3.370 \(\int \frac{e^{\tanh ^{-1}(x)}}{(1+x)^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Rubi [A]  time = 0.0181478, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6129, 37} \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]/(1 + x)^2,x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

Rule 6129

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[c^p, Int[(u*(1 + (d*x)/c)
^p*(1 + a*x)^(n/2))/(1 - a*x)^(n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ
[p] || GtQ[c, 0])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{\tanh ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx\\ &=-\frac{\sqrt{1-x}}{\sqrt{1+x}}\\ \end{align*}

Mathematica [A]  time = 0.0039239, size = 18, normalized size = 1. \[ -\frac{\sqrt{1-x}}{\sqrt{x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[x]/(1 + x)^2,x]

[Out]

-(Sqrt[1 - x]/Sqrt[1 + x])

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Maple [A]  time = 0.029, size = 14, normalized size = 0.8 \begin{align*}{(-1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.41851, size = 22, normalized size = 1.22 \begin{align*} -\frac{\sqrt{-x^{2} + 1}}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)/(x + 1)

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Fricas [A]  time = 1.73147, size = 47, normalized size = 2.61 \begin{align*} -\frac{x + \sqrt{-x^{2} + 1} + 1}{x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + sqrt(-x^2 + 1) + 1)/(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27452, size = 28, normalized size = 1.56 \begin{align*} \frac{2}{\frac{\sqrt{-x^{2} + 1} - 1}{x} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/((sqrt(-x^2 + 1) - 1)/x - 1)