Integrand size = 6, antiderivative size = 41 \[ \int (1+\coth (x))^5 \, dx=16 x-8 \coth (x)-2 (1+\coth (x))^2-\frac {2}{3} (1+\coth (x))^3-\frac {1}{4} (1+\coth (x))^4+16 \log (\sinh (x)) \]
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.29 \[ \int (1+\coth (x))^5 \, dx=\frac {(1+\coth (x))^5 \sinh (x) \left (-3 \cosh ^4(x)-20 \cosh ^3(x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\tanh ^2(x)\right ) \sinh (x)-66 \cosh ^2(x) \sinh ^2(x)-120 \cosh (x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\tanh ^2(x)\right ) \sinh ^3(x)+12 (x+16 \log (\cosh (x))+16 \log (\tanh (x))) \sinh ^4(x)\right )}{12 (\cosh (x)+\sinh (x))^5} \]
((1 + Coth[x])^5*Sinh[x]*(-3*Cosh[x]^4 - 20*Cosh[x]^3*Hypergeometric2F1[-3 /2, 1, -1/2, Tanh[x]^2]*Sinh[x] - 66*Cosh[x]^2*Sinh[x]^2 - 120*Cosh[x]*Hyp ergeometric2F1[-1/2, 1, 1/2, Tanh[x]^2]*Sinh[x]^3 + 12*(x + 16*Log[Cosh[x] ] + 16*Log[Tanh[x]])*Sinh[x]^4))/(12*(Cosh[x] + Sinh[x])^5)
Time = 0.43 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.000, Rules used = {3042, 3959, 3042, 3959, 3042, 3959, 3042, 3958, 26, 3042, 26, 3956}
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.
\(\displaystyle \int (\coth (x)+1)^5 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^5dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \int (\coth (x)+1)^4dx-\frac {1}{4} (\coth (x)+1)^4\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{4} (\coth (x)+1)^4+2 \int \left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^4dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \left (2 \int (\coth (x)+1)^3dx-\frac {1}{3} (\coth (x)+1)^3\right )-\frac {1}{4} (\coth (x)+1)^4\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{4} (\coth (x)+1)^4+2 \left (-\frac {1}{3} (\coth (x)+1)^3+2 \int \left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^3dx\right )\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \left (2 \left (2 \int (\coth (x)+1)^2dx-\frac {1}{2} (\coth (x)+1)^2\right )-\frac {1}{3} (\coth (x)+1)^3\right )-\frac {1}{4} (\coth (x)+1)^4\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{4} (\coth (x)+1)^4+2 \left (-\frac {1}{3} (\coth (x)+1)^3+2 \left (-\frac {1}{2} (\coth (x)+1)^2+2 \int \left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^2dx\right )\right )\) |
\(\Big \downarrow \) 3958 |
\(\displaystyle -\frac {1}{4} (\coth (x)+1)^4+2 \left (-\frac {1}{3} (\coth (x)+1)^3+2 \left (-\frac {1}{2} (\coth (x)+1)^2+2 (-2 i \int i \coth (x)dx+2 x-\coth (x))\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle 2 \left (2 \left (2 (2 \int \coth (x)dx+2 x-\coth (x))-\frac {1}{2} (\coth (x)+1)^2\right )-\frac {1}{3} (\coth (x)+1)^3\right )-\frac {1}{4} (\coth (x)+1)^4\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{4} (\coth (x)+1)^4+2 \left (-\frac {1}{3} (\coth (x)+1)^3+2 \left (-\frac {1}{2} (\coth (x)+1)^2+2 \left (2 \int -i \tan \left (i x+\frac {\pi }{2}\right )dx+2 x-\coth (x)\right )\right )\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {1}{4} (\coth (x)+1)^4+2 \left (-\frac {1}{3} (\coth (x)+1)^3+2 \left (-\frac {1}{2} (\coth (x)+1)^2+2 \left (-2 i \int \tan \left (i x+\frac {\pi }{2}\right )dx+2 x-\coth (x)\right )\right )\right )\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle 2 \left (2 \left (2 (2 x-\coth (x)+2 \log (\sinh (x)))-\frac {1}{2} (\coth (x)+1)^2\right )-\frac {1}{3} (\coth (x)+1)^3\right )-\frac {1}{4} (\coth (x)+1)^4\) |
