Integrand size = 9, antiderivative size = 50 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=-\frac {16}{15} \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {2}{5} \cosh ^2(x) \coth (x) \sqrt {\cosh (x) \coth (x)}+\frac {64}{15} \sqrt {\cosh (x) \coth (x)} \tanh (x) \]
-16/15*coth(x)*(cosh(x)*coth(x))^(1/2)+2/5*cosh(x)^2*coth(x)*(cosh(x)*coth (x))^(1/2)+64/15*(cosh(x)*coth(x))^(1/2)*tanh(x)
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {1}{15} \sqrt {\cosh (x) \coth (x)} \left (-10 \coth (x)+6 \cosh (x) \sinh (x)+57 \text {csch}(x) \text {sech}(x) \left (-\sinh ^2(x)\right )^{3/4}+64 \tanh (x)\right ) \]
(Sqrt[Cosh[x]*Coth[x]]*(-10*Coth[x] + 6*Cosh[x]*Sinh[x] + 57*Csch[x]*Sech[ x]*(-Sinh[x]^2)^(3/4) + 64*Tanh[x]))/15
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.98, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {3042, 4898, 3042, 4900, 3042, 3078, 3042, 3074, 3042, 3069}
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.
| \(\displaystyle \int (\cosh (x) \coth (x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (i \cos (i x) \cot (i x))^{5/2}dx\) |
| \(\Big \downarrow \) 4898 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \int (-i \cosh (x) \coth (x))^{5/2}dx}{\sqrt {-i \cosh (x) \coth (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \int (\cos (i x) \cot (i x))^{5/2}dx}{\sqrt {-i \cosh (x) \coth (x)}}\) |
| \(\Big \downarrow \) 4900 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \int \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{5/2}dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \int \sin \left (i x+\frac {\pi }{2}\right )^{5/2} \left (-\tan \left (i x+\frac {\pi }{2}\right )\right )^{5/2}dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
| \(\Big \downarrow \) 3078 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \left (\frac {8}{5} \int \sqrt {\cosh (x)} (-i \coth (x))^{5/2}dx-\frac {2}{5} i \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{3/2}\right )}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \left (\frac {8}{5} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )} \left (-\tan \left (i x+\frac {\pi }{2}\right )\right )^{5/2}dx-\frac {2}{5} i \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{3/2}\right )}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
| \(\Big \downarrow \) 3074 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \left (\frac {8}{5} \left (\frac {2}{3} i \sqrt {\cosh (x)} (-i \coth (x))^{3/2}-\frac {4}{3} \int \sqrt {\cosh (x)} \sqrt {-i \coth (x)}dx\right )-\frac {2}{5} i \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{3/2}\right )}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {\cosh (x) \coth (x)} \left (\frac {8}{5} \left (\frac {2}{3} i \sqrt {\cosh (x)} (-i \coth (x))^{3/2}-\frac {4}{3} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )} \sqrt {-\tan \left (i x+\frac {\pi }{2}\right )}dx\right )-\frac {2}{5} i \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{3/2}\right )}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
| \(\Big \downarrow \) 3069 |
| \(\displaystyle -\frac {\left (\frac {8}{5} \left (\frac {2}{3} i \sqrt {\cosh (x)} (-i \coth (x))^{3/2}+\frac {8 i \sqrt {\cosh (x)}}{3 \sqrt {-i \coth (x)}}\right )-\frac {2}{5} i \cosh ^{\frac {5}{2}}(x) (-i \coth (x))^{3/2}\right ) \sqrt {\cosh (x) \coth (x)}}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}}\) |
-((((8*((((8*I)/3)*Sqrt[Cosh[x]])/Sqrt[(-I)*Coth[x]] + ((2*I)/3)*Sqrt[Cosh [x]]*((-I)*Coth[x])^(3/2)))/5 - ((2*I)/5)*Cosh[x]^(5/2)*((-I)*Coth[x])^(3/ 2))*Sqrt[Cosh[x]*Coth[x]])/(Sqrt[Cosh[x]]*Sqrt[(-I)*Coth[x]]))
3.6.65.3.1 Defintions of rubi rules used
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n _), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f* m)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] - Simp[b^2*((m + n - 1)/(n - 1)) Int[(a*Sin[e + f*x])^m*(b*Ta n[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && In tegersQ[2*m, 2*n] && !(GtQ[m, 1] && !IntegerQ[(m - 1)/2])
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[(-b)*(a*Sin[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/( f*m)), x] + Simp[a^2*((m + n - 1)/m) Int[(a*Sin[e + f*x])^(m - 2)*(b*Tan[ e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1 ] && EqQ[n, 1/2])) && IntegersQ[2*m, 2*n]
Int[(u_.)*((a_)*(v_))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = A ctivateTrig[v]}, Simp[a^IntPart[p]*((a*vv)^FracPart[p]/vv^FracPart[p]) In t[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] && !IntegerQ[p] && !InertTrigFreeQ [v]
Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTri g[u], vv = ActivateTrig[v], ww = ActivateTrig[w]}, Simp[(vv^m*ww^n)^FracPar t[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])) Int[uu*vv^(m*p)*ww^(n*p), x] , x]] /; FreeQ[{m, n, p}, x] && !IntegerQ[p] && ( !InertTrigFreeQ[v] || ! InertTrigFreeQ[w])
\[\int \left (\coth \left (x \right ) \cosh \left (x \right )\right )^{\frac {5}{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (38) = 76\).
