### 3.334 $$\int \frac{e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx$$

Optimal. Leaf size=90 $-\frac{\sqrt{\frac{1}{x}+1} x \sqrt{1-\frac{1}{x}}}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}$

[Out]

-((Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]*x)/(1 - x)^(3/2)) - ((1 - x^(-1))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/Sq
rt[1 + x^(-1)]])/(Sqrt[2]*(1 - x)^(3/2)*(x^(-1))^(3/2))

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Rubi [A]  time = 0.109552, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.357, Rules used = {6176, 6181, 94, 93, 206} $-\frac{\sqrt{\frac{1}{x}+1} x \sqrt{1-\frac{1}{x}}}{(1-x)^{3/2}}-\frac{\left (1-\frac{1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{2} (1-x)^{3/2} \left (\frac{1}{x}\right )^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[x]/(1 - x)^(3/2),x]

[Out]

-((Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]*x)/(1 - x)^(3/2)) - ((1 - x^(-1))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/Sq
rt[1 + x^(-1)]])/(Sqrt[2]*(1 - x)^(3/2)*(x^(-1))^(3/2))

Rule 6176

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

Rule 6181

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

Rule 94

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

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

Mathematica [A]  time = 0.0907516, size = 58, normalized size = 0.64 $\frac{\frac{2}{\sqrt{\frac{1}{x+1}}}+\sqrt{2} (x-1) \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{x+1}}\right )}{2 \sqrt{-\frac{(x-1)^2}{x^2}} x}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[x]/(1 - x)^(3/2),x]

[Out]

(2/Sqrt[(1 + x)^(-1)] + Sqrt[2]*(-1 + x)*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]])/(2*Sqrt[-((-1 + x)^2/x^2)]*x)

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Maple [A]  time = 0.11, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{-2+2\,x}\sqrt{1-x} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1-x}} \right ) x-\sqrt{2}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-1-x}} \right ) +2\,\sqrt{-1-x} \right ){\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}{\frac{1}{\sqrt{-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)^(3/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-x + 1\right )}^{\frac{3}{2}} \sqrt{\frac{x - 1}{x + 1}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/2)/(1-x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-x + 1)^(3/2)*sqrt((x - 1)/(x + 1))), x)

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Fricas [A]  time = 1.57579, size = 208, normalized size = 2.31 \begin{align*} -\frac{\sqrt{2}{\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{x - 1}\right ) + 2 \,{\left (x + 1\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{2 \,{\left (x^{2} - 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)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))**(1/2)/(1-x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.13293, size = 59, normalized size = 0.66 \begin{align*} \frac{{\left (\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-x - 1}\right ) + \frac{2 \, \sqrt{-x - 1}}{x - 1}\right )} \mathrm{sgn}\left (x\right )}{2 \, \mathrm{sgn}\left (-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)^(3/2),x, algorithm="giac")

[Out]

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