### 3.281 $$\int e^{\coth ^{-1}(x)} (1-x) x \, dx$$

Optimal. Leaf size=18 $-\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3$

[Out]

-((1 - x^(-2))^(3/2)*x^3)/3

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Rubi [A]  time = 0.0525726, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.273, Rules used = {6175, 6178, 264} $-\frac{1}{3} \left (1-\frac{1}{x^2}\right )^{3/2} x^3$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[x]*(1 - x)*x,x]

[Out]

-((1 - x^(-2))^(3/2)*x^3)/3

Rule 6175

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 6178

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]

Rule 264

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]

Rubi steps

\begin{align*} \int e^{\coth ^{-1}(x)} (1-x) x \, dx &=-\int e^{\coth ^{-1}(x)} \left (1-\frac{1}{x}\right ) x^2 \, dx\\ &=\operatorname{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]  time = 0.0199014, size = 21, normalized size = 1.17 $-\frac{1}{3} \sqrt{1-\frac{1}{x^2}} x \left (x^2-1\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^ArcCoth[x]*(1 - x)*x,x]

[Out]

-(Sqrt[1 - x^(-2)]*x*(-1 + x^2))/3

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Maple [A]  time = 0.06, size = 22, normalized size = 1.2 \begin{align*} -{\frac{ \left ( 1+x \right ) \left ( -1+x \right ) ^{2}}{3}{\frac{1}{\sqrt{{\frac{-1+x}{1+x}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01459, size = 68, normalized size = 3.78 \begin{align*} \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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.83827, size = 65, normalized size = 3.61 \begin{align*} -\frac{1}{3} \,{\left (x^{3} + x^{2} - x - 1\right )} \sqrt{\frac{x - 1}{x + 1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{x}{\sqrt{\frac{x}{x + 1} - \frac{1}{x + 1}}}\, dx - \int \frac{x^{2}}{\sqrt{\frac{x}{x + 1} - \frac{1}{x + 1}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [B]  time = 1.16763, size = 39, normalized size = 2.17 \begin{align*} \frac{8}{3 \,{\left (\sqrt{\frac{x - 1}{x + 1}} - \frac{1}{\sqrt{\frac{x - 1}{x + 1}}}\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3/(sqrt((x - 1)/(x + 1)) - 1/sqrt((x - 1)/(x + 1)))^3