3.4.34 \(\int \frac {e^{\coth ^{-1}(x)}}{(1-x)^{3/2}} \, dx\) [334]

Optimal. Leaf size=90 \[ -\frac {\sqrt {1-\frac {1}{x}} \sqrt {1+\frac {1}{x}} x}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{x}}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6311, 6316, 96, 95, 212} \begin {gather*} -\frac {\sqrt {\frac {1}{x}+1} x \sqrt {1-\frac {1}{x}}}{(1-x)^{3/2}}-\frac {\left (1-\frac {1}{x}\right )^{3/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {\frac {1}{x}+1}}\right )}{\sqrt {2} (1-x)^{3/2} \left (\frac {1}{x}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^ArcCoth[x]/(1 - x)^(3/2),x]

[Out]

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

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

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Mathematica [A]
time = 0.06, size = 58, normalized size = 0.64 \begin {gather*} \frac {\frac {2}{\sqrt {\frac {1}{1+x}}}+\sqrt {2} (-1+x) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {1}{1+x}}\right )}{2 \sqrt {-\frac {(-1+x)^2}{x^2}} x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcCoth[x]/(1 - x)^(3/2),x]

[Out]

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

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Maple [A]
time = 0.10, size = 79, normalized size = 0.88

method	result	size
default	\(-\frac {\sqrt {1-x}\, \left (\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) x -\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right )+2 \sqrt {-1-x}\right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (-1+x \right ) \sqrt {-1-x}}\)	\(79\)
risch	\(\frac {\sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{-1+x}}}{\sqrt {-1-x}\, \sqrt {\frac {-1+x}{1+x}}\, \sqrt {1-x}}-\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {-1-x}\, \sqrt {2}}{2}\right ) \sqrt {\frac {\left (1+x \right ) \left (1-x \right )}{-1+x}}\, \left (-1+x \right )}{2 \sqrt {\frac {-1+x}{1+x}}\, \left (1+x \right ) \sqrt {1-x}}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-1+x)/(1+x))^(1/2)/(1-x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/((-1+x)/(1+x))^(1/2)/(-1+x)*(1-x)^(1/2)*(2^(1/2)*arctan(1/2*(-1-x)^(1/2)*2^(1/2))*x-2^(1/2)*arctan(1/2*(-
1-x)^(1/2)*2^(1/2))+2*(-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)/(1-x)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.33, size = 76, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {2} {\left (x^{2} - 2 \, x + 1\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{x - 1}\right ) + 2 \, {\left (x + 1\right )} \sqrt {-x + 1} \sqrt {\frac {x - 1}{x + 1}}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/2)/(1-x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 76.94, size = 70, normalized size = 0.78 \begin {gather*} - 2 \left (\begin {cases} \frac {\sqrt {2} \left (\frac {\operatorname {acos}{\left (\frac {\sqrt {2}}{\sqrt {1 - x}} \right )}}{2} - \frac {\sqrt {2} \sqrt {1 - \frac {2}{1 - x}}}{2 \sqrt {1 - x}}\right )}{2} & \text {for}\: \sqrt {1 - x} > - \sqrt {2} \wedge \sqrt {1 - x} < \sqrt {2} \end {cases}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))**(1/2)/(1-x)**(3/2),x)

[Out]

-2*Piecewise((sqrt(2)*(acos(sqrt(2)/sqrt(1 - x))/2 - sqrt(2)*sqrt(1 - 2/(1 - x))/(2*sqrt(1 - x)))/2, (sqrt(1 -
x) < sqrt(2)) & (sqrt(1 - x) > -sqrt(2))))

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Giac [A]
time = 0.38, size = 32, normalized size = 0.36 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {-x - 1}\right ) + \frac {\sqrt {-x - 1}}{x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-1+x)/(1+x))^(1/2)/(1-x)^(3/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-x - 1)) + sqrt(-x - 1)/(x - 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {\frac {x-1}{x+1}}\,{\left (1-x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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