\(\int \frac {1}{(1+\coth (x))^5} \, dx\) [69]

Optimal result

Integrand size = 6, antiderivative size = 56 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {x}{32}-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}-\frac {1}{24 (1+\coth (x))^3}-\frac {1}{32 (1+\coth (x))^2}-\frac {1}{32 (1+\coth (x))} \]

[Out]

1/32*x-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))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3560, 8} \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {x}{32}-\frac {1}{32 (\coth (x)+1)}-\frac {1}{32 (\coth (x)+1)^2}-\frac {1}{24 (\coth (x)+1)^3}-\frac {1}{16 (\coth (x)+1)^4}-\frac {1}{10 (\coth (x)+1)^5} \]

[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 8

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

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]

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {1}{480} \left (15 \text {arctanh}(\tanh (x))+\frac {128+625 \tanh (x)+1205 \tanh ^2(x)+1125 \tanh ^3(x)+465 \tanh ^4(x)}{(1+\tanh (x))^5}\right ) \]

[In]

Integrate[(1 + Coth[x])^(-5),x]

[Out]

(15*ArcTanh[Tanh[x]] + (128 + 625*Tanh[x] + 1205*Tanh[x]^2 + 1125*Tanh[x]^3 + 465*Tanh[x]^4)/(1 + Tanh[x])^5)/
480

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.62

method	result	size
risch	\(\frac {x}{32}+\frac {5 \,{\mathrm e}^{-2 x}}{64}-\frac {5 \,{\mathrm e}^{-4 x}}{64}+\frac {5 \,{\mathrm e}^{-6 x}}{96}-\frac {5 \,{\mathrm e}^{-8 x}}{256}+\frac {{\mathrm e}^{-10 x}}{320}\)	\(35\)
derivativedivides	\(-\frac {1}{10 \left (1+\coth \left (x \right )\right )^{5}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )^{4}}-\frac {1}{24 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{32 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{32 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{64}-\frac {\ln \left (\coth \left (x \right )-1\right )}{64}\)	\(56\)
default	\(-\frac {1}{10 \left (1+\coth \left (x \right )\right )^{5}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )^{4}}-\frac {1}{24 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{32 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{32 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{64}-\frac {\ln \left (\coth \left (x \right )-1\right )}{64}\)	\(56\)
parallelrisch	\(\frac {15 \tanh \left (x \right )^{5} x +\left (75 x +465\right ) \tanh \left (x \right )^{4}+\left (150 x +1125\right ) \tanh \left (x \right )^{3}+\left (150 x +1205\right ) \tanh \left (x \right )^{2}+\left (75 x +625\right ) \tanh \left (x \right )+15 x +128}{480 \left (1+\tanh \left (x \right )\right )^{5}}\)	\(59\)

[In]

int(1/(1+coth(x))^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (44) = 88\).

Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.84 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {12 \, {\left (10 \, x + 1\right )} \cosh \left (x\right )^{5} + 60 \, {\left (10 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 12 \, {\left (10 \, x - 1\right )} \sinh \left (x\right )^{5} + 15 \, {\left (8 \, {\left (10 \, x - 1\right )} \cosh \left (x\right )^{2} + 25\right )} \sinh \left (x\right )^{3} + 225 \, \cosh \left (x\right )^{3} + 15 \, {\left (8 \, {\left (10 \, x + 1\right )} \cosh \left (x\right )^{3} + 45 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 5 \, {\left (12 \, {\left (10 \, x - 1\right )} \cosh \left (x\right )^{4} + 225 \, \cosh \left (x\right )^{2} - 100\right )} \sinh \left (x\right ) - 100 \, \cosh \left (x\right )}{3840 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (51) = 102\).

Time = 0.93 (sec) , antiderivative size = 444, normalized size of antiderivative = 7.93 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {15 x \tanh ^{5}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {75 x \tanh ^{4}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {150 x \tanh ^{3}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {150 x \tanh ^{2}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {75 x \tanh {\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {15 x}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {465 \tanh ^{4}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {1125 \tanh ^{3}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {1205 \tanh ^{2}{\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {625 \tanh {\left (x \right )}}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} + \frac {128}{480 \tanh ^{5}{\left (x \right )} + 2400 \tanh ^{4}{\left (x \right )} + 4800 \tanh ^{3}{\left (x \right )} + 4800 \tanh ^{2}{\left (x \right )} + 2400 \tanh {\left (x \right )} + 480} \]

[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) + 465*tanh(x)*
*4/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 1125*tanh(x)*
*3/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 1205*tanh(x)*
*2/(480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 625*tanh(x)/(
480*tanh(x)**5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 128/(480*tanh(x)*
*5 + 2400*tanh(x)**4 + 4800*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {1}{32} \, x + \frac {5}{64} \, e^{\left (-2 \, x\right )} - \frac {5}{64} \, e^{\left (-4 \, x\right )} + \frac {5}{96} \, e^{\left (-6 \, x\right )} - \frac {5}{256} \, e^{\left (-8 \, x\right )} + \frac {1}{320} \, e^{\left (-10 \, x\right )} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {1}{3840} \, {\left (300 \, e^{\left (8 \, x\right )} - 300 \, e^{\left (6 \, x\right )} + 200 \, e^{\left (4 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 12\right )} e^{\left (-10 \, x\right )} + \frac {1}{32} \, x \]

[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

Mupad [B] (verification not implemented)

Time = 1.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {x}{32}+\frac {5\,{\mathrm {e}}^{-2\,x}}{64}-\frac {5\,{\mathrm {e}}^{-4\,x}}{64}+\frac {5\,{\mathrm {e}}^{-6\,x}}{96}-\frac {5\,{\mathrm {e}}^{-8\,x}}{256}+\frac {{\mathrm {e}}^{-10\,x}}{320} \]

[In]

int(1/(coth(x) + 1)^5,x)

[Out]

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