Integrand size = 6, antiderivative size = 46 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {x}{16}-\frac {1}{8 (1+\coth (x))^4}-\frac {1}{12 (1+\coth (x))^3}-\frac {1}{16 (1+\coth (x))^2}-\frac {1}{16 (1+\coth (x))} \]
Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {1}{48} \left (3 \text {arctanh}(\tanh (x))+\frac {16+61 \tanh (x)+84 \tanh ^2(x)+45 \tanh ^3(x)}{(1+\tanh (x))^4}\right ) \]
Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {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.
\(\displaystyle \int \frac {1}{(\coth (x)+1)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^4}dx\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle \frac {1}{2} \int \frac {1}{(\coth (x)+1)^3}dx-\frac {1}{8 (\coth (x)+1)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{8 (\coth (x)+1)^4}+\frac {1}{2} \int \frac {1}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{(\coth (x)+1)^2}dx-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{8 (\coth (x)+1)^4}+\frac {1}{2} \left (-\frac {1}{6 (\coth (x)+1)^3}+\frac {1}{2} \int \frac {1}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^2}dx\right )\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\coth (x)+1}dx-\frac {1}{4 (\coth (x)+1)^2}\right )-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{8 (\coth (x)+1)^4}+\frac {1}{2} \left (-\frac {1}{6 (\coth (x)+1)^3}+\frac {1}{2} \left (-\frac {1}{4 (\coth (x)+1)^2}+\frac {1}{2} \int \frac {1}{1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\) |
\(\Big \downarrow \) 3960 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {\int 1dx}{2}-\frac {1}{2 (\coth (x)+1)}\right )-\frac {1}{4 (\coth (x)+1)^2}\right )-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \left (\frac {x}{2}-\frac {1}{2 (\coth (x)+1)}\right )-\frac {1}{4 (\coth (x)+1)^2}\right )-\frac {1}{6 (\coth (x)+1)^3}\right )-\frac {1}{8 (\coth (x)+1)^4}\) |
-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
3.1.68.3.1 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 = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {x}{16}+\frac {{\mathrm e}^{-2 x}}{8}-\frac {3 \,{\mathrm e}^{-4 x}}{32}+\frac {{\mathrm e}^{-6 x}}{24}-\frac {{\mathrm e}^{-8 x}}{128}\) | \(29\) |
derivativedivides | \(-\frac {\ln \left (\coth \left (x \right )-1\right )}{32}-\frac {1}{8 \left (1+\coth \left (x \right )\right )^{4}}-\frac {1}{12 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{32}\) | \(48\) |
default | \(-\frac {\ln \left (\coth \left (x \right )-1\right )}{32}-\frac {1}{8 \left (1+\coth \left (x \right )\right )^{4}}-\frac {1}{12 \left (1+\coth \left (x \right )\right )^{3}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )^{2}}-\frac {1}{16 \left (1+\coth \left (x \right )\right )}+\frac {\ln \left (1+\coth \left (x \right )\right )}{32}\) | \(48\) |
parallelrisch | \(\frac {3 \tanh \left (x \right )^{4} x +\left (12 x +45\right ) \tanh \left (x \right )^{3}+\left (18 x +84\right ) \tanh \left (x \right )^{2}+\left (12 x +61\right ) \tanh \left (x \right )+3 x +16}{48 \left (1+\tanh \left (x \right )\right )^{4}}\) | \(49\) |
Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (36) = 72\).
Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.63 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {3 \, {\left (8 \, x - 1\right )} \cosh \left (x\right )^{4} + 12 \, {\left (8 \, x + 1\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 3 \, {\left (8 \, x - 1\right )} \sinh \left (x\right )^{4} + 2 \, {\left (9 \, {\left (8 \, x - 1\right )} \cosh \left (x\right )^{2} + 32\right )} \sinh \left (x\right )^{2} + 64 \, \cosh \left (x\right )^{2} + 4 \, {\left (3 \, {\left (8 \, x + 1\right )} \cosh \left (x\right )^{3} + 16 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 36}{384 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4}\right )}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (41) = 82\).
Time = 0.73 (sec) , antiderivative size = 299, normalized size of antiderivative = 6.50 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {3 x \tanh ^{4}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {12 x \tanh ^{3}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {18 x \tanh ^{2}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {12 x \tanh {\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {3 x}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {45 \tanh ^{3}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {84 \tanh ^{2}{\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {61 \tanh {\left (x \right )}}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} + \frac {16}{48 \tanh ^{4}{\left (x \right )} + 192 \tanh ^{3}{\left (x \right )} + 288 \tanh ^{2}{\left (x \right )} + 192 \tanh {\left (x \right )} + 48} \]
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*ta nh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48) + 3*x/(48*tanh(x)**4 + 192*t anh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 48) + 45*tanh(x)**3/(48*tanh(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) + 4 8) + 16/(48*tanh(x)**4 + 192*tanh(x)**3 + 288*tanh(x)**2 + 192*tanh(x) + 4 8)
Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {1}{16} \, x + \frac {1}{8} \, e^{\left (-2 \, x\right )} - \frac {3}{32} \, e^{\left (-4 \, x\right )} + \frac {1}{24} \, e^{\left (-6 \, x\right )} - \frac {1}{128} \, e^{\left (-8 \, x\right )} \]
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {1}{384} \, {\left (48 \, e^{\left (6 \, x\right )} - 36 \, e^{\left (4 \, x\right )} + 16 \, e^{\left (2 \, x\right )} - 3\right )} e^{\left (-8 \, x\right )} + \frac {1}{16} \, x \]
Time = 1.83 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(1+\coth (x))^4} \, dx=\frac {x}{16}+\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {3\,{\mathrm {e}}^{-4\,x}}{32}+\frac {{\mathrm {e}}^{-6\,x}}{24}-\frac {{\mathrm {e}}^{-8\,x}}{128} \]