Optimal. Leaf size=12 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0383122, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[(1 - x^2)*(3 + x^2)],x]
[Out]
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Rubi in Sympy [A] time = 3.23253, size = 14, normalized size = 1.17 \[ \frac{\sqrt{3} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/((-x**2+1)*(x**2+3))**(1/2),x)
[Out]
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Mathematica [C] time = 0.0249868, size = 18, normalized size = 1.5 \[ -i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right ) \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[(1 - x^2)*(3 + x^2)],x]
[Out]
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Maple [B] time = 0.01, size = 43, normalized size = 3.6 \[{\frac{{\it EllipticF} \left ( x,{\frac{i}{3}}\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/((-x^2+1)*(x^2+3))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{-x^{4} - 2 \, x^{2} + 3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (- x^{2} + 1\right ) \left (x^{2} + 3\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((-x**2+1)*(x**2+3))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="giac")
[Out]