3.4.29 \(\int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx\) [329]

Optimal. Leaf size=126 \[ \frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6311, 6316, 98, 96, 95, 212} \begin {gather*} \frac {2 \sqrt {1-\frac {1}{x}} \left (\frac {1}{x}+1\right )^{3/2} x^2}{3 \sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {\frac {1}{x}+1} x}{\sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]

[Out]

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

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6311

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

Rule 6316

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

Rubi steps

\begin {align*} \int \frac {e^{\coth ^{-1}(x)} x}{\sqrt {1-x}} \, dx &=\frac {\left (\sqrt {1-\frac {1}{x}} \sqrt {x}\right ) \int \frac {e^{\coth ^{-1}(x)} \sqrt {x}}{\sqrt {1-\frac {1}{x}}} \, dx}{\sqrt {1-x}}\\ &=-\frac {\sqrt {1-\frac {1}{x}} \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{5/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {\sqrt {1-\frac {1}{x}} \text {Subst}\left (\int \frac {\sqrt {1+x}}{(1-x) x^{3/2}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {\left (2 \sqrt {1-\frac {1}{x}}\right ) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x}} \, dx,x,\frac {1}{x}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {\left (4 \sqrt {1-\frac {1}{x}}\right ) \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ &=\frac {2 \sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{\sqrt {1-x}}+\frac {2 \sqrt {1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{3/2} x^2}{3 \sqrt {1-x}}-\frac {2 \sqrt {2} \sqrt {1-\frac {1}{x}} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {1-x} \sqrt {\frac {1}{x}}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 69, normalized size = 0.55 \begin {gather*} \frac {2 \sqrt {\frac {-1+x}{x}} x \left (\sqrt {1+\frac {1}{x}} (4+x)-3 \sqrt {2} \sqrt {\frac {1}{x}} \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )\right )}{3 \sqrt {1-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^ArcCoth[x]*x)/Sqrt[1 - x],x]

[Out]

(2*Sqrt[(-1 + x)/x]*x*(Sqrt[1 + x^(-1)]*(4 + x) - 3*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[Sqrt[2]*Sqrt[(1 + x)^(-1)]]))
/(3*Sqrt[1 - x])

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Maple [A]
time = 0.09, size = 66, normalized size = 0.52

method	result	size
default	\(\frac {2 \sqrt {1-x}\, \left (3 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )-x \sqrt {-1-x}-4 \sqrt {-1-x}\right )}{3 \sqrt {\frac {-1+x}{1+x}}\, \sqrt {-1-x}}\)	\(66\)
risch	\(\frac {2 \left (x +4\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{-1+x}}\, \left (-1+x \right )}{3 \sqrt {-1-x}\, \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1-x}}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{-1+x}}\, \left (-1+x \right )}{\sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3/((-1+x)/(1+x))^(1/2)*(1-x)^(1/2)*(3*2^(1/2)*arctan(1/2*(-1-x)^(1/2)*2^(1/2))-x*(-1-x)^(1/2)-4*(-1-x)^(1/2)
)/(-1-x)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.33, size = 72, normalized size = 0.57 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {2} {\left (x - 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) - {\left (x^{2} + 5 \, x + 4\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}\right )}}{3 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*sqrt(2)*(x - 1)*arctan(sqrt(2)*sqrt(-x + 1)*sqrt((x - 1)/(x + 1))/(x - 1)) - (x^2 + 5*x + 4)*sqrt(-x +
1)*sqrt((x - 1)/(x + 1)))/(x - 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {\frac {x - 1}{x + 1}} \sqrt {1 - x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: (-12*atan(i)+20*i)*1/3/sqrt(2)*sign(sage
VARx+1)-(-2/3*sqrt(-sageVARx-1)*(-sageVARx-1)+2*sqrt(-sageVARx-1)+1/3*(12*atan(i)-20*i)/sqrt(2)-4*atan(sqrt(-s
ageVARx-1)/sqrt(2))/s

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x}{\sqrt {\frac {x-1}{x+1}}\,\sqrt {1-x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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