### 3.329 $$\int \frac{e^{\coth ^{-1}(x)} x}{\sqrt{1-x}} \, dx$$

Optimal. Leaf size=126 $\frac{2 \sqrt{1-\frac{1}{x}} \left (\frac{1}{x}+1\right )^{3/2} x^2}{3 \sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}$

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.114503, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {6176, 6181, 96, 94, 93, 206} $\frac{2 \sqrt{1-\frac{1}{x}} \left (\frac{1}{x}+1\right )^{3/2} x^2}{3 \sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{\frac{1}{x}+1} x}{\sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{\frac{1}{x}+1}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]

[Out]

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

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 96

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] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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)} x}{\sqrt{1-x}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{x}} \sqrt{x}\right ) \int \frac{e^{\coth ^{-1}(x)} \sqrt{x}}{\sqrt{1-\frac{1}{x}}} \, dx}{\sqrt{1-x}}\\ &=-\frac{\sqrt{1-\frac{1}{x}} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{(1-x) x^{5/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{\sqrt{1-\frac{1}{x}} \operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{(1-x) x^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{\left (2 \sqrt{1-\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{(1-x) \sqrt{x} \sqrt{1+x}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{\left (4 \sqrt{1-\frac{1}{x}}\right ) \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ &=\frac{2 \sqrt{1-\frac{1}{x}} \sqrt{1+\frac{1}{x}} x}{\sqrt{1-x}}+\frac{2 \sqrt{1-\frac{1}{x}} \left (1+\frac{1}{x}\right )^{3/2} x^2}{3 \sqrt{1-x}}-\frac{2 \sqrt{2} \sqrt{1-\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{x}}}\right )}{\sqrt{1-x} \sqrt{\frac{1}{x}}}\\ \end{align*}

Mathematica [A]  time = 0.0467435, size = 69, normalized size = 0.55 $\frac{2 \sqrt{\frac{x-1}{x}} x \left (\sqrt{\frac{1}{x}+1} (x+4)-3 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{1}{x+1}}\right )\right )}{3 \sqrt{1-x}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]

[Out]

(2*Sqrt[(-1 + x)/x]*x*(Sqrt[1 + x^(-1)]*(4 + x) - 3*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))
/(3*Sqrt[1 - x])

________________________________________________________________________________________

Maple [A]  time = 0.118, size = 66, normalized size = 0.5 \begin{align*}{\frac{2}{3}\sqrt{1-x} \left ( 3\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{-1-x}\sqrt{2} \right ) -\sqrt{-1-x}x-4\,\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)*x/(1-x)^(1/2),x)

[Out]

2/3/((-1+x)/(1+x))^(1/2)*(1-x)^(1/2)*(3*2^(1/2)*arctan(1/2*(-1-x)^(1/2)*2^(1/2))-(-1-x)^(1/2)*x-4*(-1-x)^(1/2)
)/(-1-x)^(1/2)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-x + 1} \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)*x/(1-x)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.57471, size = 196, normalized size = 1.56 \begin{align*} \frac{2 \,{\left (3 \, \sqrt{2}{\left (x - 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{x - 1}\right ) -{\left (x^{2} + 5 \, x + 4\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}\right )}}{3 \,{\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)*x/(1-x)^(1/2),x, algorithm="fricas")

[Out]

2/3*(3*sqrt(2)*(x - 1)*arctan(sqrt(2)*sqrt(-x + 1)*sqrt((x - 1)/(x + 1))/(x - 1)) - (x^2 + 5*x + 4)*sqrt(-x +
1)*sqrt((x - 1)/(x + 1)))/(x - 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{\frac{x - 1}{x + 1}} \sqrt{1 - x}}\, 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)**(1/2),x)

[Out]

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

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError