3.2.35 \(\int \frac {\coth (x)}{(1+\coth (x))^{3/2}} \, dx\) [135]

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

[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 = 0.04, 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 = {3607, 3560, 3561, 212} \begin {gather*} -\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}} \end {gather*}

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 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 3560

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 3561

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 3607

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} \text {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] Result contains complex when optimal does not.
time = 0.23, size = 84, normalized size = 1.71 \begin {gather*} \left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {1+\coth (x)} \left (-\frac {i \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right )}{\sqrt {i (1+\coth (x))}}+\left (\frac {1}{6}-\frac {i}{6}\right ) (-2+\cosh (2 x)+\cosh (4 x)-\sinh (2 x)-\sinh (4 x))\right ) \end {gather*}

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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Maple [A]
time = 0.69, size = 35, normalized size = 0.71

method	result	size
derivativedivides	\(\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\)	\(35\)
default	\(\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (34) = 68\).
time = 0.36, size = 166, normalized size = 3.39 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (2 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth {\left (x \right )}}{\left (\coth {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (34) = 68\).
time = 0.40, size = 89, normalized size = 1.82 \begin {gather*} -\frac {\sqrt {2} {\left (\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}} + 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 )\right )}}{24 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \end {gather*}

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

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

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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