-1/4*(1 + Coth[x])^4 + 2*(-1/3*(1 + Coth[x])^3 + 2*(-1/2*(1 + Coth[x])^2 + 2*(2*x - Coth[x] + 2*Log[Sinh[x]])))
3.1.61.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2) *x, x] + (Simp[b^2*(Tan[c + d*x]/d), x] + Simp[2*a*b Int[Tan[c + d*x], x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[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] && GtQ[n , 1]
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(-\frac {\coth \left (x \right )^{4}}{4}-\frac {5 \coth \left (x \right )^{3}}{3}-\frac {11 \coth \left (x \right )^{2}}{2}-15 \coth \left (x \right )-16 \ln \left (\coth \left (x \right )-1\right )\) | \(31\) |
default | \(-\frac {\coth \left (x \right )^{4}}{4}-\frac {5 \coth \left (x \right )^{3}}{3}-\frac {11 \coth \left (x \right )^{2}}{2}-15 \coth \left (x \right )-16 \ln \left (\coth \left (x \right )-1\right )\) | \(31\) |
parallelrisch | \(-\frac {\coth \left (x \right )^{4}}{4}+16 \ln \left (\tanh \left (x \right )\right )-16 \ln \left (1-\tanh \left (x \right )\right )-15 \coth \left (x \right )-\frac {11 \coth \left (x \right )^{2}}{2}-\frac {5 \coth \left (x \right )^{3}}{3}\) | \(38\) |
risch | \(-\frac {4 \left (48 \,{\mathrm e}^{6 x}-108 \,{\mathrm e}^{4 x}+88 \,{\mathrm e}^{2 x}-25\right )}{3 \left ({\mathrm e}^{2 x}-1\right )^{4}}+16 \ln \left ({\mathrm e}^{2 x}-1\right )\) | \(41\) |
parts | \(x -\frac {\coth \left (x \right )^{4}}{4}-\frac {11 \coth \left (x \right )^{2}}{2}-13 \ln \left (\coth \left (x \right )-1\right )+2 \ln \left (1+\coth \left (x \right )\right )-15 \coth \left (x \right )-\frac {5 \coth \left (x \right )^{3}}{3}+5 \ln \left (\sinh \left (x \right )\right )\) | \(44\) |
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (37) = 74\).
Time = 0.26 (sec) , antiderivative size = 448, normalized size of antiderivative = 10.93 \[ \int (1+\coth (x))^5 \, dx=-\frac {4 \, {\left (48 \, \cosh \left (x\right )^{6} + 288 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 48 \, \sinh \left (x\right )^{6} + 36 \, {\left (20 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{4} - 108 \, \cosh \left (x\right )^{4} + 48 \, {\left (20 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 8 \, {\left (90 \, \cosh \left (x\right )^{4} - 81 \, \cosh \left (x\right )^{2} + 11\right )} \sinh \left (x\right )^{2} + 88 \, \cosh \left (x\right )^{2} - 12 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 16 \, {\left (18 \, \cosh \left (x\right )^{5} - 27 \, \cosh \left (x\right )^{3} + 11 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 25\right )}}{3 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 4 \, {\left (7 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 4 \, \cosh \left (x\right )^{6} + 8 \, {\left (7 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (35 \, \cosh \left (x\right )^{4} - 30 \, \cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{4} + 8 \, {\left (7 \, \cosh \left (x\right )^{5} - 10 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (7 \, \cosh \left (x\right )^{6} - 15 \, \cosh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 4 \, \cosh \left (x\right )^{2} + 8 \, {\left (\cosh \left (x\right )^{7} - 3 \, \cosh \left (x\right )^{5} + 3 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
