### 3.292 $$\int \frac{e^{\coth ^{-1}(x)}}{(1+x)^2} \, dx$$

Optimal. Leaf size=21 $\frac{\sqrt{\frac{x-1}{x}}}{\sqrt{\frac{1}{x}+1}}$

[Out]

Sqrt[(-1 + x)/x]/Sqrt[1 + x^(-1)]

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Rubi [A]  time = 0.0651539, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.3, Rules used = {6175, 6180, 37} $\frac{\sqrt{\frac{x-1}{x}}}{\sqrt{\frac{1}{x}+1}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[x]/(1 + x)^2,x]

[Out]

Sqrt[(-1 + x)/x]/Sqrt[1 + x^(-1)]

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 6180

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1
+ (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 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]) && IntegerQ[m]

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^{\coth ^{-1}(x)}}{(1+x)^2} \, dx &=\int \frac{e^{\coth ^{-1}(x)}}{\left (1+\frac{1}{x}\right )^2 x^2} \, dx\\ &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} (1+x)^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{\frac{-1+x}{x}}}{\sqrt{1+\frac{1}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0109699, size = 18, normalized size = 0.86 $\frac{\sqrt{1-\frac{1}{x^2}} x}{x+1}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[x]/(1 + x)^2,x]

[Out]

(Sqrt[1 - x^(-2)]*x)/(1 + x)

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Maple [A]  time = 0.058, size = 21, normalized size = 1. \begin{align*}{\frac{-1+x}{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)/(1+x)^2,x)

[Out]

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

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Maxima [A]  time = 1.03692, size = 15, normalized size = 0.71 \begin{align*} \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)/(1+x)^2,x, algorithm="maxima")

[Out]

sqrt((x - 1)/(x + 1))

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Fricas [A]  time = 1.59365, size = 31, normalized size = 1.48 \begin{align*} \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)/(1+x)^2,x, algorithm="fricas")

[Out]

sqrt((x - 1)/(x + 1))

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Sympy [A]  time = 37.0712, size = 8, normalized size = 0.38 \begin{align*} \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)/(1+x)**2,x)

[Out]

sqrt((x - 1)/(x + 1))

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Giac [A]  time = 1.10328, size = 15, normalized size = 0.71 \begin{align*} \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)/(1+x)^2,x, algorithm="giac")

[Out]

sqrt((x - 1)/(x + 1))