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

Optimal. Leaf size=38 \[ \sqrt{\frac{1-x}{x+1}} (x+1)-2 \tan ^{-1}\left (\sqrt{\frac{1-x}{x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0136912, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1959, 288, 204} \[ \sqrt{\frac{1-x}{x+1}} (x+1)-2 \tan ^{-1}\left (\sqrt{\frac{1-x}{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 - x)/(1 + x)],x]

[Out]

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

Rule 1959

Int[(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> With[{q = Denominator[p]
}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[(x^(q*(p + 1) - 1)*(-(a*e) + c*x^q)^(1/n - 1))/(b*e - d*x^q)^(1/n + 1),
x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && FractionQ[p] && IntegerQ[1/n
]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{\frac{1-x}{1+x}} \, dx &=-\left (4 \operatorname{Subst}\left (\int \frac{x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt{\frac{1-x}{1+x}}\right )\right )\\ &=\sqrt{\frac{1-x}{1+x}} (1+x)+2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{\frac{1-x}{1+x}}\right )\\ &=\sqrt{\frac{1-x}{1+x}} (1+x)-2 \tan ^{-1}\left (\sqrt{\frac{1-x}{1+x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0216509, size = 67, normalized size = 1.76 \[ \frac{\sqrt{\frac{1-x}{x+1}} \sqrt{x+1} \left (\sqrt{x+1} (x-1)+2 \sqrt{1-x} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right )}{x-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 - x)/(1 + x)],x]

[Out]

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

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Maple [A]  time = 0.004, size = 39, normalized size = 1. \begin{align*}{(1+x)\sqrt{-{\frac{x-1}{1+x}}} \left ( \sqrt{-{x}^{2}+1}+\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{- \left ( x-1 \right ) \left ( 1+x \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.49234, size = 58, normalized size = 1.53 \begin{align*} -\frac{2 \, \sqrt{-\frac{x - 1}{x + 1}}}{\frac{x - 1}{x + 1} - 1} - 2 \, \arctan \left (\sqrt{-\frac{x - 1}{x + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.77437, size = 90, normalized size = 2.37 \begin{align*}{\left (x + 1\right )} \sqrt{-\frac{x - 1}{x + 1}} - 2 \, \arctan \left (\sqrt{-\frac{x - 1}{x + 1}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((1 - x)/(x + 1)), x)

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Giac [A]  time = 1.12657, size = 39, normalized size = 1.03 \begin{align*} \frac{1}{2} \, \pi \mathrm{sgn}\left (x + 1\right ) + \arcsin \left (x\right ) \mathrm{sgn}\left (x + 1\right ) + \sqrt{-x^{2} + 1} \mathrm{sgn}\left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*pi*sgn(x + 1) + arcsin(x)*sgn(x + 1) + sqrt(-x^2 + 1)*sgn(x + 1)