Integrand size = 6, antiderivative size = 56 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {x}{32}-\frac {1}{10 (1+\coth (x))^5}-\frac {1}{16 (1+\coth (x))^4}-\frac {1}{24 (1+\coth (x))^3}-\frac {1}{32 (1+\coth (x))^2}-\frac {1}{32 (1+\coth (x))} \]
1/32*x-1/10/(1+coth(x))^5-1/16/(1+coth(x))^4-1/24/(1+coth(x))^3-1/32/(1+co th(x))^2-1/32/(1+coth(x))
Time = 0.50 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {1}{480} \left (15 \text {arctanh}(\tanh (x))+\frac {128+625 \tanh (x)+1205 \tanh ^2(x)+1125 \tanh ^3(x)+465 \tanh ^4(x)}{(1+\tanh (x))^5}\right ) \]
(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
Time = 0.44 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.833, Rules used = {3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 3042, 3960, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(\coth (x)+1)^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^5}dx\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{(\coth (x)+1)^4}dx-\frac {1}{10 (\coth (x)+1)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{10 (\coth (x)+1)^5}+\frac {1}{2} \int \frac {1}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{(\coth (x)+1)^3}dx-\frac {1}{8 (\coth (x)+1)^4}\right )-\frac {1}{10 (\coth (x)+1)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{10 (\coth (x)+1)^5}+\frac {1}{2} \left (-\frac {1}{8 (\coth (x)+1)^4}+\frac {1}{2} \int \frac {1}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^3}dx\right )\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{(\coth (x)+1)^2}dx-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\right )-\frac {1}{10 (\coth (x)+1)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{10 (\coth (x)+1)^5}+\frac {1}{2} \left (-\frac {1}{8 (\coth (x)+1)^4}+\frac {1}{2} \left (-\frac {1}{6 (\coth (x)+1)^3}+\frac {1}{2} \int \frac {1}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^2}dx\right )\right )\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\coth (x)+1}dx-\frac {1}{4 (\coth (x)+1)^2}\right )-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\right )-\frac {1}{10 (\coth (x)+1)^5}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{10 (\coth (x)+1)^5}+\frac {1}{2} \left (-\frac {1}{8 (\coth (x)+1)^4}+\frac {1}{2} \left (-\frac {1}{6 (\coth (x)+1)^3}+\frac {1}{2} \left (-\frac {1}{4 (\coth (x)+1)^2}+\frac {1}{2} \int \frac {1}{1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\right )\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int 1dx}{2}-\frac {1}{2 (\coth (x)+1)}\right )-\frac {1}{4 (\coth (x)+1)^2}\right )-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\right )-\frac {1}{10 (\coth (x)+1)^5}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {x}{2}-\frac {1}{2 (\coth (x)+1)}\right )-\frac {1}{4 (\coth (x)+1)^2}\right )-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\right )-\frac {1}{10 (\coth (x)+1)^5}\) |
-1/10*1/(1 + Coth[x])^5 + (-1/8*1/(1 + Coth[x])^4 + (-1/6*1/(1 + Coth[x])^ 3 + (-1/4*1/(1 + Coth[x])^2 + (x/2 - 1/(2*(1 + Coth[x])))/2)/2)/2)/2
3.1.69.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[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]
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 | \(\frac {15 \tanh \left (x \right )^{5} x +\left (75 x +465\right ) \tanh \left (x \right )^{4}+\left (150 x +1125\right ) \tanh \left (x \right )^{3}+\left (150 x +1205\right ) \tanh \left (x \right )^{2}+\left (75 x +625\right ) \tanh \left (x \right )+15 x +128}{480 \left (1+\tanh \left (x \right )\right )^{5}}\) | \(59\) |
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 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {12 \, {\left (10 \, x + 1\right )} \cosh \left (x\right )^{5} + 60 \, {\left (10 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 12 \, {\left (10 \, x - 1\right )} \sinh \left (x\right )^{5} + 15 \, {\left (8 \, {\left (10 \, x - 1\right )} \cosh \left (x\right )^{2} + 25\right )} \sinh \left (x\right )^{3} + 225 \, \cosh \left (x\right )^{3} + 15 \, {\left (8 \, {\left (10 \, x + 1\right )} \cosh \left (x\right )^{3} + 45 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 5 \, {\left (12 \, {\left (10 \, x - 1\right )} \cosh \left (x\right )^{4} + 225 \, \cosh \left (x\right )^{2} - 100\right )} \sinh \left (x\right ) - 100 \, \cosh \left (x\right )}{3840 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )}} \]
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*sinh(x)^2 + 10*cosh(x)^2*sinh(x)^3 + 5*co sh(x)*sinh(x)^4 + sinh(x)^5)
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} \]
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) + 15 0*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) + 15*x/(480*tanh(x)**5 + 2400*tanh(x)**4 + 48 00*tanh(x)**3 + 4800*tanh(x)**2 + 2400*tanh(x) + 480) + 465*tanh(x)**4/(48 0*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*ta nh(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)
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {1}{32} \, x + \frac {5}{64} \, e^{\left (-2 \, x\right )} - \frac {5}{64} \, e^{\left (-4 \, x\right )} + \frac {5}{96} \, e^{\left (-6 \, x\right )} - \frac {5}{256} \, e^{\left (-8 \, x\right )} + \frac {1}{320} \, e^{\left (-10 \, x\right )} \]
Time = 0.27 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {1}{3840} \, {\left (300 \, e^{\left (8 \, x\right )} - 300 \, e^{\left (6 \, x\right )} + 200 \, e^{\left (4 \, x\right )} - 75 \, e^{\left (2 \, x\right )} + 12\right )} e^{\left (-10 \, x\right )} + \frac {1}{32} \, x \]
Time = 1.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+\coth (x))^5} \, dx=\frac {x}{32}+\frac {5\,{\mathrm {e}}^{-2\,x}}{64}-\frac {5\,{\mathrm {e}}^{-4\,x}}{64}+\frac {5\,{\mathrm {e}}^{-6\,x}}{96}-\frac {5\,{\mathrm {e}}^{-8\,x}}{256}+\frac {{\mathrm {e}}^{-10\,x}}{320} \]