Integrand size = 10, antiderivative size = 121 \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\frac {154 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \cosh ^3(x)}}+\frac {154 \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}}+\frac {22 \text {sech}^2(x) \tanh (x)}{117 a^2 \sqrt {a \cosh ^3(x)}}+\frac {2 \text {sech}^4(x) \tanh (x)}{13 a^2 \sqrt {a \cosh ^3(x)}} \]
154/195*I*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh (1/2*x),2^(1/2))/a^2/(a*cosh(x)^3)^(1/2)+154/195*cosh(x)*sinh(x)/a^2/(a*co sh(x)^3)^(1/2)+154/585*tanh(x)/a^2/(a*cosh(x)^3)^(1/2)+22/117*sech(x)^2*ta nh(x)/a^2/(a*cosh(x)^3)^(1/2)+2/13*sech(x)^4*tanh(x)/a^2/(a*cosh(x)^3)^(1/ 2)
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\frac {462 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+462 \cosh (x) \sinh (x)+2 \left (77+55 \text {sech}^2(x)+45 \text {sech}^4(x)\right ) \tanh (x)}{585 a^2 \sqrt {a \cosh ^3(x)}} \]
((462*I)*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2] + 462*Cosh[x]*Sinh[x] + 2*(77 + 55*Sech[x]^2 + 45*Sech[x]^4)*Tanh[x])/(585*a^2*Sqrt[a*Cosh[x]^3])
Time = 0.50 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.79, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3686, 3042, 3116, 3042, 3116, 3042, 3116, 3042, 3116, 3042, 3119}
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.
| \(\displaystyle \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sin \left (\frac {\pi }{2}+i x\right )^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\cosh ^{\frac {15}{2}}(x)}dx}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \int \frac {1}{\sin \left (i x+\frac {\pi }{2}\right )^{15/2}}dx}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {11}{13} \int \frac {1}{\cosh ^{\frac {11}{2}}(x)}dx+\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}+\frac {11}{13} \int \frac {1}{\sin \left (i x+\frac {\pi }{2}\right )^{11/2}}dx\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \int \frac {1}{\cosh ^{\frac {7}{2}}(x)}dx+\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}+\frac {11}{13} \left (\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}+\frac {7}{9} \int \frac {1}{\sin \left (i x+\frac {\pi }{2}\right )^{7/2}}dx\right )\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\cosh ^{\frac {3}{2}}(x)}dx+\frac {2 \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}\right )+\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}+\frac {11}{13} \left (\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {3}{5} \int \frac {1}{\sin \left (i x+\frac {\pi }{2}\right )^{3/2}}dx\right )\right )\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}-\int \sqrt {\cosh (x)}dx\right )+\frac {2 \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}\right )+\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}+\frac {11}{13} \left (\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {3}{5} \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}-\int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\right )\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\cosh ^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{13 \cosh ^{\frac {13}{2}}(x)}+\frac {11}{13} \left (\frac {2 \sinh (x)}{9 \cosh ^{\frac {9}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {3}{5} \left (\frac {2 \sinh (x)}{\sqrt {\cosh (x)}}+2 i E\left (\left .\frac {i x}{2}\right |2\right )\right )\right )\right )\right )}{a^2 \sqrt {a \cosh ^3(x)}}\) |
(Cosh[x]^(3/2)*((2*Sinh[x])/(13*Cosh[x]^(13/2)) + (11*((2*Sinh[x])/(9*Cosh [x]^(9/2)) + (7*((2*Sinh[x])/(5*Cosh[x]^(5/2)) + (3*((2*I)*EllipticE[(I/2) *x, 2] + (2*Sinh[x])/Sqrt[Cosh[x]]))/5))/9))/13))/(a^2*Sqrt[a*Cosh[x]^3])
3.2.33.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
\[\int \frac {1}{\left (a \cosh \left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 1668, normalized size of antiderivative = 13.79 \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
2/585*(231*(sqrt(2)*cosh(x)^14 + 14*sqrt(2)*cosh(x)*sinh(x)^13 + sqrt(2)*s inh(x)^14 + 7*(13*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^12 + 7*sqrt(2)*cosh (x)^12 + 28*(13*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^11 + 7*(143 *sqrt(2)*cosh(x)^4 + 66*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x)^10 + 21*sqr t(2)*cosh(x)^10 + 14*(143*sqrt(2)*cosh(x)^5 + 110*sqrt(2)*cosh(x)^3 + 15*s qrt(2)*cosh(x))*sinh(x)^9 + 7*(429*sqrt(2)*cosh(x)^6 + 495*sqrt(2)*cosh(x) ^4 + 135*sqrt(2)*cosh(x)^2 + 5*sqrt(2))*sinh(x)^8 + 35*sqrt(2)*cosh(x)^8 + 8*(429*sqrt(2)*cosh(x)^7 + 693*sqrt(2)*cosh(x)^5 + 315*sqrt(2)*cosh(x)^3 + 35*sqrt(2)*cosh(x))*sinh(x)^7 + 7*(429*sqrt(2)*cosh(x)^8 + 924*sqrt(2)*c osh(x)^6 + 630*sqrt(2)*cosh(x)^4 + 140*sqrt(2)*cosh(x)^2 + 5*sqrt(2))*sinh (x)^6 + 35*sqrt(2)*cosh(x)^6 + 14*(143*sqrt(2)*cosh(x)^9 + 396*sqrt(2)*cos h(x)^7 + 378*sqrt(2)*cosh(x)^5 + 140*sqrt(2)*cosh(x)^3 + 15*sqrt(2)*cosh(x ))*sinh(x)^5 + 7*(143*sqrt(2)*cosh(x)^10 + 495*sqrt(2)*cosh(x)^8 + 630*sqr t(2)*cosh(x)^6 + 350*sqrt(2)*cosh(x)^4 + 75*sqrt(2)*cosh(x)^2 + 3*sqrt(2)) *sinh(x)^4 + 21*sqrt(2)*cosh(x)^4 + 28*(13*sqrt(2)*cosh(x)^11 + 55*sqrt(2) *cosh(x)^9 + 90*sqrt(2)*cosh(x)^7 + 70*sqrt(2)*cosh(x)^5 + 25*sqrt(2)*cosh (x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^3 + 7*(13*sqrt(2)*cosh(x)^12 + 66*sqrt( 2)*cosh(x)^10 + 135*sqrt(2)*cosh(x)^8 + 140*sqrt(2)*cosh(x)^6 + 75*sqrt(2) *cosh(x)^4 + 18*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 7*sqrt(2)*cosh(x) ^2 + 14*(sqrt(2)*cosh(x)^13 + 6*sqrt(2)*cosh(x)^11 + 15*sqrt(2)*cosh(x)...
Timed out. \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \cosh ^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\mathrm {cosh}\left (x\right )}^3\right )}^{5/2}} \,d x \]