3.1069 \(\int \frac{\sqrt{1+x}}{(1-x)^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x) \]

[Out]

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

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Rubi [A]  time = 0.0215233, antiderivative size = 23, 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 \[ \frac{2 \sqrt{x+1}}{\sqrt{1-x}}-\sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x]/(1 - x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.22439, size = 17, normalized size = 0.74 \[ - \operatorname{asin}{\left (x \right )} + \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**(1/2)/(1-x)**(3/2),x)

[Out]

-asin(x) + 2*sqrt(x + 1)/sqrt(-x + 1)

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Mathematica [A]  time = 0.0296442, size = 35, normalized size = 1.52 \[ -\frac{2 \sqrt{1-x^2}}{x-1}-2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x]/(1 - x)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.039, size = 64, normalized size = 2.8 \[ 2\,{\frac{\sqrt{1+x}\sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }}{\sqrt{- \left ( 1+x \right ) \left ( -1+x \right ) }\sqrt{1-x}}}-{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(1-x)^(3/2),x)

[Out]

2*(1+x)^(1/2)/(-(1+x)*(-1+x))^(1/2)*((1+x)*(1-x))^(1/2)/(1-x)^(1/2)-((1+x)*(1-x)
)^(1/2)/(1+x)^(1/2)/(1-x)^(1/2)*arcsin(x)

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Maxima [A]  time = 1.49794, size = 28, normalized size = 1.22 \[ -\frac{2 \, \sqrt{-x^{2} + 1}}{x - 1} - \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(-x + 1)^(3/2),x, algorithm="maxima")

[Out]

-2*sqrt(-x^2 + 1)/(x - 1) - arcsin(x)

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Fricas [A]  time = 0.213439, size = 82, normalized size = 3.57 \[ \frac{2 \,{\left ({\left (x + \sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right ) + 2 \, x\right )}}{x + \sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(-x + 1)^(3/2),x, algorithm="fricas")

[Out]

2*((x + sqrt(x + 1)*sqrt(-x + 1) - 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) +
2*x)/(x + sqrt(x + 1)*sqrt(-x + 1) - 1)

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Sympy [A]  time = 5.3051, size = 71, normalized size = 3.09 \[ \begin{cases} 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\- 2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**(1/2)/(1-x)**(3/2),x)

[Out]

Piecewise((2*I*acosh(sqrt(2)*sqrt(x + 1)/2) - 2*I*sqrt(x + 1)/sqrt(x - 1), Abs(x
+ 1)/2 > 1), (-2*asin(sqrt(2)*sqrt(x + 1)/2) + 2*sqrt(x + 1)/sqrt(-x + 1), True
))

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GIAC/XCAS [A]  time = 0.20739, size = 45, normalized size = 1.96 \[ -\frac{2 \, \sqrt{x + 1} \sqrt{-x + 1}}{x - 1} - 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x + 1)/(-x + 1)^(3/2),x, algorithm="giac")

[Out]

-2*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 2*arcsin(1/2*sqrt(2)*sqrt(x + 1))