x^4-2 x^2+3}{\left (\sqrt{6} x^2+3\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{2}{3}} x\right )|\frac{1}{12} \left (6+\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{2 x^4-2 x^2+3}} \]
[Out]
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Rubi [A] time = 0.0526938, antiderivative size = 90, 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 = 3.93937, size = 88, normalized size = 0.98 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{2 x^{4} - 2 x^{2} + 3}{\left (\frac{\sqrt{6} x^{2}}{3} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{3} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{2} \cdot 3^{\frac{3}{4}} x}{3} \right )}\middle | \frac{\sqrt{6}}{12} + \frac{1}{2}\right )}{12 \sqrt{2 x^{4} - 2 x^{2} + 3}} \]
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 = 0.136139, size = 144, normalized size = 1.6 \[ -\frac{i \sqrt{1-\frac{2 x^2}{1-i \sqrt{5}}} \sqrt{1-\frac{2 x^2}{1+i \sqrt{5}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{1-i \sqrt{5}}} x\right )|\frac{1-i \sqrt{5}}{1+i \sqrt{5}}\right )}{\sqrt{2} \sqrt{-\frac{1}{1-i \sqrt{5}}} \sqrt{2 x^4-2 x^2+3}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[3 - 2*x^2 + 2*x^4],x]
[Out]
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Maple [C] time = 0.116, size = 87, normalized size = 1. \[ 3\,{\frac{\sqrt{1- \left ( 1/3+i/3\sqrt{5} \right ){x}^{2}}\sqrt{1- \left ( 1/3-i/3\sqrt{5} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{3+3\,i\sqrt{5}},1/3\,\sqrt{-6-3\,i\sqrt{5}} \right ) }{\sqrt{3+3\,i\sqrt{5}}\sqrt{2\,{x}^{4}-2\,{x}^{2}+3}}} \]
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. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} - 2 \, x^{2} + 3}}, x\right ) \]
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]