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

Optimal. Leaf size=13 \[ -\frac{2}{3} (1-x)^{3/2} \]

[Out]

(-2*(1 - x)^(3/2))/3

_______________________________________________________________________________________

Rubi [A]  time = 0.00614335, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ -\frac{2}{3} (1-x)^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 - x)^(3/2))/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.24227, size = 10, normalized size = 0.77 \[ - \frac{2 \left (- x + 1\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(-x + 1)**(3/2)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0110106, size = 25, normalized size = 1.92 \[ \frac{2 (x-1) \sqrt{1-x^2}}{3 \sqrt{x+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1 + x)*Sqrt[1 - x^2])/(3*Sqrt[1 + x])

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 20, normalized size = 1.5 \[{\frac{2\,x-2}{3}\sqrt{-{x}^{2}+1}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(-1+x)*(-x^2+1)^(1/2)/(1+x)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.691604, size = 16, normalized size = 1.23 \[ \frac{2}{3} \,{\left (x - 1\right )} \sqrt{-x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(x - 1)*sqrt(-x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.265076, size = 39, normalized size = 3. \[ -\frac{2 \,{\left (x^{3} - x^{2} - x + 1\right )}}{3 \, \sqrt{-x^{2} + 1} \sqrt{x + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(x^3 - x^2 - x + 1)/(sqrt(-x^2 + 1)*sqrt(x + 1))

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt{x + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/sqrt(x + 1), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.267156, size = 20, normalized size = 1.54 \[ -\frac{2}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} + \frac{4}{3} \, \sqrt{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*(-x + 1)^(3/2) + 4/3*sqrt(2)