Time = 0.27 (sec) , antiderivative size = 259, normalized size of antiderivative = 5.18 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (3 \, \cosh \left (x\right )^{8} + 24 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 3 \, \sinh \left (x\right )^{8} + 12 \, {\left (7 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{6} + 108 \, \cosh \left (x\right )^{6} + 24 \, {\left (7 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (105 \, \cosh \left (x\right )^{4} + 810 \, \cosh \left (x\right )^{2} - 151\right )} \sinh \left (x\right )^{4} - 302 \, \cosh \left (x\right )^{4} + 8 \, {\left (21 \, \cosh \left (x\right )^{5} + 270 \, \cosh \left (x\right )^{3} - 151 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 12 \, {\left (7 \, \cosh \left (x\right )^{6} + 135 \, \cosh \left (x\right )^{4} - 151 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{2} + 108 \, \cosh \left (x\right )^{2} + 8 \, {\left (3 \, \cosh \left (x\right )^{7} + 81 \, \cosh \left (x\right )^{5} - 151 \, \cosh \left (x\right )^{3} + 27 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3\right )}}{30 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}} \]
1/30*sqrt(1/2)*(3*cosh(x)^8 + 24*cosh(x)*sinh(x)^7 + 3*sinh(x)^8 + 12*(7*c osh(x)^2 + 9)*sinh(x)^6 + 108*cosh(x)^6 + 24*(7*cosh(x)^3 + 27*cosh(x))*si nh(x)^5 + 2*(105*cosh(x)^4 + 810*cosh(x)^2 - 151)*sinh(x)^4 - 302*cosh(x)^ 4 + 8*(21*cosh(x)^5 + 270*cosh(x)^3 - 151*cosh(x))*sinh(x)^3 + 12*(7*cosh( x)^6 + 135*cosh(x)^4 - 151*cosh(x)^2 + 9)*sinh(x)^2 + 108*cosh(x)^2 + 8*(3 *cosh(x)^7 + 81*cosh(x)^5 - 151*cosh(x)^3 + 27*cosh(x))*sinh(x) + 3)/((cos h(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - c osh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x))*sqrt(cosh(x)^3 + 3*cosh(x)*s inh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sinh(x) - cosh(x)))
Timed out. \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (38) = 76\).
Time = 0.31 (sec) , antiderivative size = 163, normalized size of antiderivative = 3.26 \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\frac {\sqrt {2} e^{\left (\frac {5}{2} \, x\right )}}{20 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {7 \, \sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{4 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {41 \, \sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {41 \, \sqrt {2} e^{\left (-\frac {7}{2} \, x\right )}}{6 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {7 \, \sqrt {2} e^{\left (-\frac {11}{2} \, x\right )}}{4 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} - \frac {\sqrt {2} e^{\left (-\frac {15}{2} \, x\right )}}{20 \, {\left (e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}} {\left (-e^{\left (-x\right )} + 1\right )}^{\frac {5}{2}}} \]
1/20*sqrt(2)*e^(5/2*x)/((e^(-x) + 1)^(5/2)*(-e^(-x) + 1)^(5/2)) + 7/4*sqrt (2)*e^(1/2*x)/((e^(-x) + 1)^(5/2)*(-e^(-x) + 1)^(5/2)) - 41/6*sqrt(2)*e^(- 3/2*x)/((e^(-x) + 1)^(5/2)*(-e^(-x) + 1)^(5/2)) + 41/6*sqrt(2)*e^(-7/2*x)/ ((e^(-x) + 1)^(5/2)*(-e^(-x) + 1)^(5/2)) - 7/4*sqrt(2)*e^(-11/2*x)/((e^(-x ) + 1)^(5/2)*(-e^(-x) + 1)^(5/2)) - 1/20*sqrt(2)*e^(-15/2*x)/((e^(-x) + 1) ^(5/2)*(-e^(-x) + 1)^(5/2))
\[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\int { \left (\cosh \left (x\right ) \coth \left (x\right )\right )^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (\cosh (x) \coth (x))^{5/2} \, dx=\int {\left (\mathrm {cosh}\left (x\right )\,\mathrm {coth}\left (x\right )\right )}^{5/2} \,d x \]