3.858 \(\int \frac{\sqrt{-1+\frac{1}{x}} \sqrt{\frac{1}{x}} \sqrt{x}}{\sqrt{1+x}} \, dx\)

Optimal. Leaf size=24 \[ -\frac{2 \sqrt{-x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{x}} \]

[Out]

(-2*Sqrt[-x]*EllipticE[ArcSin[Sqrt[-x]], -1])/Sqrt[x]

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Rubi [B]  time = 0.0747845, antiderivative size = 49, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{2 \sqrt{\frac{1}{x}-1} \sqrt{\frac{1}{x}} \sqrt{-x} \sqrt{x} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]  Int[(Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[x])/Sqrt[1 + x],x]

[Out]

(-2*Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[-x]*Sqrt[x]*EllipticE[ArcSin[Sqrt[-x]],
-1])/Sqrt[1 - x]

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Rubi in Sympy [A]  time = 34.4422, size = 80, normalized size = 3.33 \[ 2 \sqrt{x} \sqrt{-1 + \frac{1}{x}} \sqrt{x + 1} \sqrt{\frac{1}{x}} - \frac{2 \sqrt{1 - \frac{1}{x}} \sqrt{x + 1} \sqrt{\frac{1}{x}} E\left (\operatorname{asin}{\left (\frac{1}{\sqrt{x}} \right )}\middle | -1\right )}{\sqrt{-1 + \frac{1}{x}} \sqrt{1 + \frac{1}{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-1+1/x)**(1/2)*(1/x)**(1/2)*x**(1/2)/(1+x)**(1/2),x)

[Out]

2*sqrt(x)*sqrt(-1 + 1/x)*sqrt(x + 1)*sqrt(1/x) - 2*sqrt(1 - 1/x)*sqrt(x + 1)*sqr
t(1/x)*elliptic_e(asin(1/sqrt(x)), -1)/(sqrt(-1 + 1/x)*sqrt(1 + 1/x))

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Mathematica [A]  time = 0.056796, size = 46, normalized size = 1.92 \[ \frac{2 \sqrt{\frac{1}{x}-1} \left (\frac{1}{x}\right )^{5/2} \left (-x^2\right )^{3/2} E\left (\left .\sin ^{-1}\left (\sqrt{-x}\right )\right |-1\right )}{\sqrt{1-x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(Sqrt[-1 + x^(-1)]*Sqrt[x^(-1)]*Sqrt[x])/Sqrt[1 + x],x]

[Out]

(2*Sqrt[-1 + x^(-1)]*(x^(-1))^(5/2)*(-x^2)^(3/2)*EllipticE[ArcSin[Sqrt[-x]], -1]
)/Sqrt[1 - x]

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Maple [B]  time = 0.036, size = 49, normalized size = 2. \[ -2\,{\frac{\sqrt{{x}^{-1}}\sqrt{x}{\it EllipticE} \left ( \sqrt{1+x},1/2\,\sqrt{2} \right ) \sqrt{-x}\sqrt{2-2\,x}}{-1+x}\sqrt{-{\frac{-1+x}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-1+1/x)^(1/2)*(1/x)^(1/2)*x^(1/2)/(1+x)^(1/2),x)

[Out]

-2*(1/x)^(1/2)*x^(1/2)*(-(-1+x)/x)^(1/2)*EllipticE((1+x)^(1/2),1/2*2^(1/2))*(-x)
^(1/2)*(2-2*x)^(1/2)/(-1+x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{1}{x} - 1}}{\sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/x - 1)/sqrt(x + 1),x, algorithm="maxima")

[Out]

integrate(sqrt(1/x - 1)/sqrt(x + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-\frac{x - 1}{x}}}{\sqrt{x + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/x - 1)/sqrt(x + 1),x, algorithm="fricas")

[Out]

integral(sqrt(-(x - 1)/x)/sqrt(x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-1+1/x)**(1/2)*(1/x)**(1/2)*x**(1/2)/(1+x)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{1}{x} - 1}}{\sqrt{x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(1/x - 1)/sqrt(x + 1),x, algorithm="giac")

[Out]

integrate(sqrt(1/x - 1)/sqrt(x + 1), x)