Integrand size = 7, antiderivative size = 22 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=-\frac {2}{(1+\cosh (x))^2}+\frac {4}{1+\cosh (x)}+\log (1+\cosh (x)) \] Output:
-2/(1+cosh(x))^2+4/(1+cosh(x))+ln(1+cosh(x))
Time = 0.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=2 \log \left (\cosh \left (\frac {x}{2}\right )\right )+2 \text {sech}^2\left (\frac {x}{2}\right )-\frac {1}{2} \text {sech}^4\left (\frac {x}{2}\right ) \] Input:
Integrate[(Coth[x] + Csch[x])^(-5),x]
Output:
2*Log[Cosh[x/2]] + 2*Sech[x/2]^2 - Sech[x/2]^4/2
Time = 0.30 (sec) , antiderivative size = 22, 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.
\(\displaystyle \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(i \cot (i x)+i \csc (i x))^5}dx\) |
\(\Big \downarrow \) 4892 |
\(\displaystyle \int \frac {i \sinh ^5(x)}{(i \cosh (x)+i)^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int -\frac {i \sinh ^5(x)}{(\cosh (x)+1)^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\sinh ^5(x)}{(\cosh (x)+1)^5}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \cos \left (-\frac {\pi }{2}+i x\right )^5}{\left (1-\sin \left (-\frac {\pi }{2}+i x\right )\right )^5}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\cos \left (i x-\frac {\pi }{2}\right )^5}{\left (1-\sin \left (i x-\frac {\pi }{2}\right )\right )^5}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle \int \frac {(1-\cosh (x))^2}{(\cosh (x)+1)^3}d\cosh (x)\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \int \left (\frac {1}{\cosh (x)+1}-\frac {4}{(\cosh (x)+1)^2}+\frac {4}{(\cosh (x)+1)^3}\right )d\cosh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{\cosh (x)+1}-\frac {2}{(\cosh (x)+1)^2}+\log (\cosh (x)+1)\) |
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 = 19.17 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.09
method | result | size |
parallelrisch | \(0\) | \(2\) |
risch | \(-x +\frac {8 \,{\mathrm e}^{x} \left ({\mathrm e}^{2 x}+{\mathrm e}^{x}+1\right )}{\left (1+{\mathrm e}^{x}\right )^{4}}+2 \ln \left (1+{\mathrm e}^{x}\right )\) | \(30\) |
default | \(-\frac {\tanh \left (\frac {x}{2}\right )^{4}}{2}-\tanh \left (\frac {x}{2}\right )^{2}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\ln \left (1+\tanh \left (\frac {x}{2}\right )\right )\) | \(36\) |
Input:
int(1/(coth(x)+csch(x))^5,x,method=_RETURNVERBOSE)
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (22) = 44\).
Time = 0.10 (sec) , antiderivative size = 266, normalized size of antiderivative = 12.09 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=-\frac {x \cosh \left (x\right )^{4} + x \sinh \left (x\right )^{4} + 4 \, {\left (x - 2\right )} \cosh \left (x\right )^{3} + 4 \, {\left (x \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right )^{3} + 2 \, {\left (3 \, x - 4\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, x \cosh \left (x\right )^{2} + 6 \, {\left (x - 2\right )} \cosh \left (x\right ) + 3 \, x - 4\right )} \sinh \left (x\right )^{2} + 4 \, {\left (x - 2\right )} \cosh \left (x\right ) - 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 4 \, {\left (x \cosh \left (x\right )^{3} + 3 \, {\left (x - 2\right )} \cosh \left (x\right )^{2} + {\left (3 \, x - 4\right )} \cosh \left (x\right ) + x - 2\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} + 6 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right )^{2} + 6 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} + 3 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 4 \, \cosh \left (x\right ) + 1} \] Input:
integrate(1/(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)
\[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=\int \frac {1}{\left (\coth {\left (x \right )} + \operatorname {csch}{\left (x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(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. 52 vs. \(2 (22) = 44\).
Time = 0.06 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.36 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=x + \frac {8 \, {\left (e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + e^{\left (-3 \, x\right )}\right )}}{4 \, e^{\left (-x\right )} + 6 \, e^{\left (-2 \, x\right )} + 4 \, e^{\left (-3 \, x\right )} + e^{\left (-4 \, x\right )} + 1} + 2 \, \log \left (e^{\left (-x\right )} + 1\right ) \] Input:
integrate(1/(coth(x)+csch(x))^5,x, algorithm="maxima")
Output:
x + 8*(e^(-x) + e^(-2*x) + e^(-3*x))/(4*e^(-x) + 6*e^(-2*x) + 4*e^(-3*x) + e^(-4*x) + 1) + 2*log(e^(-x) + 1)
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=-x + \frac {8 \, {\left (e^{\left (3 \, x\right )} + e^{\left (2 \, x\right )} + e^{x}\right )}}{{\left (e^{x} + 1\right )}^{4}} + 2 \, \log \left (e^{x} + 1\right ) \] Input:
integrate(1/(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(e^x + 1)
Time = 0.79 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.59 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=2\,\ln \left ({\mathrm {e}}^x+1\right )-x-\frac {16}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {8}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}+\frac {8}{{\mathrm {e}}^x+1}+\frac {16}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1} \] Input:
int(1/(coth(x) + 1/sinh(x))^5,x)
Output:
2*log(exp(x) + 1) - x - 16/(exp(2*x) + 2*exp(x) + 1) - 8/(6*exp(2*x) + 4*e xp(3*x) + exp(4*x) + 4*exp(x) + 1) + 8/(exp(x) + 1) + 16/(3*exp(2*x) + exp (3*x) + 3*exp(x) + 1)
Time = 0.20 (sec) , antiderivative size = 136, normalized size of antiderivative = 6.18 \[ \int \frac {1}{(\coth (x)+\text {csch}(x))^5} \, dx=\frac {2 e^{4 x} \mathrm {log}\left (e^{x}+1\right )-e^{4 x} x -2 e^{4 x}+8 e^{3 x} \mathrm {log}\left (e^{x}+1\right )-4 e^{3 x} x +12 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-6 e^{2 x} x -4 e^{2 x}+8 e^{x} \mathrm {log}\left (e^{x}+1\right )-4 e^{x} x +2 \,\mathrm {log}\left (e^{x}+1\right )-x -2}{e^{4 x}+4 e^{3 x}+6 e^{2 x}+4 e^{x}+1} \] Input:
int(1/(coth(x)+csch(x))^5,x)
Output:
(2*e**(4*x)*log(e**x + 1) - e**(4*x)*x - 2*e**(4*x) + 8*e**(3*x)*log(e**x + 1) - 4*e**(3*x)*x + 12*e**(2*x)*log(e**x + 1) - 6*e**(2*x)*x - 4*e**(2*x ) + 8*e**x*log(e**x + 1) - 4*e**x*x + 2*log(e**x + 1) - x - 2)/(e**(4*x) + 4*e**(3*x) + 6*e**(2*x) + 4*e**x + 1)