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

Optimal. Leaf size=9 \[ 2 \sqrt{x+1} \]

[Out]

2*Sqrt[1 + x]

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Rubi [A]  time = 0.0011453, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {26, 32} \[ 2 \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x]/Sqrt[1 - x^2],x]

[Out]

2*Sqrt[1 + x]

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}}{\sqrt{1-x^2}} \, dx &=\int \frac{1}{\sqrt{1+x}} \, dx\\ &=2 \sqrt{1+x}\\ \end{align*}

Mathematica [B]  time = 0.0226464, size = 25, normalized size = 2.78 \[ \frac{2 \sqrt{1-x} (x+1)}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - x]*(1 + x))/Sqrt[1 - x^2]

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Maple [B]  time = 0.002, size = 22, normalized size = 2.4 \begin{align*} 2\,{\frac{ \left ( 1+x \right ) \sqrt{1-x}}{\sqrt{-{x}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-x)^(1/2)/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.47036, size = 9, normalized size = 1. \begin{align*} 2 \, \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x + 1)

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Fricas [C]  time = 1.43803, size = 54, normalized size = 6. \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1} \sqrt{-x + 1}}{x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-x)**(1/2)/(-x**2+1)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.12649, size = 18, normalized size = 2. \begin{align*} -2 \, \sqrt{2} + 2 \, \sqrt{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(2) + 2*sqrt(x + 1)