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

Optimal. Leaf size=36 \[ \frac{x}{8}-\frac{1}{8 (\coth (x)+1)}-\frac{1}{8 (\coth (x)+1)^2}-\frac{1}{6 (\coth (x)+1)^3} \]

[Out]

x/8 - 1/(6*(1 + Coth[x])^3) - 1/(8*(1 + Coth[x])^2) - 1/(8*(1 + Coth[x]))

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Rubi [A]  time = 0.0274045, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac{x}{8}-\frac{1}{8 (\coth (x)+1)}-\frac{1}{8 (\coth (x)+1)^2}-\frac{1}{6 (\coth (x)+1)^3} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^(-3),x]

[Out]

x/8 - 1/(6*(1 + Coth[x])^3) - 1/(8*(1 + Coth[x])^2) - 1/(8*(1 + Coth[x]))

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{1}{(1+\coth (x))^3} \, dx &=-\frac{1}{6 (1+\coth (x))^3}+\frac{1}{2} \int \frac{1}{(1+\coth (x))^2} \, dx\\ &=-\frac{1}{6 (1+\coth (x))^3}-\frac{1}{8 (1+\coth (x))^2}+\frac{1}{4} \int \frac{1}{1+\coth (x)} \, dx\\ &=-\frac{1}{6 (1+\coth (x))^3}-\frac{1}{8 (1+\coth (x))^2}-\frac{1}{8 (1+\coth (x))}+\frac{\int 1 \, dx}{8}\\ &=\frac{x}{8}-\frac{1}{6 (1+\coth (x))^3}-\frac{1}{8 (1+\coth (x))^2}-\frac{1}{8 (1+\coth (x))}\\ \end{align*}

Mathematica [A]  time = 0.0870341, size = 44, normalized size = 1.22 \[ \frac{1}{96} (12 x-18 \sinh (2 x)+9 \sinh (4 x)-2 \sinh (6 x)+18 \cosh (2 x)-9 \cosh (4 x)+2 \cosh (6 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^(-3),x]

[Out]

(12*x + 18*Cosh[2*x] - 9*Cosh[4*x] + 2*Cosh[6*x] - 18*Sinh[2*x] + 9*Sinh[4*x] - 2*Sinh[6*x])/96

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Maple [A]  time = 0.017, size = 40, normalized size = 1.1 \begin{align*} -{\frac{1}{6\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{3}}}-{\frac{1}{8\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}}}-{\frac{1}{8+8\,{\rm coth} \left (x\right )}}+{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{16}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/(1+coth(x))^3-1/8/(1+coth(x))^2-1/8/(1+coth(x))+1/16*ln(1+coth(x))-1/16*ln(coth(x)-1)

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Maxima [A]  time = 1.05014, size = 30, normalized size = 0.83 \begin{align*} \frac{1}{8} \, x + \frac{3}{16} \, e^{\left (-2 \, x\right )} - \frac{3}{32} \, e^{\left (-4 \, x\right )} + \frac{1}{48} \, e^{\left (-6 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x + 3/16*e^(-2*x) - 3/32*e^(-4*x) + 1/48*e^(-6*x)

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Fricas [B]  time = 2.26864, size = 278, normalized size = 7.72 \begin{align*} \frac{2 \,{\left (6 \, x + 1\right )} \cosh \left (x\right )^{3} + 6 \,{\left (6 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 2 \,{\left (6 \, x - 1\right )} \sinh \left (x\right )^{3} + 3 \,{\left (2 \,{\left (6 \, x - 1\right )} \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right ) + 9 \, \cosh \left (x\right )}{96 \,{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(2*(6*x + 1)*cosh(x)^3 + 6*(6*x + 1)*cosh(x)*sinh(x)^2 + 2*(6*x - 1)*sinh(x)^3 + 3*(2*(6*x - 1)*cosh(x)^2
+ 9)*sinh(x) + 9*cosh(x))/(cosh(x)^3 + 3*cosh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)

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Sympy [B]  time = 1.47227, size = 182, normalized size = 5.06 \begin{align*} \frac{3 x \tanh ^{3}{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} + \frac{9 x \tanh ^{2}{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} + \frac{9 x \tanh{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} + \frac{3 x}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} - \frac{7 \tanh ^{3}{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} + \frac{6 \tanh{\left (x \right )}}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} + \frac{3}{24 \tanh ^{3}{\left (x \right )} + 72 \tanh ^{2}{\left (x \right )} + 72 \tanh{\left (x \right )} + 24} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x*tanh(x)**3/(24*tanh(x)**3 + 72*tanh(x)**2 + 72*tanh(x) + 24) + 9*x*tanh(x)**2/(24*tanh(x)**3 + 72*tanh(x)*
*2 + 72*tanh(x) + 24) + 9*x*tanh(x)/(24*tanh(x)**3 + 72*tanh(x)**2 + 72*tanh(x) + 24) + 3*x/(24*tanh(x)**3 + 7
2*tanh(x)**2 + 72*tanh(x) + 24) - 7*tanh(x)**3/(24*tanh(x)**3 + 72*tanh(x)**2 + 72*tanh(x) + 24) + 6*tanh(x)/(
24*tanh(x)**3 + 72*tanh(x)**2 + 72*tanh(x) + 24) + 3/(24*tanh(x)**3 + 72*tanh(x)**2 + 72*tanh(x) + 24)

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Giac [A]  time = 1.14846, size = 32, normalized size = 0.89 \begin{align*} \frac{1}{96} \,{\left (18 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 2\right )} e^{\left (-6 \, x\right )} + \frac{1}{8} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(18*e^(4*x) - 9*e^(2*x) + 2)*e^(-6*x) + 1/8*x