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]

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

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Rubi [A]  time = 0.01, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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])

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 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
]

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*}

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Mathematica [A]  time = 0.02, 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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fricas [A]  time = 0.43, size = 32, normalized size = 0.78 \[ {\left (x - 1\right )} \sqrt {-\frac {x + 1}{x - 1}} + 2 \, \arctan \left (\sqrt {-\frac {x + 1}{x - 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.13, size = 30, normalized size = 0.73 \[ \frac {1}{2} \, \pi \mathrm {sgn}\left (x - 1\right ) - \arcsin \relax (x) \mathrm {sgn}\left (x - 1\right ) + \sqrt {-x^{2} + 1} \mathrm {sgn}\left (x - 1\right ) \]

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)

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maple [A]  time = 0.01, size = 41, normalized size = 1.00 \[ \frac {\sqrt {-\frac {x +1}{x -1}}\, \left (x -1\right ) \left (-\arcsin \relax (x )+\sqrt {-x^{2}+1}\right )}{\sqrt {-\left (x -1\right ) \left (x +1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.20, size = 43, normalized size = 1.05 \[ \frac {2 \, \sqrt {-\frac {x + 1}{x - 1}}}{\frac {x + 1}{x - 1} - 1} + 2 \, \arctan \left (\sqrt {-\frac {x + 1}{x - 1}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/(1-x))^(1/2),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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mupad [B]  time = 0.05, size = 43, normalized size = 1.05 \[ 2\,\mathrm {atan}\left (\sqrt {-\frac {x+1}{x-1}}\right )+\frac {2\,\sqrt {-\frac {x+1}{x-1}}}{\frac {x+1}{x-1}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-(x + 1)/(x - 1))^(1/2),x)

[Out]

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

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

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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