\(\int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx\) [356]

Optimal result

Integrand size = 15, antiderivative size = 126 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=-\frac {\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x^2}{(1-x)^{3/2}}+\frac {3 \left (1-\frac {1}{x}\right )^{3/2} \sqrt {1+\frac {1}{x}} x^2}{(1-x)^{3/2}}-\frac {5 \left (1-\frac {1}{x}\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=-\frac {\sqrt {\frac {-1+x}{x}} x \left (2 \sqrt {1+\frac {1}{x}} (3-2 x)+5 \sqrt {2} (-1+x) \sqrt {\frac {1}{x}} \text {arctanh}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{2 (1-x)^{3/2}} \] Input:

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

Output:

-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)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6730, 107, 105, 104, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6730

\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x}\right )^2 \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\)

\(\Big \downarrow \) 107

\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\frac {5}{4} \int \frac {\sqrt {1+\frac {1}{x}}}{\left (1-\frac {1}{x}\right ) \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}+\frac {\left (\frac {1}{x}+1\right )^{3/2}}{2 \left (1-\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\)

\(\Big \downarrow \) 105

\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\frac {5}{4} \left (2 \int \frac {1}{\left (1-\frac {1}{x}\right ) \sqrt {1+\frac {1}{x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )+\frac {\left (\frac {1}{x}+1\right )^{3/2}}{2 \left (1-\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\frac {5}{4} \left (4 \int \frac {1}{1-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )+\frac {\left (\frac {1}{x}+1\right )^{3/2}}{2 \left (1-\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\left (1-\frac {1}{x}\right )^{3/2} \left (\frac {5}{4} \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )-\frac {2 \sqrt {\frac {1}{x}+1}}{\sqrt {\frac {1}{x}}}\right )+\frac {\left (\frac {1}{x}+1\right )^{3/2}}{2 \left (1-\frac {1}{x}\right ) \sqrt {\frac {1}{x}}}\right )}{(1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}}\)

Input:

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

Output:

-(((1 - x^(-1))^(3/2)*((1 + x^(-1))^(3/2)/(2*(1 - x^(-1))*Sqrt[x^(-1)]) + 
(5*((-2*Sqrt[1 + x^(-1)])/Sqrt[x^(-1)] + 2*Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[x 
^(-1)])/Sqrt[1 + x^(-1)]]))/4))/((1 - x)^(3/2)*(x^(-1))^(3/2)))
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[(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] || SumSimplerQ[m, 1])
 

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 6730

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(p 
_), x_Symbol] :> Simp[(-(e*x)^m)*(1/x)^(m + p)*((c + d*x)^p/(1 + c/(d*x))^p 
)   Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(m + p + 2)))/(1 - x/a)^( 
n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, e, m, n, p}, x] && EqQ[a^2*c^2 - d 
^2, 0] &&  !IntegerQ[p]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\left (x -1\right ) \left (-5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right ) x +4 \sqrt {-x -1}\, x +5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right )-6 \sqrt {-x -1}\right )}{2 \sqrt {\frac {x -1}{1+x}}\, \left (1-x \right )^{\frac {3}{2}} \sqrt {-x -1}}\)	\(88\)
risch	\(-\frac {\left (2 x^{2}-x -3\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}}{\sqrt {-x -1}\, \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}-\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {-x -1}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{x -1}}\, \left (x -1\right )}{2 \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\)	\(120\)

Input:

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

Output:

1/2*(x-1)*(-5*2^(1/2)*arctan(1/2*(-x-1)^(1/2)*2^(1/2))*x+4*(-x-1)^(1/2)*x+ 
5*2^(1/2)*arctan(1/2*(-x-1)^(1/2)*2^(1/2))-6*(-x-1)^(1/2))/((x-1)/(1+x))^( 
1/2)/(1-x)^(3/2)/(-x-1)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.70 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=-\frac {5 \, \sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {2} {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x - 1\right )}}\right ) - 2 \, {\left (2 \, x^{2} - x - 3\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \] Input:

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

Output:

-1/2*(5*sqrt(2)*(x^2 - 2*x + 1)*arctan(1/2*sqrt(2)*(x + 1)*sqrt(-x + 1)*sq 
rt((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 = 87.89 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=2 \left (\begin {cases} \sqrt {2} \left (\frac {\sqrt {2} \sqrt {- x - 1}}{2} - \operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}\right ) & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) - 2 \left (\begin {cases} \frac {\sqrt {2} \left (\frac {\operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}}{2} - \frac {\sqrt {2} \sqrt {1 - \frac {2}{1 - x}}}{2 \sqrt {1 - x}}\right )}{2} & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \] Input:

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

Output:

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]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.40 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=\frac {5 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right ) - 4 \, \sqrt {-x - 1} + \frac {2 \, \sqrt {-x - 1}}{x - 1}}{2 \, \mathrm {sgn}\left (x + 1\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.58 \[ \int \frac {e^{\coth ^{-1}(x)} x}{(1-x)^{3/2}} \, dx=\frac {i \left (-16 \sqrt {x +1}\, x +24 \sqrt {x +1}-20 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )\right ) x +20 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )\right )+17 \sqrt {2}\, x -17 \sqrt {2}\right )}{8 x -8} \] Input:

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

Output:

(i*( - 16*sqrt(x + 1)*x + 24*sqrt(x + 1) - 20*sqrt(2)*log(tan(asin(sqrt( - 
 x + 1)/sqrt(2))/2))*x + 20*sqrt(2)*log(tan(asin(sqrt( - x + 1)/sqrt(2))/2 
)) + 17*sqrt(2)*x - 17*sqrt(2)))/(8*(x - 1))