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

Optimal. Leaf size=18 \[ 2 \tan ^{-1}\left (\sqrt {\frac {1-x}{x+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {1961, 204} \[ 2 \tan ^{-1}\left (\sqrt {\frac {1-x}{x+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 - x)/(1 + x)]/(-1 + x),x]

[Out]

2*ArcTan[Sqrt[(1 - x)/(1 + 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])

Rule 1961

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 34, normalized size = 1.89 \[ \frac {\sqrt {\frac {1-x}{x+1}} \sqrt {1-x^2} \sin ^{-1}(x)}{x-1} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 - x)/(1 + x)]/(-1 + x),x]

[Out]

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

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fricas [A]  time = 0.41, size = 15, normalized size = 0.83 \[ 2 \, \arctan \left (\sqrt {-\frac {x - 1}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(-(x - 1)/(x + 1)))

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giac [A]  time = 0.31, size = 16, normalized size = 0.89 \[ -\frac {1}{2} \, \pi \mathrm {sgn}\left (x + 1\right ) - \arcsin \relax (x) \mathrm {sgn}\left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 30, normalized size = 1.67 \[ -\frac {\sqrt {-\frac {x -1}{x +1}}\, \left (x +1\right ) \arcsin \relax (x )}{\sqrt {-\left (x -1\right ) \left (x +1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.54, size = 15, normalized size = 0.83 \[ 2 \, \arctan \left (\sqrt {-\frac {x - 1}{x + 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(sqrt(-(x - 1)/(x + 1)))

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mupad [B]  time = 3.14, size = 15, normalized size = 0.83 \[ 2\,\mathrm {atan}\left (\sqrt {-\frac {x-1}{x+1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atan((-(x - 1)/(x + 1))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \frac {x - 1}{x + 1}}}{x - 1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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