3.1068 \(\int \frac{\sqrt{1+x}}{\sqrt{1-x}} \, dx\)

Optimal. Leaf size=21 \[ \sin ^{-1}(x)-\sqrt{1-x} \sqrt{x+1} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0215368, antiderivative size = 21, 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 \[ \sin ^{-1}(x)-\sqrt{1-x} \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x]/Sqrt[1 - x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.91407, size = 15, normalized size = 0.71 \[ - \sqrt{- x + 1} \sqrt{x + 1} + \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x + 1)*sqrt(x + 1) + asin(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0133682, size = 30, normalized size = 1.43 \[ 2 \sin ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{2}}\right )-\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x]/Sqrt[1 - x],x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.006, size = 42, normalized size = 2. \[ -\sqrt{1-x}\sqrt{1+x}+{\arcsin \left ( x \right ) \sqrt{ \left ( 1+x \right ) \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(1-x)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.4858, size = 19, normalized size = 0.9 \[ -\sqrt{-x^{2} + 1} + \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1) + arcsin(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.210893, size = 80, normalized size = 3.81 \[ \frac{x^{2} - 2 \,{\left (\sqrt{x + 1} \sqrt{-x + 1} - 1\right )} \arctan \left (\frac{\sqrt{x + 1} \sqrt{-x + 1} - 1}{x}\right )}{\sqrt{x + 1} \sqrt{-x + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 5.94904, size = 100, normalized size = 4.76 \[ \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} - \frac{i \left (x + 1\right )^{\frac{3}{2}}}{\sqrt{x - 1}} + \frac{2 i \sqrt{x + 1}}{\sqrt{x - 1}} & \text{for}\: \frac{\left |{x + 1}\right |}{2} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{x + 1}}{2} \right )} + \frac{\left (x + 1\right )^{\frac{3}{2}}}{\sqrt{- x + 1}} - \frac{2 \sqrt{x + 1}}{\sqrt{- x + 1}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-2*I*acosh(sqrt(2)*sqrt(x + 1)/2) - I*(x + 1)**(3/2)/sqrt(x - 1) + 2*
I*sqrt(x + 1)/sqrt(x - 1), Abs(x + 1)/2 > 1), (2*asin(sqrt(2)*sqrt(x + 1)/2) + (
x + 1)**(3/2)/sqrt(-x + 1) - 2*sqrt(x + 1)/sqrt(-x + 1), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.20717, size = 38, normalized size = 1.81 \[ -\sqrt{x + 1} \sqrt{-x + 1} + 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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