\(\int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx\) [294]

Optimal result

Integrand size = 12, antiderivative size = 24 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{3 \left (1-\frac {1}{x}\right )^3} \]

[Out]

-1/3*(1-1/x^2)^(3/2)/(1-1/x)^3

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6310, 6313, 665} \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{3 \left (1-\frac {1}{x}\right )^3} \]

[In]

Int[E^ArcCoth[x]/(1 - x)^2,x]

[Out]

-1/3*(1 - x^(-2))^(3/2)/(1 - x^(-1))^3

Rule 665

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

Rule 6310

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

Rule 6313

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^n, Subst[Int[(c +
d*x)^(p - n)*((1 - x^2/a^2)^(n/2)/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a*d, 0] &&
IntegerQ[(n - 1)/2] && IntegerQ[m] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p, n/2 + 1] || LtQ[-5, m, -1]) && In
tegerQ[2*p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{\coth ^{-1}(x)}}{\left (1-\frac {1}{x}\right )^2 x^2} \, dx \\ & = -\text {Subst}\left (\int \frac {\sqrt {1-x^2}}{(1-x)^3} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\left (1-\frac {1}{x^2}\right )^{3/2}}{3 \left (1-\frac {1}{x}\right )^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {\sqrt {1-\frac {1}{x^2}} x (1+x)}{3 (-1+x)^2} \]

[In]

Integrate[E^ArcCoth[x]/(1 - x)^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

method	result	size
gosper	\(-\frac {1+x}{3 \left (x -1\right ) \sqrt {\frac {x -1}{1+x}}}\)	\(22\)
trager	\(-\frac {\left (1+x \right )^{2} \sqrt {-\frac {1-x}{1+x}}}{3 \left (x -1\right )^{2}}\)	\(27\)
risch	\(-\frac {x^{2}+2 x +1}{3 \sqrt {\frac {x -1}{1+x}}\, \left (1+x \right ) \left (x -1\right )}\)	\(32\)
default	\(-\frac {\left (x^{2}-1\right )^{\frac {3}{2}}}{3 \sqrt {\frac {x -1}{1+x}}\, \left (x -1\right )^{2} \sqrt {\left (x -1\right ) \left (1+x \right )}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {{\left (x^{2} + 2 \, x + 1\right )} \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (x^{2} - 2 \, x + 1\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(1/(sqrt((x - 1)/(x + 1))*(x - 1)**2), x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {1}{3 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

-1/3/((x - 1)/(x + 1))^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=\frac {2 \, {\left (3 \, {\left (x - \sqrt {x^{2} - 1}\right )}^{2} + 1\right )}}{3 \, {\left (x - \sqrt {x^{2} - 1} - 1\right )}^{3} \mathrm {sgn}\left (x + 1\right )} + \frac {1}{3} \, \mathrm {sgn}\left (x + 1\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {e^{\coth ^{-1}(x)}}{(1-x)^2} \, dx=-\frac {1}{3\,{\left (\frac {x-1}{x+1}\right )}^{3/2}} \]

[In]

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

[Out]

-1/(3*((x - 1)/(x + 1))^(3/2))