Integrand size = 7, antiderivative size = 28 \[ \int (\coth (x)+\text {csch}(x))^5 \, dx=-\frac {2}{(1-\cosh (x))^2}+\frac {4}{1-\cosh (x)}+\log (1-\cosh (x)) \] Output:
-2/(1-cosh(x))^2+4/(1-cosh(x))+ln(1-cosh(x))
Time = 0.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int (\coth (x)+\text {csch}(x))^5 \, dx=-2 \text {csch}^2\left (\frac {x}{2}\right )-\frac {1}{2} \text {csch}^4\left (\frac {x}{2}\right )+2 \log \left (\sinh \left (\frac {x}{2}\right )\right ) \] Input:
Integrate[(Coth[x] + Csch[x])^5,x]
Output:
-2*Csch[x/2]^2 - Csch[x/2]^4/2 + 2*Log[Sinh[x/2]]
Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.286, Rules used = {3042, 4892, 26, 26, 3042, 26, 3146, 49, 2009}
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.
| \(\displaystyle \int (\coth (x)+\text {csch}(x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (i \cot (i x)+i \csc (i x))^5dx\) |
| \(\Big \downarrow \) 4892 |
| \(\displaystyle \int -i (i \cosh (x)+i)^5 \text {csch}^5(x)dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int i (\cosh (x)+1)^5 \text {csch}^5(x)dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int (\cosh (x)+1)^5 \text {csch}^5(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \left (1-\sin \left (-\frac {\pi }{2}+i x\right )\right )^5}{\cos \left (-\frac {\pi }{2}+i x\right )^5}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\left (1-\sin \left (i x-\frac {\pi }{2}\right )\right )^5}{\cos \left (i x-\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 3146 |
| \(\displaystyle -\int \frac {(\cosh (x)+1)^2}{(1-\cosh (x))^3}d\cosh (x)\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle -\int \left (-\frac {4}{(\cosh (x)-1)^2}-\frac {4}{(\cosh (x)-1)^3}+\frac {1}{1-\cosh (x)}\right )d\cosh (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4}{1-\cosh (x)}-\frac {2}{(1-\cosh (x))^2}+\log (1-\cosh (x))\) |
Input:
Int[(Coth[x] + Csch[x])^5,x]
Output:
-2/(1 - Cosh[x])^2 + 4/(1 - Cosh[x]) + Log[1 - Cosh[x]]
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[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Int[(cot[(c_.) + (d_.)*(x_)]^(n_.)*(a_.) + csc[(c_.) + (d_.)*(x_)]^(n_.)*(b _.))^(p_)*(u_.), x_Symbol] :> Int[ActivateTrig[u]*Csc[c + d*x]^(n*p)*(b + a *Cos[c + d*x]^n)^p, x] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p]
Time = 8.84 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
| method | result | size |
| risch | \(-x -\frac {8 \,{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-{\mathrm e}^{x}+1\right )}{\left ({\mathrm e}^{x}-1\right )^{4}}+2 \ln \left ({\mathrm e}^{x}-1\right )\) | \(32\) |
| default | \(\ln \left (\sinh \left (x \right )\right )-\frac {\coth \left (x \right )^{2}}{2}-\frac {\coth \left (x \right )^{4}}{4}-\frac {5 \cosh \left (x \right )^{3}}{\sinh \left (x \right )^{4}}+\frac {5 \cosh \left (x \right )}{3 \sinh \left (x \right )^{4}}+\frac {8 \left (-\frac {\operatorname {csch}\left (x \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (x \right )}{8}\right ) \coth \left (x \right )}{3}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{x}\right )-\frac {5 \cosh \left (x \right )^{2}}{\sinh \left (x \right )^{4}}+\frac {5}{4 \sinh \left (x \right )^{4}}\) | \(71\) |
| parts | \(-\frac {11 \coth \left (x \right )^{4}}{4}-\frac {\coth \left (x \right )^{2}}{2}-\frac {\ln \left (\coth \left (x \right )-1\right )}{2}-\frac {\ln \left (\coth \left (x \right )+1\right )}{2}+\frac {8 \left (-\frac {\operatorname {csch}\left (x \right )^{3}}{4}+\frac {3 \,\operatorname {csch}\left (x \right )}{8}\right ) \coth \left (x \right )}{3}-2 \,\operatorname {arctanh}\left ({\mathrm e}^{x}\right )-\frac {5 \cosh \left (x \right )^{3}}{\sinh \left (x \right )^{4}}+\frac {5 \cosh \left (x \right )}{3 \sinh \left (x \right )^{4}}-\frac {5 \operatorname {csch}\left (x \right )^{4}}{4}\) | \(72\) |
Input:
int((coth(x)+csch(x))^5,x,method=_RETURNVERBOSE)
Output:
-x-8*exp(x)*(exp(2*x)-exp(x)+1)/(exp(x)-1)^4+2*ln(exp(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (24) = 48\).
