3.287 \(\int \frac{e^{\coth ^{-1}(x)} x}{1+x} \, dx\)

Optimal. Leaf size=22 \[ \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x \]

[Out]

Sqrt[1 + x^(-1)]*Sqrt[(-1 + x)/x]*x

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Rubi [A]  time = 0.0536301, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6175, 6179, 95} \[ \sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}} x \]

Antiderivative was successfully verified.

[In]

Int[(E^ArcCoth[x]*x)/(1 + x),x]

[Out]

Sqrt[1 + x^(-1)]*Sqrt[(-1 + x)/x]*x

Rule 6175

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

Rule 6179

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

Rule 95

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{e^{\coth ^{-1}(x)} x}{1+x} \, dx &=\int \frac{e^{\coth ^{-1}(x)}}{1+\frac{1}{x}} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2 \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )\\ &=\sqrt{1+\frac{1}{x}} \sqrt{\frac{-1+x}{x}} x\\ \end{align*}

Mathematica [A]  time = 0.0233141, size = 15, normalized size = 0.68 \[ x \sqrt{\frac{x^2-1}{x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcCoth[x]*x)/(1 + x),x]

[Out]

x*Sqrt[(-1 + x^2)/x^2]

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Maple [A]  time = 0.067, size = 16, normalized size = 0.7 \begin{align*}{(-1+x){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02722, size = 35, normalized size = 1.59 \begin{align*} -\frac{2 \, \sqrt{\frac{x - 1}{x + 1}}}{\frac{x - 1}{x + 1} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.53454, size = 42, normalized size = 1.91 \begin{align*}{\left (x + 1\right )} \sqrt{\frac{x - 1}{x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*sqrt((x - 1)/(x + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16971, size = 39, normalized size = 1.77 \begin{align*} -\frac{2}{\sqrt{\frac{x - 1}{x + 1}} - \frac{1}{\sqrt{\frac{x - 1}{x + 1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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