3.69 \(\int \frac{1}{(1+\coth (x))^5} \, dx\)

Optimal. Leaf size=56 \[ \frac{x}{32}-\frac{1}{32 (\coth (x)+1)}-\frac{1}{32 (\coth (x)+1)^2}-\frac{1}{24 (\coth (x)+1)^3}-\frac{1}{16 (\coth (x)+1)^4}-\frac{1}{10 (\coth (x)+1)^5} \]

[Out]

x/32 - 1/(10*(1 + Coth[x])^5) - 1/(16*(1 + Coth[x])^4) - 1/(24*(1 + Coth[x])^3) - 1/(32*(1 + Coth[x])^2) - 1/(
32*(1 + Coth[x]))

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Rubi [A]  time = 0.0462366, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3479, 8} \[ \frac{x}{32}-\frac{1}{32 (\coth (x)+1)}-\frac{1}{32 (\coth (x)+1)^2}-\frac{1}{24 (\coth (x)+1)^3}-\frac{1}{16 (\coth (x)+1)^4}-\frac{1}{10 (\coth (x)+1)^5} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^(-5),x]

[Out]

x/32 - 1/(10*(1 + Coth[x])^5) - 1/(16*(1 + Coth[x])^4) - 1/(24*(1 + Coth[x])^3) - 1/(32*(1 + Coth[x])^2) - 1/(
32*(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))^5} \, dx &=-\frac{1}{10 (1+\coth (x))^5}+\frac{1}{2} \int \frac{1}{(1+\coth (x))^4} \, dx\\ &=-\frac{1}{10 (1+\coth (x))^5}-\frac{1}{16 (1+\coth (x))^4}+\frac{1}{4} \int \frac{1}{(1+\coth (x))^3} \, dx\\ &=-\frac{1}{10 (1+\coth (x))^5}-\frac{1}{16 (1+\coth (x))^4}-\frac{1}{24 (1+\coth (x))^3}+\frac{1}{8} \int \frac{1}{(1+\coth (x))^2} \, dx\\ &=-\frac{1}{10 (1+\coth (x))^5}-\frac{1}{16 (1+\coth (x))^4}-\frac{1}{24 (1+\coth (x))^3}-\frac{1}{32 (1+\coth (x))^2}+\frac{1}{16} \int \frac{1}{1+\coth (x)} \, dx\\ &=-\frac{1}{10 (1+\coth (x))^5}-\frac{1}{16 (1+\coth (x))^4}-\frac{1}{24 (1+\coth (x))^3}-\frac{1}{32 (1+\coth (x))^2}-\frac{1}{32 (1+\coth (x))}+\frac{\int 1 \, dx}{32}\\ &=\frac{x}{32}-\frac{1}{10 (1+\coth (x))^5}-\frac{1}{16 (1+\coth (x))^4}-\frac{1}{24 (1+\coth (x))^3}-\frac{1}{32 (1+\coth (x))^2}-\frac{1}{32 (1+\coth (x))}\\ \end{align*}

Mathematica [A]  time = 0.138169, size = 62, normalized size = 1.11 \[ \frac{(\cosh (5 x)-\sinh (5 x)) (-500 \sinh (x)+375 \sinh (3 x)+120 x \sinh (5 x)-12 \sinh (5 x)-100 \cosh (x)+225 \cosh (3 x)+120 x \cosh (5 x)+12 \cosh (5 x))}{3840} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^(-5),x]

[Out]

((Cosh[5*x] - Sinh[5*x])*(-100*Cosh[x] + 225*Cosh[3*x] + 12*Cosh[5*x] + 120*x*Cosh[5*x] - 500*Sinh[x] + 375*Si
nh[3*x] - 12*Sinh[5*x] + 120*x*Sinh[5*x]))/3840

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Maple [A]  time = 0.024, size = 56, normalized size = 1. \begin{align*} -{\frac{1}{10\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{5}}}-{\frac{1}{16\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{4}}}-{\frac{1}{24\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{3}}}-{\frac{1}{32\, \left ( 1+{\rm coth} \left (x\right ) \right ) ^{2}}}-{\frac{1}{32+32\,{\rm coth} \left (x\right )}}+{\frac{\ln \left ( 1+{\rm coth} \left (x\right ) \right ) }{64}}-{\frac{\ln \left ({\rm coth} \left (x\right )-1 \right ) }{64}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+coth(x))^5,x)

