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

Optimal. Leaf size=11 \[ -2 \sqrt{1-x} \]

[Out]

-2*Sqrt[1 - x]

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Rubi [A]  time = 0.001246, antiderivative size = 11, 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} \[ -2 \sqrt{1-x} \]

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 = 0.02142, size = 23, normalized size = 2.09 \[ \frac{2 (x-1) \sqrt{x+1}}{\sqrt{1-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.002, size = 20, normalized size = 1.8 \begin{align*} 2\,{\frac{ \left ( x-1 \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*(x-1)*(1+x)^(1/2)/(-x^2+1)^(1/2)

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Maxima [A]  time = 1.42184, size = 16, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (x - 1\right )}}{\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*(x - 1)/sqrt(-x + 1)

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Fricas [C]  time = 1.45357, size = 42, normalized size = 3.82 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{\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="fricas")

[Out]

-2*sqrt(-x^2 + 1)/sqrt(x + 1)

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

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Giac [A]  time = 1.08708, size = 20, normalized size = 1.82 \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)