Optimal. Leaf size=110 \[ \frac{\sqrt{\frac{\left (4-\sqrt{10}\right ) x^2+3}{\left (4+\sqrt{10}\right ) x^2+3}} \left (\left (4+\sqrt{10}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right )|-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{2 x^4+8 x^2+3}} \]
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Rubi [A] time = 0.184747, antiderivative size = 110, 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 \[ \frac{\sqrt{\frac{\left (4-\sqrt{10}\right ) x^2+3}{\left (4+\sqrt{10}\right ) x^2+3}} \left (\left (4+\sqrt{10}\right ) x^2+3\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{3} \left (4+\sqrt{10}\right )} x\right )|-\frac{2}{3} \left (5-2 \sqrt{10}\right )\right )}{\sqrt{3 \left (4+\sqrt{10}\right )} \sqrt{2 x^4+8 x^2+3}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[3 + 8*x^2 + 2*x^4],x]
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Rubi in Sympy [A] time = 5.06709, size = 107, normalized size = 0.97 \[ \frac{\sqrt{3} \sqrt{\frac{x^{2} \left (- 2 \sqrt{10} + 8\right ) + 6}{x^{2} \left (2 \sqrt{10} + 8\right ) + 6}} \left (x^{2} \left (2 \sqrt{10} + 8\right ) + 6\right ) F\left (\operatorname{atan}{\left (\frac{\sqrt{3} x \sqrt{\sqrt{10} + 4}}{3} \right )}\middle | - \frac{10}{3} + \frac{4 \sqrt{10}}{3}\right )}{6 \sqrt{\sqrt{10} + 4} \sqrt{2 x^{4} + 8 x^{2} + 3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2*x**4+8*x**2+3)**(1/2),x)
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Mathematica [C] time = 0.140025, size = 98, normalized size = 0.89 \[ -\frac{i \sqrt{\frac{-2 x^2+\sqrt{10}-4}{\sqrt{10}-4}} \sqrt{2 x^2+\sqrt{10}+4} F\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{4+\sqrt{10}}} x\right )|\frac{13}{3}+\frac{4 \sqrt{10}}{3}\right )}{\sqrt{4 x^4+16 x^2+6}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/Sqrt[3 + 8*x^2 + 2*x^4],x]
[Out]
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Maple [A] time = 0.234, size = 82, normalized size = 0.8 \[ 3\,{\frac{\sqrt{1- \left ( -4/3+1/3\,\sqrt{10} \right ){x}^{2}}\sqrt{1- \left ( -4/3-1/3\,\sqrt{10} \right ){x}^{2}}{\it EllipticF} \left ( 1/3\,x\sqrt{-12+3\,\sqrt{10}},2/3\,\sqrt{6}+1/3\,\sqrt{15} \right ) }{\sqrt{-12+3\,\sqrt{10}}\sqrt{2\,{x}^{4}+8\,{x}^{2}+3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2*x^4+8*x^2+3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 + 8*x^2 + 3),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 + 8*x^2 + 3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 x^{4} + 8 x^{2} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2*x**4+8*x**2+3)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{2 \, x^{4} + 8 \, x^{2} + 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(2*x^4 + 8*x^2 + 3),x, algorithm="giac")
[Out]