\(\int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx\) [333]

Optimal result

Integrand size = 15, antiderivative size = 130 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \]

[Out]

-5/2*(1-1/x)^(3/2)*arctanh(2^(1/2)*(1/x)^(1/2)/(1+1/x)^(1/2))/(1-x)^(3/2)/(1/x)^(3/2)*2^(1/2)-1/2*(1+1/x)^(3/2
)*x^2*(1-1/x)^(1/2)/(1-x)^(3/2)+5/2*(1-1/x)^(3/2)*x^2*(1+1/x)^(1/2)/(1-x)^(3/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6311, 6316, 98, 96, 95, 212} \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2}{2 (1-x)^{3/2}}+\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {\frac {1}{x}+1} x^2}{2 (1-x)^{3/2}} \]

[In]

Int[(E^ArcCoth[x]*x)/(1 - x)^(3/2),x]

[Out]

(5*(1 - x^(-1))^(3/2)*Sqrt[1 + x^(-1)]*x^2)/(2*(1 - x)^(3/2)) - (Sqrt[1 - x^(-1)]*(1 + x^(-1))^(3/2)*x^2)/(2*(
1 - x)^(3/2)) - (5*(1 - x^(-1))^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/Sqrt[1 + x^(-1)]])/(Sqrt[2]*(1 - x)^(3/2)
*(x^(-1))^(3/2))

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 98

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 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 (\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} \sqrt {x}} \, dx}{(1-x)^{3/2}} \\ & = -\frac {\left (1-\frac {1}{x}\right )^{3/2} \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x)^2 x^{3/2}} \, dx,x,\frac {1}{x}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \\ & = -\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {\left (5 \left (1-\frac {1}{x}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{3/2}} \, dx,x,\frac {1}{x}\right )}{4 (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \\ & = \frac {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {\left (5 \left (1-\frac {1}{x}\right )^{3/2}\right ) \text {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 {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {\left (5 \left (1-\frac {1}{x}\right )^{3/2}\right ) \text {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 {5 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{2 (1-x)^{3/2}}-\frac {\sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{2 (1-x)^{3/2}}-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \text {arctanh}\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] (verified)

Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=-\frac {\sqrt {\frac {-1+x}{x}} x \left (2 \sqrt {1+\frac {1}{x}} (3-2 x)+5 \sqrt {2} (-1+x) \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{2 (1-x)^{3/2}} \]

[In]

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

[Out]

-1/2*(Sqrt[(-1 + x)/x]*x*(2*Sqrt[1 + x^(-1)]*(3 - 2*x) + 5*Sqrt[2]*(-1 + x)*Sqrt[x^(-1)]*ArcTanh[Sqrt[2]*Sqrt[
(1 + x)^(-1)]]))/(1 - x)^(3/2)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {\sqrt {1-x}\, \left (5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) x -5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-4 \sqrt {-1-x}\, x +6 \sqrt {-1-x}\right )}{2 \sqrt {\frac {x -1}{1+x}}\, \left (x -1\right ) \sqrt {-1-x}}\)	\(90\)
risch	\(-\frac {\left (2 x^{2}-x -3\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}}{\sqrt {-1-x}\, \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{2 \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\)	\(120\)

[In]

int(1/((x-1)/(1+x))^(1/2)*x/(1-x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/((x-1)/(1+x))^(1/2)/(x-1)*(1-x)^(1/2)*(5*2^(1/2)*arctan(1/2*(-1-x)^(1/2)*2^(1/2))*x-5*2^(1/2)*arctan(1/2*
(-1-x)^(1/2)*2^(1/2))-4*(-1-x)^(1/2)*x+6*(-1-x)^(1/2))/(-1-x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=-\frac {5 \, \sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - 2 \, {\left (2 \, x^{2} - x - 3\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 101.00 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=2 \left (\begin {cases} \sqrt {2} \left (\frac {\sqrt {2} \sqrt {- x - 1}}{2} - \operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}\right ) & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) - 2 \left (\begin {cases} \frac {\sqrt {2} \left (\frac {\operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}}{2} - \frac {\sqrt {2} \sqrt {1 - \frac {2}{1 - x}}}{2 \sqrt {1 - x}}\right )}{2} & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \]

[In]

integrate(1/((-1+x)/(1+x))**(1/2)*x/(1-x)**(3/2),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)))) - 2*Piecewise((sqrt(2)*(acos(sqrt(2)/sqrt(1 - x))/2 - sqrt(2)*sqrt(1 - 2/(1 - x))/(2*sqrt(1
- x)))/2, (sqrt(1 - x) < sqrt(2)) & (sqrt(1 - x) > -sqrt(2))))

Maxima [F]

\[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=\int { \frac {x}{{\left (-x + 1\right )}^{\frac {3}{2}} \sqrt {\frac {x - 1}{x + 1}}} \,d x } \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.32 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=\frac {5}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right ) - 2 \, \sqrt {-x - 1} + \frac {\sqrt {-x - 1}}{x - 1} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=\int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,{\left (1-x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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