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]

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

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.70 \[ \frac {2 \left (\frac {1}{x}+1\right )^{3/2} \sqrt {1-x} x}{3 \sqrt {1-\frac {1}{x}}} \]

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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fricas [A]  time = 0.60, size = 33, normalized size = 1.65 \[ \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{3 \, {\left (x - 1\right )}} \]

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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giac [C]  time = 0.15, size = 27, normalized size = 1.35 \[ \frac {1}{3} \, {\left (-4 i \, \sqrt {2} + \frac {2 \, {\left (-x - 1\right )}^{\frac {3}{2}}}{\mathrm {sgn}\left (-x - 1\right )}\right )} \mathrm {sgn}\relax (x) \]

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]

1/3*(-4*I*sqrt(2) + 2*(-x - 1)^(3/2)/sgn(-x - 1))*sgn(x)

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maple [A]  time = 0.03, size = 24, normalized size = 1.20 \[ \frac {2 \left (1+x \right ) \sqrt {1-x}}{3 \sqrt {\frac {-1+x}{1+x}}} \]

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 = 0.50, size = 12, normalized size = 0.60 \[ \frac {1}{3} \, \sqrt {x + 1} {\left (2 i \, x + 2 i\right )} \]

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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mupad [B]  time = 1.27, size = 25, normalized size = 1.25 \[ -\frac {2\,\sqrt {\frac {x-1}{x+1}}\,{\left (x+1\right )}^2}{3\,\sqrt {1-x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 6.80, size = 29, normalized size = 1.45 \[ - \frac {2 i x}{3 \sqrt {\frac {1}{x + 1}}} - \frac {2 i}{3 \sqrt {\frac {1}{x + 1}}} \]

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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