3.8.38 \(\int \frac {\sqrt {\frac {1-x}{1+x}}}{-1+x} \, dx\) [738]

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

[Out]

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

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Rubi [A]
time = 0.01, 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 = {1983, 210} \begin {gather*} 2 \text {ArcTan}\left (\sqrt {\frac {1-x}{x+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 - x)/(1 + x)]/(-1 + x),x]

[Out]

2*ArcTan[Sqrt[(1 - x)/(1 + x)]]

Rule 210

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

Rule 1983

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)/(b*e - d*x^q)^(1/n + 1))*(u /. x -> ((-a)*e + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r, 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 \text {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 [B] Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(18)=36\).
time = 0.04, size = 51, normalized size = 2.83 \begin {gather*} -\frac {2 \sqrt {\frac {1-x}{1+x}} \sqrt {1-x^2} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right )}{-1+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.36, size = 30, normalized size = 1.67

method	result	size
default	\(-\frac {\sqrt {-\frac {-1+x}{1+x}}\, \left (1+x \right ) \arcsin \left (x \right )}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}}\)	\(30\)
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1+x}{1+x}}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-\frac {-1+x}{1+x}}+x \right )\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1-x)/(1+x))^(1/2)/(-1+x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 15, normalized size = 0.83 \begin {gather*} 2 \, \arctan \left (\sqrt {-\frac {x - 1}{x + 1}}\right ) \end {gather*}

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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Fricas [A]
time = 0.35, size = 15, normalized size = 0.83 \begin {gather*} 2 \, \arctan \left (\sqrt {-\frac {x - 1}{x + 1}}\right ) \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- \frac {x - 1}{x + 1}}}{x - 1}\, dx \end {gather*}

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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Giac [A]
time = 4.09, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{2} \, \pi \mathrm {sgn}\left (x + 1\right ) - \arcsin \left (x\right ) \mathrm {sgn}\left (x + 1\right ) \end {gather*}

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

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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