\(\int e^{\coth ^{-1}(x)} (1-x) x \, dx\) [281]

Optimal result

Integrand size = 11, antiderivative size = 18 \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6310, 6313, 270} \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=-\frac {1}{3} \left (1-\frac {1}{x^2}\right )^{3/2} x^3 \]

[In]

Int[E^ArcCoth[x]*(1 - x)*x,x]

[Out]

-1/3*((1 - x^(-2))^(3/2)*x^3)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.17 \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=-\frac {1}{3} \sqrt {1-\frac {1}{x^2}} x \left (-1+x^2\right ) \]

[In]

Integrate[E^ArcCoth[x]*(1 - x)*x,x]

[Out]

-1/3*(Sqrt[1 - x^(-2)]*x*(-1 + x^2))

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22

method	result	size
gosper	\(-\frac {\left (x -1\right )^{2} \left (1+x \right )}{3 \sqrt {\frac {x -1}{1+x}}}\)	\(22\)
default	\(-\frac {\left (x -1\right )^{2} \left (1+x \right )}{3 \sqrt {\frac {x -1}{1+x}}}\)	\(22\)
risch	\(-\frac {\left (x^{2}-1\right ) \left (x -1\right )}{3 \sqrt {\frac {x -1}{1+x}}}\)	\(22\)
trager	\(-\frac {\left (1+x \right ) \left (x^{2}-1\right ) \sqrt {-\frac {1-x}{1+x}}}{3}\)	\(25\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=-\frac {1}{3} \, {\left (x^{3} + x^{2} - x - 1\right )} \sqrt {\frac {x - 1}{x + 1}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (14) = 28\).

Time = 0.20 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.78 \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=\frac {8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {3}{2}}}{3 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} \]

[In]

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

[Out]

8/3*((x - 1)/(x + 1))^(3/2)/(3*(x - 1)/(x + 1) - 3*(x - 1)^2/(x + 1)^2 + (x - 1)^3/(x + 1)^3 - 1)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=-\frac {{\left (x^{2} - 1\right )}^{\frac {3}{2}}}{3 \, \mathrm {sgn}\left (x + 1\right )} \]

[In]

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

[Out]

-1/3*(x^2 - 1)^(3/2)/sgn(x + 1)

Mupad [B] (verification not implemented)

Time = 4.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int e^{\coth ^{-1}(x)} (1-x) x \, dx=-\frac {{\left (\frac {x-1}{x+1}\right )}^{3/2}\,{\left (x+1\right )}^3}{3} \]

[In]

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

[Out]

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