3.130 \(\int \frac{\coth ^3(x)}{1+\coth (x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{3 x}{2}+\frac{\coth ^2(x)}{2 (\coth (x)+1)}-\frac{3 \coth (x)}{2}-\log (\sinh (x)) \]

[Out]

(3*x)/2 - (3*Coth[x])/2 + Coth[x]^2/(2*(1 + Coth[x])) - Log[Sinh[x]]

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Rubi [A]  time = 0.0553428, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {3550, 3525, 3475} \[ \frac{3 x}{2}+\frac{\coth ^2(x)}{2 (\coth (x)+1)}-\frac{3 \coth (x)}{2}-\log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(1 + Coth[x]),x]

[Out]

(3*x)/2 - (3*Coth[x])/2 + Coth[x]^2/(2*(1 + Coth[x])) - Log[Sinh[x]]

Rule 3550

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

Rule 3525

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

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\coth ^3(x)}{1+\coth (x)} \, dx &=\frac{\coth ^2(x)}{2 (1+\coth (x))}-\frac{1}{2} \int (2-3 \coth (x)) \coth (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \coth (x)}{2}+\frac{\coth ^2(x)}{2 (1+\coth (x))}-\int \coth (x) \, dx\\ &=\frac{3 x}{2}-\frac{3 \coth (x)}{2}+\frac{\coth ^2(x)}{2 (1+\coth (x))}-\log (\sinh (x))\\ \end{align*}

Mathematica [A]  time = 0.0413891, size = 27, normalized size = 0.87 \[ \frac{1}{4} (6 x+\sinh (2 x)-\cosh (2 x)-4 \coth (x)-4 \log (\sinh (x))) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(1 + Coth[x]),x]

[Out]

(6*x - Cosh[2*x] - 4*Coth[x] - 4*Log[Sinh[x]] + Sinh[2*x])/4

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Maple [A]  time = 0.017, size = 28, normalized size = 0.9 \begin{align*} -{\rm coth} \left (x\right )+{\frac{1}{2+2\,{\rm coth} \left (x\right )}}+{\frac{5\,\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{4}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-coth(x)+1/2/(1+coth(x))+5/4*ln(1+coth(x))-1/4*ln(coth(x)-1)

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Maxima [A]  time = 1.09329, size = 51, normalized size = 1.65 \begin{align*} \frac{1}{2} \, x + \frac{2}{e^{\left (-2 \, x\right )} - 1} - \frac{1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 2/(e^(-2*x) - 1) - 1/4*e^(-2*x) - log(e^(-x) + 1) - log(e^(-x) - 1)

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Fricas [B]  time = 3.01997, size = 632, normalized size = 20.39 \begin{align*} \frac{10 \, x \cosh \left (x\right )^{4} + 40 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 10 \, x \sinh \left (x\right )^{4} -{\left (10 \, x + 9\right )} \cosh \left (x\right )^{2} +{\left (60 \, x \cosh \left (x\right )^{2} - 10 \, x - 9\right )} \sinh \left (x\right )^{2} - 4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \,{\left (20 \, x \cosh \left (x\right )^{3} -{\left (10 \, x + 9\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} +{\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \,{\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(10*x*cosh(x)^4 + 40*x*cosh(x)*sinh(x)^3 + 10*x*sinh(x)^4 - (10*x + 9)*cosh(x)^2 + (60*x*cosh(x)^2 - 10*x
- 9)*sinh(x)^2 - 4*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*
(2*cosh(x)^3 - cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))) + 2*(20*x*cosh(x)^3 - (10*x + 9)*cosh(x))*
sinh(x) + 1)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cos
h(x)^3 - cosh(x))*sinh(x))

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Sympy [B]  time = 1.3806, size = 162, normalized size = 5.23 \begin{align*} \frac{x \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{x \tanh{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{2 \log{\left (\tanh{\left (x \right )} + 1 \right )} \tanh{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} - \frac{2 \log{\left (\tanh{\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} - \frac{2 \log{\left (\tanh{\left (x \right )} \right )} \tanh{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} + \frac{3 \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} - \frac{2}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*tanh(x)**2/(2*tanh(x)**2 + 2*tanh(x)) + x*tanh(x)/(2*tanh(x)**2 + 2*tanh(x)) + 2*log(tanh(x) + 1)*tanh(x)**2
/(2*tanh(x)**2 + 2*tanh(x)) + 2*log(tanh(x) + 1)*tanh(x)/(2*tanh(x)**2 + 2*tanh(x)) - 2*log(tanh(x))*tanh(x)**
2/(2*tanh(x)**2 + 2*tanh(x)) - 2*log(tanh(x))*tanh(x)/(2*tanh(x)**2 + 2*tanh(x)) + 3*tanh(x)**2/(2*tanh(x)**2
+ 2*tanh(x)) - 2/(2*tanh(x)**2 + 2*tanh(x))

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Giac [A]  time = 1.13294, size = 49, normalized size = 1.58 \begin{align*} \frac{5}{2} \, x - \frac{{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{4 \,{\left (e^{\left (2 \, x\right )} - 1\right )}} - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*x - 1/4*(9*e^(2*x) - 1)*e^(-2*x)/(e^(2*x) - 1) - log(abs(e^(2*x) - 1))