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

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

[Out]

(-2*(1 - x)^(3/2))/3

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Rubi [A]  time = 0.0013626, 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, Rules used = {26, 32} \[ -\frac{2}{3} (1-x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/Sqrt[1 + x],x]

[Out]

(-2*(1 - x)^(3/2))/3

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-(b^2/d))^m, Int[
u/(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d,
0] && GtQ[a, 0] && LtQ[d, 0]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-x^2}}{\sqrt{1+x}} \, dx &=\int \sqrt{1-x} \, dx\\ &=-\frac{2}{3} (1-x)^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0196129, 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])

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

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

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Maxima [A]  time = 1.11251, size = 16, normalized size = 1.23 \begin{align*} \frac{2}{3} \,{\left (x - 1\right )} \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1+x)^(1/2),x, algorithm="maxima")

[Out]

2/3*(x - 1)*sqrt(-x + 1)

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Fricas [B]  time = 1.46389, size = 54, normalized size = 4.15 \begin{align*} \frac{2 \, \sqrt{-x^{2} + 1}{\left (x - 1\right )}}{3 \, \sqrt{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1+x)^(1/2),x, algorithm="fricas")

[Out]

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

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

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)

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Giac [A]  time = 1.1058, size = 20, normalized size = 1.54 \begin{align*} -\frac{2}{3} \,{\left (-x + 1\right )}^{\frac{3}{2}} + \frac{4}{3} \, \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^2+1)^(1/2)/(1+x)^(1/2),x, algorithm="giac")

[Out]

-2/3*(-x + 1)^(3/2) + 4/3*sqrt(2)