### 3.326 $$\int e^{\coth ^{-1}(x)} \sqrt{1-x} \, dx$$

Optimal. Leaf size=20 $\frac{2}{3} \sqrt{1-x} (x+1) e^{\coth ^{-1}(x)}$

[Out]

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

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Rubi [A]  time = 0.0196056, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {6174} $\frac{2}{3} \sqrt{1-x} (x+1) e^{\coth ^{-1}(x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^ArcCoth[x]*Sqrt[1 - x],x]

[Out]

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

Rule 6174

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

Rubi steps

\begin{align*} \int e^{\coth ^{-1}(x)} \sqrt{1-x} \, dx &=\frac{2}{3} e^{\coth ^{-1}(x)} \sqrt{1-x} (1+x)\\ \end{align*}

Mathematica [A]  time = 0.0124854, size = 34, normalized size = 1.7 $\frac{2 \left (\frac{1}{x}+1\right )^{3/2} \sqrt{1-x} x}{3 \sqrt{1-\frac{1}{x}}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[x]*Sqrt[1 - x],x]

[Out]

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

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Maple [A]  time = 0.056, size = 24, normalized size = 1.2 \begin{align*}{\frac{2+2\,x}{3}\sqrt{1-x}{\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)^(1/2),x)

[Out]

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

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Maxima [C]  time = 1.09669, size = 16, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \sqrt{x + 1}{\left (2 i \, x + 2 i\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(x + 1)*(2*I*x + 2*I)

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Fricas [A]  time = 1.59786, size = 86, normalized size = 4.3 \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \sqrt{-x + 1} \sqrt{\frac{x - 1}{x + 1}}}{3 \,{\left (x - 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)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 48.8617, size = 29, normalized size = 1.45 \begin{align*} - \frac{2 i x}{3 \sqrt{\frac{1}{x + 1}}} - \frac{2 i}{3 \sqrt{\frac{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)**(1/2),x)

[Out]

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

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Giac [C]  time = 1.14028, size = 36, normalized size = 1.8 \begin{align*} -\frac{2}{3} \,{\left (2 i \, \sqrt{2} - \frac{{\left (-x - 1\right )}^{\frac{3}{2}}}{\mathrm{sgn}\left (-x - 1\right )}\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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