[Out]

-1/10/(1+coth(x))^5-1/16/(1+coth(x))^4-1/24/(1+coth(x))^3-1/32/(1+coth(x))^2-1/32/(1+coth(x))+1/64*ln(1+coth(x
))-1/64*ln(coth(x)-1)

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Maxima [A]  time = 1.03789, size = 46, normalized size = 0.82 \begin{align*} \frac{1}{32} \, x + \frac{5}{64} \, e^{\left (-2 \, x\right )} - \frac{5}{64} \, e^{\left (-4 \, x\right )} + \frac{5}{96} \, e^{\left (-6 \, x\right )} - \frac{5}{256} \, e^{\left (-8 \, x\right )} + \frac{1}{320} \, e^{\left (-10 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^5,x, algorithm="maxima")

[Out]

1/32*x + 5/64*e^(-2*x) - 5/64*e^(-4*x) + 5/96*e^(-6*x) - 5/256*e^(-8*x) + 1/320*e^(-10*x)

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Fricas [B]  time = 2.2716, size = 540, normalized size = 9.64 \begin{align*} \frac{12 \,{\left (10 \, x + 1\right )} \cosh \left (x\right )^{5} + 60 \,{\left (10 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 12 \,{\left (10 \, x - 1\right )} \sinh \left (x\right )^{5} + 15 \,{\left (8 \,{\left (10 \, x - 1\right )} \cosh \left (x\right )^{2} + 25\right )} \sinh \left (x\right )^{3} + 225 \, \cosh \left (x\right )^{3} + 15 \,{\left (8 \,{\left (10 \, x + 1\right )} \cosh \left (x\right )^{3} + 45 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 5 \,{\left (12 \,{\left (10 \, x - 1\right )} \cosh \left (x\right )^{4} + 225 \, \cosh \left (x\right )^{2} - 100\right )} \sinh \left (x\right ) - 100 \, \cosh \left (x\right )}{3840 \,{\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^5,x, algorithm="fricas")

[Out]

1/3840*(12*(10*x + 1)*cosh(x)^5 + 60*(10*x + 1)*cosh(x)*sinh(x)^4 + 12*(10*x - 1)*sinh(x)^5 + 15*(8*(10*x - 1)
*cosh(x)^2 + 25)*sinh(x)^3 + 225*cosh(x)^3 + 15*(8*(10*x + 1)*cosh(x)^3 + 45*cosh(x))*sinh(x)^2 + 5*(12*(10*x
- 1)*cosh(x)^4 + 225*cosh(x)^2 - 100)*sinh(x) - 100*cosh(x))/(cosh(x)^5 + 5*cosh(x)^4*sinh(x) + 10*cosh(x)^3*s
inh(x)^2 + 10*cosh(x)^2*sinh(x)^3 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5)

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Sympy [B]  time = 3.23362, size = 444, normalized size = 7.93 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))**5,x)

[Out]

15*x*tanh(x)**5/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) +
75*x*tanh(x)**4/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) +
150*x*tanh(x)**3/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) +
150*x*tanh(x)**2/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480)
+ 75*x*tanh(x)/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 1
5*x/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) - 93*tanh(x)**
5/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 195*tanh(x)**3
/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 275*tanh(x)**2/
(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 160*tanh(x)/(480
*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 35/(480*tanh(x)**5 +
2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480)

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Giac [A]  time = 1.14209, size = 49, normalized size = 0.88 \begin{align*} \frac{1}{3840} \,{\left (300 \, e^{\left (8 \, x\right )} - 300 \, e^{\left (6 \, x\right )} + 200 \, e^{\left (4 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 12\right )} e^{\left (-10 \, x\right )} + \frac{1}{32} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^5,x, algorithm="giac")

[Out]

1/3840*(300*e^(8*x) - 300*e^(6*x) + 200*e^(4*x) - 75*e^(2*x) + 12)*e^(-10*x) + 1/32*x