Integrand size = 8, antiderivative size = 67 \[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\frac {6 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{5 a^4 \sqrt {\cosh (x)}}+\frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac {6 \sinh (x)}{5 a^3 \sqrt {a \cosh (x)}} \]
2/5*sinh(x)/a/(a*cosh(x))^(5/2)+6/5*sinh(x)/a^3/(a*cosh(x))^(1/2)+6/5*I*(c osh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*cosh(x ))^(1/2)/a^4/cosh(x)^(1/2)
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\frac {2 \left (3 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+3 \cosh (x) \sinh (x)+\tanh (x)\right )}{5 a^2 (a \cosh (x))^{3/2}} \]
(2*((3*I)*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2] + 3*Cosh[x]*Sinh[x] + Tanh[x ]))/(5*a^2*(a*Cosh[x])^(3/2))
Time = 0.36 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3116, 3042, 3116, 3042, 3121, 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.
| \(\displaystyle \int \frac {1}{(a \cosh (x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sin \left (\frac {\pi }{2}+i x\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {3 \int \frac {1}{(a \cosh (x))^{3/2}}dx}{5 a^2}+\frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac {3 \int \frac {1}{\left (a \sin \left (i x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{5 a^2}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {3 \left (\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\int \sqrt {a \cosh (x)}dx}{a^2}\right )}{5 a^2}+\frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac {3 \left (\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\int \sqrt {a \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {3 \left (\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\sqrt {a \cosh (x)} \int \sqrt {\cosh (x)}dx}{a^2 \sqrt {\cosh (x)}}\right )}{5 a^2}+\frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac {3 \left (\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\sqrt {a \cosh (x)} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2 \sqrt {\cosh (x)}}\right )}{5 a^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {2 \sinh (x)}{5 a (a \cosh (x))^{5/2}}+\frac {3 \left (\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}+\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{a^2 \sqrt {\cosh (x)}}\right )}{5 a^2}\) |
(2*Sinh[x])/(5*a*(a*Cosh[x])^(5/2)) + (3*(((2*I)*Sqrt[a*Cosh[x]]*EllipticE [(I/2)*x, 2])/(a^2*Sqrt[Cosh[x]]) + (2*Sinh[x])/(a*Sqrt[a*Cosh[x]])))/(5*a ^2)
3.1.22.3.1 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[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Leaf count of result is larger than twice the leaf count of optimal. \(253\) vs. \(2(68)=136\).
Time = 0.68 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.79
| method | result | size |
| default | \(\frac {2 \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \left (\frac {\cosh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}}{20 a \left (\cosh \left (\frac {x}{2}\right )^{2}-\frac {1}{2}\right )^{3}}+\frac {6 \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )}{5 \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}}+\frac {3 \sqrt {2}\, \sqrt {-2 \cosh \left (\frac {x}{2}\right )^{2}+1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )}{10 \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}}-\frac {3 \sqrt {2}\, \sqrt {-2 \cosh \left (\frac {x}{2}\right )^{2}+1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \left (\operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticE}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )\right )}{5 \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}}\right )}{a^{3} \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\) | \(254\) |
2*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)/a^3*(1/20*cosh(1/2*x)/a*(a*( 2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2)/(cosh(1/2*x)^2-1/2)^3+6/5*sinh(1/2*x )^2*cosh(1/2*x)/(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)+3/10*2^(1/2)*( -2*cosh(1/2*x)^2+1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(a*(2*sinh(1/2*x)^4+sinh( 1/2*x)^2))^(1/2)*EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-3/5*2^(1/2)*(- 2*cosh(1/2*x)^2+1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)/(a*(2*sinh(1/2*x)^4+sinh(1 /2*x)^2))^(1/2)*(EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))-EllipticE(2^(1 /2)*cosh(1/2*x),1/2*2^(1/2))))/sinh(1/2*x)/(a*(2*cosh(1/2*x)^2-1))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 422, normalized size of antiderivative = 6.30 \[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{4} + 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, {\left (3 \, \cosh \left (x\right )^{6} + 18 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 3 \, \sinh \left (x\right )^{6} + {\left (45 \, \cosh \left (x\right )^{2} + 8\right )} \sinh \left (x\right )^{4} + 8 \, \cosh \left (x\right )^{4} + 4 \, {\left (15 \, \cosh \left (x\right )^{3} + 8 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (45 \, \cosh \left (x\right )^{4} + 48 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \, {\left (9 \, \cosh \left (x\right )^{5} + 16 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}\right )}}{5 \, {\left (a^{4} \cosh \left (x\right )^{6} + 6 \, a^{4} \cosh \left (x\right ) \sinh \left (x\right )^{5} + a^{4} \sinh \left (x\right )^{6} + 3 \, a^{4} \cosh \left (x\right )^{4} + 3 \, a^{4} \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{4} + a^{4} + 4 \, {\left (5 \, a^{4} \cosh \left (x\right )^{3} + 3 \, a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, a^{4} \cosh \left (x\right )^{4} + 6 \, a^{4} \cosh \left (x\right )^{2} + a^{4}\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a^{4} \cosh \left (x\right )^{5} + 2 \, a^{4} \cosh \left (x\right )^{3} + a^{4} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]
2/5*(3*(sqrt(2)*cosh(x)^6 + 6*sqrt(2)*cosh(x)*sinh(x)^5 + sqrt(2)*sinh(x)^ 6 + 3*(5*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^4 + 3*sqrt(2)*cosh(x)^4 + 4* (5*sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x))*sinh(x)^3 + 3*(5*sqrt(2)*cosh(x) ^4 + 6*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x)^2 + 3*sqrt(2)*cosh(x)^2 + 6*(s qrt(2)*cosh(x)^5 + 2*sqrt(2)*cosh(x)^3 + sqrt(2)*cosh(x))*sinh(x) + sqrt(2 ))*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sin h(x))) + 2*(3*cosh(x)^6 + 18*cosh(x)*sinh(x)^5 + 3*sinh(x)^6 + (45*cosh(x) ^2 + 8)*sinh(x)^4 + 8*cosh(x)^4 + 4*(15*cosh(x)^3 + 8*cosh(x))*sinh(x)^3 + (45*cosh(x)^4 + 48*cosh(x)^2 + 1)*sinh(x)^2 + cosh(x)^2 + 2*(9*cosh(x)^5 + 16*cosh(x)^3 + cosh(x))*sinh(x))*sqrt(a*cosh(x)))/(a^4*cosh(x)^6 + 6*a^4 *cosh(x)*sinh(x)^5 + a^4*sinh(x)^6 + 3*a^4*cosh(x)^4 + 3*a^4*cosh(x)^2 + 3 *(5*a^4*cosh(x)^2 + a^4)*sinh(x)^4 + a^4 + 4*(5*a^4*cosh(x)^3 + 3*a^4*cosh (x))*sinh(x)^3 + 3*(5*a^4*cosh(x)^4 + 6*a^4*cosh(x)^2 + a^4)*sinh(x)^2 + 6 *(a^4*cosh(x)^5 + 2*a^4*cosh(x)^3 + a^4*cosh(x))*sinh(x))
Timed out. \[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )\right )^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\int { \frac {1}{\left (a \cosh \left (x\right )\right )^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(a \cosh (x))^{7/2}} \, dx=\int \frac {1}{{\left (a\,\mathrm {cosh}\left (x\right )\right )}^{7/2}} \,d x \]