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

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

[Out]

-(Sqrt[1 - x]*Sqrt[1 + x])

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Rubi [A]  time = 0.0340615, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6129, 74} \[ -\sqrt{1-x} \sqrt{x+1} \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcTanh[x]*x)/(1 + x),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 74

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

Rubi steps

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

Mathematica [A]  time = 0.0052176, size = 13, normalized size = 0.72 \[ -\sqrt{1-x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcTanh[x]*x)/(1 + x),x]

[Out]

-Sqrt[1 - x^2]

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Maple [A]  time = 0.027, size = 17, normalized size = 0.9 \begin{align*}{ \left ( 1+x \right ) \left ( -1+x \right ){\frac{1}{\sqrt{-{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.937585, size = 15, normalized size = 0.83 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Fricas [C]  time = 1.75239, size = 23, normalized size = 1.28 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)

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Sympy [A]  time = 0.139766, size = 8, normalized size = 0.44 \begin{align*} - \sqrt{1 - x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(1 - x**2)

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Giac [A]  time = 1.14251, size = 15, normalized size = 0.83 \begin{align*} -\sqrt{-x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(-x^2 + 1)