\(\int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx\) [330]

Optimal result

Integrand size = 14, antiderivative size = 90 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6311, 6316, 96, 95, 212} \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]

[In]

Int[E^ArcCoth[x]/Sqrt[1 - x],x]

[Out]

(2*Sqrt[1 - x^(-1)]*Sqrt[1 + x^(-1)]*x)/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 95

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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 212

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

Rule 6311

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 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1-\frac {1}{x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-\frac {1}{x}} \sqrt {x}} \, dx}{\sqrt {1-x}} \\ & = -\frac {\sqrt {1-\frac {1}{x}} \text {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 {\left (2 \sqrt {1-\frac {1}{x}}\right ) \text {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 {\left (4 \sqrt {1-\frac {1}{x}}\right ) \text {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 {2} \sqrt {1-\frac {1}{x}} \text {arctanh}\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] (verified)

Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \sqrt {\frac {-1+x}{x}} x \left (\sqrt {1+\frac {1}{x}}-\sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{\sqrt {1-x}} \]

[In]

Integrate[E^ArcCoth[x]/Sqrt[1 - x],x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {2 \sqrt {1-x}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-\sqrt {-1-x}\right )}{\sqrt {\frac {x -1}{1+x}}\, \sqrt {-1-x}}\)	\(55\)
risch	\(\frac {2 \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {-1-x}\, \sqrt {\frac {x -1}{1+x}}\, \sqrt {1-x}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{\sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\)	\(108\)

[In]

int(1/((x-1)/(1+x))^(1/2)/(1-x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\frac {2 \, {\left (\sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{x - 1} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 13.99 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=- 2 \left (\begin {cases} \sqrt {2} \left (\frac {\sqrt {2} \sqrt {- x - 1}}{2} - \operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}\right ) & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \]

[In]

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

[Out]

-2*Piecewise((sqrt(2)*(sqrt(2)*sqrt(-x - 1)/2 - acos(sqrt(2)/sqrt(1 - x))), (sqrt(1 - x) < sqrt(2)) & (sqrt(1
- x) > -sqrt(2))))

Maxima [F]

\[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\int { \frac {1}{\sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\text {Exception raised: NotImplementedError} \]

[In]

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

[Out]

Exception raised: NotImplementedError >> unable to parse Giac output: (-4*atan(i)+4*i)/sqrt(2)*sign(sageVARx+1
)-(2*sqrt(-sageVARx-1)+(4*atan(i)-4*i)/sqrt(2)-4*atan(sqrt(-sageVARx-1)/sqrt(2))/sqrt(2))*sign(sageVARx)/sign(
sageVARx+1)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(x)}}{\sqrt {1-x}} \, dx=\int \frac {1}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \]

[In]

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

[Out]

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