-4/3*(48*cosh(x)^6 + 288*cosh(x)*sinh(x)^5 + 48*sinh(x)^6 + 36*(20*cosh(x) ^2 - 3)*sinh(x)^4 - 108*cosh(x)^4 + 48*(20*cosh(x)^3 - 9*cosh(x))*sinh(x)^ 3 + 8*(90*cosh(x)^4 - 81*cosh(x)^2 + 11)*sinh(x)^2 + 88*cosh(x)^2 - 12*(co sh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6*cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*cosh(x)^6 - 15*cosh(x)^4 + 9*cosh(x)^2 - 1)* sinh(x)^2 - 4*cosh(x)^2 + 8*(cosh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh( x))*sinh(x) + 1)*log(2*sinh(x)/(cosh(x) - sinh(x))) + 16*(18*cosh(x)^5 - 2 7*cosh(x)^3 + 11*cosh(x))*sinh(x) - 25)/(cosh(x)^8 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8 + 4*(7*cosh(x)^2 - 1)*sinh(x)^6 - 4*cosh(x)^6 + 8*(7*cosh(x)^3 - 3*cosh(x))*sinh(x)^5 + 2*(35*cosh(x)^4 - 30*cosh(x)^2 + 3)*sinh(x)^4 + 6 *cosh(x)^4 + 8*(7*cosh(x)^5 - 10*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 4*(7*c osh(x)^6 - 15*cosh(x)^4 + 9*cosh(x)^2 - 1)*sinh(x)^2 - 4*cosh(x)^2 + 8*(co sh(x)^7 - 3*cosh(x)^5 + 3*cosh(x)^3 - cosh(x))*sinh(x) + 1)
Time = 0.67 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17 \[ \int (1+\coth (x))^5 \, dx=32 x - 16 \log {\left (\tanh {\left (x \right )} + 1 \right )} + 16 \log {\left (\tanh {\left (x \right )} \right )} - \frac {15}{\tanh {\left (x \right )}} - \frac {11}{2 \tanh ^{2}{\left (x \right )}} - \frac {5}{3 \tanh ^{3}{\left (x \right )}} - \frac {1}{4 \tanh ^{4}{\left (x \right )}} \]
32*x - 16*log(tanh(x) + 1) + 16*log(tanh(x)) - 15/tanh(x) - 11/(2*tanh(x)* *2) - 5/(3*tanh(x)**3) - 1/(4*tanh(x)**4)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (37) = 74\).
Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.41 \[ \int (1+\coth (x))^5 \, dx=27 \, x - \frac {20 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \, {\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac {4 \, {\left (e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} + \frac {20 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {20}{e^{\left (-2 \, x\right )} - 1} + 11 \, \log \left (e^{\left (-x\right )} + 1\right ) + 11 \, \log \left (e^{\left (-x\right )} - 1\right ) + 5 \, \log \left (\sinh \left (x\right )\right ) \]
27*x - 20/3*(3*e^(-2*x) - 3*e^(-4*x) - 2)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6 *x) - 1) + 4*(e^(-2*x) - e^(-4*x) + e^(-6*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4 *e^(-6*x) - e^(-8*x) - 1) + 20*e^(-2*x)/(2*e^(-2*x) - e^(-4*x) - 1) + 20/( e^(-2*x) - 1) + 11*log(e^(-x) + 1) + 11*log(e^(-x) - 1) + 5*log(sinh(x))
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00 \[ \int (1+\coth (x))^5 \, dx=-\frac {4 \, {\left (48 \, e^{\left (6 \, x\right )} - 108 \, e^{\left (4 \, x\right )} + 88 \, e^{\left (2 \, x\right )} - 25\right )}}{3 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} + 16 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \]
-4/3*(48*e^(6*x) - 108*e^(4*x) + 88*e^(2*x) - 25)/(e^(2*x) - 1)^4 + 16*log (abs(e^(2*x) - 1))
Time = 1.87 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.15 \[ \int (1+\coth (x))^5 \, dx=16\,\ln \left ({\mathrm {e}}^{2\,x}-1\right )-\frac {64}{3\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {48}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {4}{6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}-\frac {64}{{\mathrm {e}}^{2\,x}-1} \]