3.68 \(\int \frac{1}{(1+\coth (x))^4} \, dx\)

Optimal. Leaf size=46 \[ \frac{x}{16}-\frac{1}{16 (\coth (x)+1)}-\frac{1}{16 (\coth (x)+1)^2}-\frac{1}{12 (\coth (x)+1)^3}-\frac{1}{8 (\coth (x)+1)^4} \]

[Out]

x/16 - 1/(8*(1 + Coth[x])^4) - 1/(12*(1 + Coth[x])^3) - 1/(16*(1 + Coth[x])^2) - 1/(16*(1 + Coth[x]))

________________________________________________________________________________________

Rubi [A]  time = 0.0359077, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac{x}{16}-\frac{1}{16 (\coth (x)+1)}-\frac{1}{16 (\coth (x)+1)^2}-\frac{1}{12 (\coth (x)+1)^3}-\frac{1}{8 (\coth (x)+1)^4} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^(-4),x]

[Out]

x/16 - 1/(8*(1 + Coth[x])^4) - 1/(12*(1 + Coth[x])^3) - 1/(16*(1 + Coth[x])^2) - 1/(16*(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))^4} \, dx &=-\frac{1}{8 (1+\coth (x))^4}+\frac{1}{2} \int \frac{1}{(1+\coth (x))^3} \, dx\\ &=-\frac{1}{8 (1+\coth (x))^4}-\frac{1}{12 (1+\coth (x))^3}+\frac{1}{4} \int \frac{1}{(1+\coth (x))^2} \, dx\\ &=-\frac{1}{8 (1+\coth (x))^4}-\frac{1}{12 (1+\coth (x))^3}-\frac{1}{16 (1+\coth (x))^2}+\frac{1}{8} \int \frac{1}{1+\coth (x)} \, dx\\ &=-\frac{1}{8 (1+\coth (x))^4}-\frac{1}{12 (1+\coth (x))^3}-\frac{1}{16 (1+\coth (x))^2}-\frac{1}{16 (1+\coth (x))}+\frac{\int 1 \, dx}{16}\\ &=\frac{x}{16}-\frac{1}{8 (1+\coth (x))^4}-\frac{1}{12 (1+\coth (x))^3}-\frac{1}{16 (1+\coth (x))^2}-\frac{1}{16 (1+\coth (x))}\\ \end{align*}

Mathematica [A]  time = 0.121232, size = 53, normalized size = 1.15 \[ \frac{1}{384} (\cosh (4 x)-\sinh (4 x)) (32 \sinh (2 x)+24 x \sinh (4 x)+3 \sinh (4 x)+64 \cosh (2 x)+3 (8 x-1) \cosh (4 x)-36) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^(-4),x]

[Out]

((Cosh[4*x] - Sinh[4*x])*(-36 + 64*Cosh[2*x] + 3*(-1 + 8*x)*Cosh[4*x] + 32*Sinh[2*x] + 3*Sinh[4*x] + 24*x*Sinh
[4*x]))/384

________________________________________________________________________________________

Maple [A]  time = 0.017, size = 48, normalized size = 1. \begin{align*} -{\frac{1}{8\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{4}}}-{\frac{1}{12\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{3}}}-{\frac{1}{16\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}}}-{\frac{1}{16+16\,{\rm coth} \left (x\right )}}+{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{32}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+coth(x))^4,x)

[Out]

-1/8/(1+coth(x))^4-1/12/(1+coth(x))^3-1/16/(1+coth(x))^2-1/16/(1+coth(x))+1/32*ln(1+coth(x))-1/32*ln(coth(x)-1
)

________________________________________________________________________________________

Maxima [A]  time = 1.17884, size = 38, normalized size = 0.83 \begin{align*} \frac{1}{16} \, x + \frac{1}{8} \, e^{\left (-2 \, x\right )} - \frac{3}{32} \, e^{\left (-4 \, x\right )} + \frac{1}{24} \, e^{\left (-6 \, x\right )} - \frac{1}{128} \, e^{\left (-8 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^4,x, algorithm="maxima")

[Out]

1/16*x + 1/8*e^(-2*x) - 3/32*e^(-4*x) + 1/24*e^(-6*x) - 1/128*e^(-8*x)

________________________________________________________________________________________

Fricas [B]  time = 2.37733, size = 394, normalized size = 8.57 \begin{align*} \frac{3 \,{\left (8 \, x - 1\right )} \cosh \left (x\right )^{4} + 12 \,{\left (8 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \,{\left (8 \, x - 1\right )} \sinh \left (x\right )^{4} + 2 \,{\left (9 \,{\left (8 \, x - 1\right )} \cosh \left (x\right )^{2} + 32\right )} \sinh \left (x\right )^{2} + 64 \, \cosh \left (x\right )^{2} + 4 \,{\left (3 \,{\left (8 \, x + 1\right )} \cosh \left (x\right )^{3} + 16 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 36}{384 \,{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^4,x, algorithm="fricas")

[Out]

1/384*(3*(8*x - 1)*cosh(x)^4 + 12*(8*x + 1)*cosh(x)*sinh(x)^3 + 3*(8*x - 1)*sinh(x)^4 + 2*(9*(8*x - 1)*cosh(x)
^2 + 32)*sinh(x)^2 + 64*cosh(x)^2 + 4*(3*(8*x + 1)*cosh(x)^3 + 16*cosh(x))*sinh(x) - 36)/(cosh(x)^4 + 4*cosh(x
)^3*sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4)

________________________________________________________________________________________

Sympy [B]  time = 2.55462, size = 299, normalized size = 6.5 \begin{align*} \frac{3 x \tanh ^{4}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{12 x \tanh ^{3}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{18 x \tanh ^{2}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{12 x \tanh{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{3 x}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{45 \tanh ^{3}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{84 \tanh ^{2}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{61 \tanh{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} + \frac{16}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh{\left (x \right )} + 48} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))**4,x)

[Out]

3*x*tanh(x)**4/(48*tanh(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48) + 12*x*tanh(x)**3/(48*tanh
(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48) + 18*x*tanh(x)**2/(48*tanh(x)**4 + 192*tanh(x)**3
+ 288*tanh(x)**2 + 192*tanh(x) + 48) + 12*x*tanh(x)/(48*tanh(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*ta
nh(x) + 48) + 3*x/(48*tanh(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48) + 45*tanh(x)**3/(48*tan
h(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48) + 84*tanh(x)**2/(48*tanh(x)**4 + 192*tanh(x)**3
+ 288*tanh(x)**2 + 192*tanh(x) + 48) + 61*tanh(x)/(48*tanh(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(
x) + 48) + 16/(48*tanh(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48)

________________________________________________________________________________________

Giac [A]  time = 1.15168, size = 41, normalized size = 0.89 \begin{align*} \frac{1}{384} \,{\left (48 \, e^{\left (6 \, x\right )} - 36 \, e^{\left (4 \, x\right )} + 16 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-8 \, x\right )} + \frac{1}{16} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^4,x, algorithm="giac")

[Out]

1/384*(48*e^(6*x) - 36*e^(4*x) + 16*e^(2*x) - 3)*e^(-8*x) + 1/16*x