Optimal. Leaf size=48 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.140866, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353 \[ \frac{1}{3} \sqrt{-x^4-2 x^2+3} x+\frac{4 F\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}}-\frac{2 E\left (\sin ^{-1}(x)|-\frac{1}{3}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(1 - x^2)*(3 + x^2)],x]
[Out]
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Rubi in Sympy [A] time = 12.1868, size = 49, normalized size = 1.02 \[ \frac{x \sqrt{- x^{4} - 2 x^{2} + 3}}{3} - \frac{2 \sqrt{3} E\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} + \frac{4 \sqrt{3} F\left (\operatorname{asin}{\left (x \right )}\middle | - \frac{1}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((-x**2+1)*(x**2+3))**(1/2),x)
[Out]
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Mathematica [C] time = 0.0951991, size = 59, normalized size = 1.23 \[ \frac{1}{3} \left (\sqrt{-x^4-2 x^2+3} x-4 i F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )-2 i E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right )\right |-3\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(1 - x^2)*(3 + x^2)],x]
[Out]
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Maple [B] time = 0.016, size = 114, normalized size = 2.4 \[{\frac{x}{3}\sqrt{-{x}^{4}-2\,{x}^{2}+3}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}}+{\frac{2\,{\it EllipticF} \left ( x,i/3\sqrt{3} \right ) -2\,{\it EllipticE} \left ( x,i/3\sqrt{3} \right ) }{3}\sqrt{-{x}^{2}+1}\sqrt{3\,{x}^{2}+9}{\frac{1}{\sqrt{-{x}^{4}-2\,{x}^{2}+3}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((-x^2+1)*(x^2+3))^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{-x^{4} - 2 \, x^{2} + 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (- x^{2} + 1\right ) \left (x^{2} + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((-x**2+1)*(x**2+3))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-{\left (x^{2} + 3\right )}{\left (x^{2} - 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-(x^2 + 3)*(x^2 - 1)),x, algorithm="giac")
[Out]