3.135 \(\int \frac {\coth (x)}{(1+\coth (x))^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {1}{2 \sqrt {\coth (x)+1}}+\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]

[Out]

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

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Rubi [A]  time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3526, 3479, 3480, 206} \[ -\frac {1}{2 \sqrt {\coth (x)+1}}+\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]/(1 + Coth[x])^(3/2),x]

[Out]

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

Rule 206

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

Rule 3479

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(a + b*Tan[c + d*x])^n)/(2*b*d*n), x] +
Dist[1/(2*a), Int[(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0]

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3526

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^m)/(2*a*f*m), x] + Dist[(b*c + a*d)/(2*a*b), Int[(a + b*Tan[e + f*x])^(m + 1),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0]

Rubi steps

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

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Mathematica [C]  time = 0.35, size = 84, normalized size = 1.71 \[ \left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {\coth (x)+1} \left (\left (\frac {1}{6}-\frac {i}{6}\right ) (-\sinh (2 x)-\sinh (4 x)+\cosh (2 x)+\cosh (4 x)-2)-\frac {i \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )}{\sqrt {i (\coth (x)+1)}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]/(1 + Coth[x])^(3/2),x]

[Out]

(1/4 + I/4)*Sqrt[1 + Coth[x]]*(((-I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]])/Sqrt[I*(1 + Coth[x])] + (1/6 -
I/6)*(-2 + Cosh[2*x] + Cosh[4*x] - Sinh[2*x] - Sinh[4*x]))

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fricas [B]  time = 0.40, size = 166, normalized size = 3.39 \[ -\frac {2 \, \sqrt {2} {\left (2 \, \sqrt {2} \cosh \relax (x)^{2} + 4 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x) + 2 \, \sqrt {2} \sinh \relax (x)^{2} + \sqrt {2}\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 3 \, {\left (\sqrt {2} \cosh \relax (x)^{3} + 3 \, \sqrt {2} \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{2} + \sqrt {2} \sinh \relax (x)^{3}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{24 \, {\left (\cosh \relax (x)^{3} + 3 \, \cosh \relax (x)^{2} \sinh \relax (x) + 3 \, \cosh \relax (x) \sinh \relax (x)^{2} + \sinh \relax (x)^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(2*sqrt(2)*(2*sqrt(2)*cosh(x)^2 + 4*sqrt(2)*cosh(x)*sinh(x) + 2*sqrt(2)*sinh(x)^2 + sqrt(2))*sqrt(sinh(x
)/(cosh(x) - sinh(x))) - 3*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)^2*sinh(x) + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sq
rt(2)*sinh(x)^3)*log(2*sqrt(2)*sqrt(sinh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x)) + 2*cosh(x)^2 + 4*cosh(x)
*sinh(x) + 2*sinh(x)^2 - 1))/(cosh(x)^3 + 3*cosh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)

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giac [B]  time = 0.22, size = 107, normalized size = 2.18 \[ -\frac {1}{24} \, \sqrt {2} {\left (\frac {3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} + \frac {2 \, {\left (3 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 3 \, e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - 4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*sqrt(2)*(3*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))/sgn(e^(2*x) - 1) + 2*(3*sqrt(e^(4*x) - e^
(2*x)) - 3*e^(2*x) + 1)/((sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^3*sgn(e^(2*x) - 1)) - 4*sgn(e^(2*x) - 1))

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maple [A]  time = 0.08, size = 35, normalized size = 0.71 \[ \frac {1}{3 \left (1+\coth \relax (x )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(1+coth(x))^(3/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth \relax (x)}{{\left (\coth \relax (x) + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)/(1+coth(x))^(3/2),x, algorithm="maxima")

[Out]

integrate(coth(x)/(coth(x) + 1)^(3/2), x)

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mupad [B]  time = 1.22, size = 32, normalized size = 0.65 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )}{4}-\frac {\frac {\mathrm {coth}\relax (x)}{2}+\frac {1}{6}}{{\left (\mathrm {coth}\relax (x)+1\right )}^{3/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)/(coth(x) + 1)^(3/2),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2))/4 - (coth(x)/2 + 1/6)/(coth(x) + 1)^(3/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\relax (x )}}{\left (\coth {\relax (x )} + 1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(x)/(coth(x) + 1)**(3/2), x)

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