Optimal. Leaf size=88 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+x^2+2}} \]
[Out]
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Rubi [A] time = 0.0588196, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ \frac{\left (\sqrt{6} x^2+2\right ) \sqrt{\frac{3 x^4+x^2+2}{\left (\sqrt{6} x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{\frac{3}{2}} x\right )|\frac{1}{24} \left (12-\sqrt{6}\right )\right )}{2 \sqrt [4]{6} \sqrt{3 x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[2 + x^2 + 3*x^4],x]
[Out]
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Rubi in Sympy [A] time = 2.99494, size = 85, normalized size = 0.97 \[ \frac{6^{\frac{3}{4}} \sqrt{\frac{3 x^{4} + x^{2} + 2}{\left (\frac{\sqrt{6} x^{2}}{2} + 1\right )^{2}}} \left (\frac{\sqrt{6} x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{2^{\frac{3}{4}} \sqrt [4]{3} x}{2} \right )}\middle | - \frac{\sqrt{6}}{24} + \frac{1}{2}\right )}{12 \sqrt{3 x^{4} + x^{2} + 2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(3*x**4+x**2+2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.144619, size = 142, normalized size = 1.61 \[ -\frac{i \sqrt{1-\frac{6 x^2}{-1-i \sqrt{23}}} \sqrt{1-\frac{6 x^2}{-1+i \sqrt{23}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{6}{-1-i \sqrt{23}}} x\right )|\frac{-1-i \sqrt{23}}{-1+i \sqrt{23}}\right )}{\sqrt{6} \sqrt{-\frac{1}{-1-i \sqrt{23}}} \sqrt{3 x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[2 + x^2 + 3*x^4],x]
[Out]
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Maple [C] time = 0.154, size = 85, normalized size = 1. \[ 2\,{\frac{\sqrt{1- \left ( -1/4+i/4\sqrt{23} \right ){x}^{2}}\sqrt{1- \left ( -1/4-i/4\sqrt{23} \right ){x}^{2}}{\it EllipticF} \left ( 1/2\,x\sqrt{-1+i\sqrt{23}},1/6\,\sqrt{-33+3\,i\sqrt{23}} \right ) }{\sqrt{-1+i\sqrt{23}}\sqrt{3\,{x}^{4}+{x}^{2}+2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(3*x^4+x^2+2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{4} + x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(3*x^4 + x^2 + 2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{3 \, x^{4} + x^{2} + 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(3*x^4 + x^2 + 2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 x^{4} + x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(3*x**4+x**2+2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{3 \, x^{4} + x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(3*x^4 + x^2 + 2),x, algorithm="giac")
[Out]