Optimal. Leaf size=64 \[ \frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
[Out]
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Rubi [A] time = 0.0212322, antiderivative size = 64, 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 \[ \frac{\left (x^2+2\right ) \sqrt{\frac{x^4+3 x^2+4}{\left (x^2+2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )|\frac{1}{8}\right )}{2 \sqrt{2} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[4 + 3*x^2 + x^4],x]
[Out]
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Rubi in Sympy [A] time = 9.96456, size = 63, normalized size = 0.98 \[ \frac{\sqrt{2} \sqrt{\frac{x^{4} + 3 x^{2} + 4}{\left (\frac{x^{2}}{2} + 1\right )^{2}}} \left (\frac{x^{2}}{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | \frac{1}{8}\right )}{4 \sqrt{x^{4} + 3 x^{2} + 4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**4+3*x**2+4)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0955351, size = 142, normalized size = 2.22 \[ -\frac{i \sqrt{1-\frac{2 x^2}{-3-i \sqrt{7}}} \sqrt{1-\frac{2 x^2}{-3+i \sqrt{7}}} F\left (i \sinh ^{-1}\left (\sqrt{-\frac{2}{-3-i \sqrt{7}}} x\right )|\frac{-3-i \sqrt{7}}{-3+i \sqrt{7}}\right )}{\sqrt{2} \sqrt{-\frac{1}{-3-i \sqrt{7}}} \sqrt{x^4+3 x^2+4}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[4 + 3*x^2 + x^4],x]
[Out]
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Maple [C] time = 0.004, size = 85, normalized size = 1.3 \[ 4\,{\frac{\sqrt{1- \left ( -3/8+i/8\sqrt{7} \right ){x}^{2}}\sqrt{1- \left ( -3/8-i/8\sqrt{7} \right ){x}^{2}}{\it EllipticF} \left ( 1/4\,x\sqrt{-6+2\,i\sqrt{7}},1/4\,\sqrt{2+6\,i\sqrt{7}} \right ) }{\sqrt{-6+2\,i\sqrt{7}}\sqrt{{x}^{4}+3\,{x}^{2}+4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^4+3*x^2+4)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x^4 + 3*x^2 + 4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x^4 + 3*x^2 + 4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} + 3 x^{2} + 4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**4+3*x**2+4)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{4} + 3 \, x^{2} + 4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(x^4 + 3*x^2 + 4),x, algorithm="giac")
[Out]