### 3.288 $$\int \frac{e^{\coth ^{-1}(x)}}{1+x} \, dx$$

Optimal. Leaf size=22 $\tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right )$

[Out]

ArcTanh[Sqrt[1 + x^(-1)]*Sqrt[(-1 + x)/x]]

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Rubi [A]  time = 0.0660964, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6175, 6180, 92, 206} $\tanh ^{-1}\left (\sqrt{\frac{1}{x}+1} \sqrt{\frac{x-1}{x}}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[x]/(1 + x),x]

[Out]

ArcTanh[Sqrt[1 + x^(-1)]*Sqrt[(-1 + 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 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 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0094405, size = 18, normalized size = 0.82 $\log \left (x \left (\sqrt{\frac{x^2-1}{x^2}}+1\right )\right )$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[x]/(1 + x),x]

[Out]

Log[x*(1 + Sqrt[(-1 + x^2)/x^2])]

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Maple [A]  time = 0.126, size = 35, normalized size = 1.6 \begin{align*}{(-1+x)\ln \left ( x+\sqrt{{x}^{2}-1} \right ){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{ \left ( 1+x \right ) \left ( -1+x \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00396, size = 42, normalized size = 1.91 \begin{align*} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.57594, size = 88, normalized size = 4. \begin{align*} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \log \left (\sqrt{\frac{x - 1}{x + 1}} - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 20.6437, size = 29, normalized size = 1.32 \begin{align*} - \log{\left (-1 + \frac{1}{\sqrt{1 - \frac{2}{x + 1}}} \right )} + \log{\left (1 + \frac{1}{\sqrt{1 - \frac{2}{x + 1}}} \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14995, size = 43, normalized size = 1.95 \begin{align*} \log \left (\sqrt{\frac{x - 1}{x + 1}} + 1\right ) - \log \left ({\left | \sqrt{\frac{x - 1}{x + 1}} - 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt((x - 1)/(x + 1)) + 1) - log(abs(sqrt((x - 1)/(x + 1)) - 1))