Optimal. Leaf size=10 \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]
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Rubi [A] time = 0.0271326, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 F\left (\left .\sin ^{-1}\left (\sqrt{x}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]),x]
[Out]
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Rubi in Sympy [A] time = 2.99924, size = 10, normalized size = 1. \[ 2 F\left (\operatorname{asin}{\left (\sqrt{x} \right )}\middle | -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1-x)**(1/2)/x**(1/2)/(1+x)**(1/2),x)
[Out]
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Mathematica [C] time = 0.115731, size = 66, normalized size = 6.6 \[ \frac{2 i \sqrt{\frac{1}{x-1}+1} \sqrt{\frac{2}{x-1}+1} (x-1)^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac{1}{\sqrt{x-1}}\right )\right |2\right )}{\sqrt{-(x-1) x} \sqrt{x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]),x]
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Maple [B] time = 0.045, size = 24, normalized size = 2.4 \[{\sqrt{2}\sqrt{-x}{\it EllipticF} \left ( \sqrt{1+x},{\frac{\sqrt{2}}{2}} \right ){\frac{1}{\sqrt{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1-x)^(1/2)/x^(1/2)/(1+x)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.8399, size = 66, normalized size = 6.6 \[ \frac{i{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{2}, 1, 1 & \frac{3}{4}, \frac{3}{4}, \frac{5}{4} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4} & 0 \end{matrix} \middle |{\frac{1}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i{G_{6, 6}^{3, 5}\left (\begin{matrix} - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4} & 1 \\0, \frac{1}{2}, 0 & - \frac{1}{4}, \frac{1}{4}, \frac{1}{4} \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1-x)**(1/2)/x**(1/2)/(1+x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 1} \sqrt{x} \sqrt{-x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x + 1)*sqrt(x)*sqrt(-x + 1)),x, algorithm="giac")
[Out]