Time = 0.10 (sec) , antiderivative size = 270, normalized size of antiderivative = 9.64 \[ \int (\coth (x)+\text {csch}(x))^5 \, dx=-\frac {x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} - 4 \, {\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \, {\left (x \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, x \cosh \left (x\right )^{2} - 6 \, {\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} - 4 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \, {\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 4 \, {\left (x \cosh \left (x\right )^{3} - 3 \, {\left (x - 2\right )} \cosh \left (x\right )^{2} + {\left (3 \, x - 4\right )} \cosh \left (x\right ) - x + 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 4 \, \cosh \left (x\right )^{3} + 6 \, {\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) - 4 \, \cosh \left (x\right ) + 1} \] Input:
integrate((coth(x)+csch(x))^5,x, algorithm="fricas")
Output:
-(x*cosh(x)^4 + x*sinh(x)^4 - 4*(x - 2)*cosh(x)^3 + 4*(x*cosh(x) - x + 2)* sinh(x)^3 + 2*(3*x - 4)*cosh(x)^2 + 2*(3*x*cosh(x)^2 - 6*(x - 2)*cosh(x) + 3*x - 4)*sinh(x)^2 - 4*(x - 2)*cosh(x) - 2*(cosh(x)^4 + 4*(cosh(x) - 1)*s inh(x)^3 + sinh(x)^4 - 4*cosh(x)^3 + 6*(cosh(x)^2 - 2*cosh(x) + 1)*sinh(x) ^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x)^2 + 3*cosh(x) - 1)*sinh(x) - 4 *cosh(x) + 1)*log(cosh(x) + sinh(x) - 1) + 4*(x*cosh(x)^3 - 3*(x - 2)*cosh (x)^2 + (3*x - 4)*cosh(x) - x + 2)*sinh(x) + x)/(cosh(x)^4 + 4*(cosh(x) - 1)*sinh(x)^3 + sinh(x)^4 - 4*cosh(x)^3 + 6*(cosh(x)^2 - 2*cosh(x) + 1)*sin h(x)^2 + 6*cosh(x)^2 + 4*(cosh(x)^3 - 3*cosh(x)^2 + 3*cosh(x) - 1)*sinh(x) - 4*cosh(x) + 1)
\[ \int (\coth (x)+\text {csch}(x))^5 \, dx=\int \left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{5}\, dx \] Input:
integrate((coth(x)+csch(x))**5,x)
Output:
Integral((coth(x) + csch(x))**5, x)
Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (24) = 48\).
Time = 0.06 (sec) , antiderivative size = 236, normalized size of antiderivative = 8.43 \[ \int (\coth (x)+\text {csch}(x))^5 \, dx=-\frac {5}{2} \, \coth \left (x\right )^{4} + x + \frac {5 \, {\left (5 \, e^{\left (-x\right )} + 3 \, e^{\left (-3 \, x\right )} + 3 \, e^{\left (-5 \, x\right )} + 5 \, e^{\left (-7 \, x\right )}\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} - \frac {3 \, e^{\left (-x\right )} - 11 \, e^{\left (-3 \, x\right )} - 11 \, e^{\left (-5 \, x\right )} + 3 \, e^{\left (-7 \, x\right )}}{4 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {5 \, {\left (e^{\left (-x\right )} + 7 \, e^{\left (-3 \, x\right )} + 7 \, e^{\left (-5 \, x\right )} + e^{\left (-7 \, x\right )}\right )}}{2 \, {\left (4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1\right )}} + \frac {4 \, {\left (e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )}\right )}}{4 \, e^{\left (-2 \, x\right )} - 6 \, e^{\left (-4 \, x\right )} + 4 \, e^{\left (-6 \, x\right )} - e^{\left (-8 \, x\right )} - 1} - \frac {20}{{\left (e^{\left (-x\right )} - e^{x}\right )}^{4}} + 2 \, \log \left (e^{\left (-x\right )} - 1\right ) \] Input:
