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

Optimal. Leaf size=11 \[ -2 \sqrt{1-x} \]

[Out]

-2*Sqrt[1 - x]

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Rubi [A]  time = 0.0192077, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6129, 32} \[ -2 \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Int[E^ArcTanh[x]/Sqrt[1 + x],x]

[Out]

-2*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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.003727, size = 11, normalized size = 1. \[ -2 \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcTanh[x]/Sqrt[1 + x],x]

[Out]

-2*Sqrt[1 - x]

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Maple [B]  time = 0.029, size = 20, normalized size = 1.8 \begin{align*} 2\,{\frac{ \left ( -1+x \right ) \sqrt{1+x}}{\sqrt{-{x}^{2}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969603, size = 16, normalized size = 1.45 \begin{align*} \frac{2 \,{\left (x - 1\right )}}{\sqrt{-x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(x - 1)/sqrt(-x + 1)

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Fricas [C]  time = 1.84967, size = 42, normalized size = 3.82 \begin{align*} -\frac{2 \, \sqrt{-x^{2} + 1}}{\sqrt{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt(-x^2 + 1)/sqrt(x + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15086, size = 20, normalized size = 1.82 \begin{align*} 2 \, \sqrt{2} - 2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(2) - 2*sqrt(-x + 1)