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

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

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [A]  time = 0.01, 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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fricas [C]  time = 0.49, size = 11, normalized size = 0.61 \[ -\sqrt {-x^{2} + 1} \]

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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giac [C]  time = 0.29, size = 11, normalized size = 0.61 \[ -\sqrt {-x^{2} + 1} \]

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)

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maple [A]  time = 0.03, size = 17, normalized size = 0.94 \[ \frac {\left (1+x \right ) \left (-1+x \right )}{\sqrt {-x^{2}+1}} \]

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 [C]  time = 0.39, size = 11, normalized size = 0.61 \[ -\sqrt {-x^{2} + 1} \]

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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mupad [B]  time = 0.11, size = 11, normalized size = 0.61 \[ -\sqrt {1-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(1 - x^2)^(1/2),x)

[Out]

-(1 - x^2)^(1/2)

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sympy [A]  time = 0.12, size = 8, normalized size = 0.44 \[ - \sqrt {1 - x^{2}} \]

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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