integrate((coth(x)+csch(x))^5,x, algorithm="maxima")
Output:
-5/2*coth(x)^4 + x + 5/4*(5*e^(-x) + 3*e^(-3*x) + 3*e^(-5*x) + 5*e^(-7*x)) /(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*x) - 1) - 1/4*(3*e^(-x) - 1 1*e^(-3*x) - 11*e^(-5*x) + 3*e^(-7*x))/(4*e^(-2*x) - 6*e^(-4*x) + 4*e^(-6* x) - e^(-8*x) - 1) + 5/2*(e^(-x) + 7*e^(-3*x) + 7*e^(-5*x) + e^(-7*x))/(4* e^(-2*x) - 6*e^(-4*x) + 4*e^(-6*x) - e^(-8*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^(-x) - e^x)^4 + 2*log(e^(-x) - 1)
Time = 0.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int (\coth (x)+\text {csch}(x))^5 \, dx=-x - \frac {8 \, {\left (e^{\left (3 \, x\right )} - e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} - 1\right )}^{4}} + 2 \, \log \left ({\left | e^{x} - 1 \right |}\right ) \] Input:
integrate((coth(x)+csch(x))^5,x, algorithm="giac")
Output:
-x - 8*(e^(3*x) - e^(2*x) + e^x)/(e^x - 1)^4 + 2*log(abs(e^x - 1))
Time = 0.79 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int (\coth (x)+\text {csch}(x))^5 \, dx=2\,\ln \left ({\mathrm {e}}^x-1\right )-x+\frac {16}{3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1}-\frac {16}{{\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1}-\frac {8}{6\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^x+1}-\frac {8}{{\mathrm {e}}^x-1} \] Input:
int((coth(x) + 1/sinh(x))^5,x)
Output:
2*log(exp(x) - 1) - x + 16/(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1) - 16/(ex p(2*x) - 2*exp(x) + 1) - 8/(6*exp(2*x) - 4*exp(3*x) + exp(4*x) - 4*exp(x) + 1) - 8/(exp(x) - 1)
\[ \int (\coth (x)+\text {csch}(x))^5 \, dx=\frac {e^{8 x} \left (\int \coth \left (x \right )^{5}d x \right )+e^{8 x} \mathrm {log}\left (e^{x}-1\right )-e^{8 x} \mathrm {log}\left (e^{x}+1\right )-5 e^{8 x}-8 e^{7 x}-4 e^{6 x} \left (\int \coth \left (x \right )^{5}d x \right )-4 e^{6 x} \mathrm {log}\left (e^{x}-1\right )+4 e^{6 x} \mathrm {log}\left (e^{x}+1\right )-24 e^{5 x}+6 e^{4 x} \left (\int \coth \left (x \right )^{5}d x \right )+6 e^{4 x} \mathrm {log}\left (e^{x}-1\right )-6 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-50 e^{4 x}-24 e^{3 x}-4 e^{2 x} \left (\int \coth \left (x \right )^{5}d x \right )-4 e^{2 x} \mathrm {log}\left (e^{x}-1\right )+4 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-8 e^{x}+\int \coth \left (x \right )^{5}d x +\mathrm {log}\left (e^{x}-1\right )-\mathrm {log}\left (e^{x}+1\right )-5}{e^{8 x}-4 e^{6 x}+6 e^{4 x}-4 e^{2 x}+1} \] Input:
int((coth(x)+csch(x))^5,x)
Output:
(e**(8*x)*int(coth(x)**5,x) + e**(8*x)*log(e**x - 1) - e**(8*x)*log(e**x + 1) - 5*e**(8*x) - 8*e**(7*x) - 4*e**(6*x)*int(coth(x)**5,x) - 4*e**(6*x)* log(e**x - 1) + 4*e**(6*x)*log(e**x + 1) - 24*e**(5*x) + 6*e**(4*x)*int(co th(x)**5,x) + 6*e**(4*x)*log(e**x - 1) - 6*e**(4*x)*log(e**x + 1) - 50*e** (4*x) - 24*e**(3*x) - 4*e**(2*x)*int(coth(x)**5,x) - 4*e**(2*x)*log(e**x - 1) + 4*e**(2*x)*log(e**x + 1) - 8*e**x + int(coth(x)**5,x) + log(e**x - 1 ) - log(e**x + 1) - 5)/(e**(8*x) - 4*e**(6*x) + 6*e**(4*x) - 4*e**(2*x) + 1)