### 3.322 $$\int \sqrt{\frac{1+x}{1-x}} \, dx$$

Optimal. Leaf size=41 $2 \tan ^{-1}\left (\sqrt{\frac{x+1}{1-x}}\right )-(1-x) \sqrt{\frac{x+1}{1-x}}$

[Out]

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

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Rubi [A]  time = 0.0130075, antiderivative size = 41, 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, 203} $2 \tan ^{-1}\left (\sqrt{\frac{x+1}{1-x}}\right )-(1-x) \sqrt{\frac{x+1}{1-x}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[(1 + x)/(1 - x)],x]

[Out]

-((1 - x)*Sqrt[(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 203

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

Rubi steps

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

Mathematica [A]  time = 0.0228433, size = 62, normalized size = 1.51 $\frac{\sqrt{\frac{x+1}{1-x}} \left ((x-1) \sqrt{x+1}-2 \sqrt{1-x} \sin ^{-1}\left (\frac{\sqrt{1-x}}{\sqrt{2}}\right )\right )}{\sqrt{x+1}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[(1 + x)/(1 - x)],x]

[Out]

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

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Maple [A]  time = 0.008, size = 41, normalized size = 1. \begin{align*}{(-1+x)\sqrt{-{\frac{1+x}{-1+x}}} \left ( \sqrt{-{x}^{2}+1}-\arcsin \left ( x \right ) \right ){\frac{1}{\sqrt{- \left ( -1+x \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]

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

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Maxima [A]  time = 1.41076, size = 58, normalized size = 1.41 \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 = 2.27597, size = 90, normalized size = 2.2 \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{x + 1}{1 - x}}\, 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((x + 1)/(1 - x)), x)

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Giac [A]  time = 1.07219, size = 41, normalized size = 1. \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)