Integrand size = 10, antiderivative size = 77 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=-\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}} \]
-14/15*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2) )/a/cosh(x)^(3/2)/(a*sech(x)^3)^(1/2)+14/45*sinh(x)/a/(a*sech(x)^3)^(1/2)+ 2/9*cosh(x)^2*sinh(x)/a/(a*sech(x)^3)^(1/2)
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\frac {-\frac {84 i E\left (\left .\frac {i x}{2}\right |2\right )}{\cosh ^{\frac {3}{2}}(x)}+33 \sinh (x)+5 \sinh (3 x)}{90 a \sqrt {a \text {sech}^3(x)}} \]
(((-84*I)*EllipticE[(I/2)*x, 2])/Cosh[x]^(3/2) + 33*Sinh[x] + 5*Sinh[3*x]) /(90*a*Sqrt[a*Sech[x]^3])
Time = 0.44 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4611, 3042, 4256, 3042, 4256, 3042, 4258, 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.
| \(\displaystyle \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sec (i x)^3\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {9}{2}}(x)}dx}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{9/2}}dx}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {7}{9} \int \frac {1}{\text {sech}^{\frac {5}{2}}(x)}dx+\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}+\frac {7}{9} \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{5/2}}dx\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {7}{9} \left (\frac {3}{5} \int \frac {1}{\sqrt {\text {sech}(x)}}dx+\frac {2 \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}\right )+\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}+\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {7}{9} \left (\frac {3}{5} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \sqrt {\cosh (x)}dx+\frac {2 \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}\right )+\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}+\frac {3}{5} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx\right )\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}-\frac {6}{5} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )\right )}{a \sqrt {a \text {sech}^3(x)}}\) |
(Sech[x]^(3/2)*((2*Sinh[x])/(9*Sech[x]^(7/2)) + (7*(((-6*I)/5)*Sqrt[Cosh[x ]]*EllipticE[(I/2)*x, 2]*Sqrt[Sech[x]] + (2*Sinh[x])/(5*Sech[x]^(3/2))))/9 ))/(a*Sqrt[a*Sech[x]^3])
3.1.43.3.1 Defintions of rubi rules used
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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
\[\int \frac {1}{\left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {3}{2}}}d x\]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 5.29 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=-\frac {672 \, \sqrt {2} {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - \sqrt {2} {\left (5 \, \cosh \left (x\right )^{10} + 50 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + 5 \, \sinh \left (x\right )^{10} + {\left (225 \, \cosh \left (x\right )^{2} + 43\right )} \sinh \left (x\right )^{8} + 43 \, \cosh \left (x\right )^{8} + 8 \, {\left (75 \, \cosh \left (x\right )^{3} + 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 2 \, {\left (525 \, \cosh \left (x\right )^{4} + 602 \, \cosh \left (x\right )^{2} - 149\right )} \sinh \left (x\right )^{6} - 298 \, \cosh \left (x\right )^{6} + 4 \, {\left (315 \, \cosh \left (x\right )^{5} + 602 \, \cosh \left (x\right )^{3} - 447 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (525 \, \cosh \left (x\right )^{6} + 1505 \, \cosh \left (x\right )^{4} - 2235 \, \cosh \left (x\right )^{2} - 187\right )} \sinh \left (x\right )^{4} - 374 \, \cosh \left (x\right )^{4} + 8 \, {\left (75 \, \cosh \left (x\right )^{7} + 301 \, \cosh \left (x\right )^{5} - 745 \, \cosh \left (x\right )^{3} - 187 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (225 \, \cosh \left (x\right )^{8} + 1204 \, \cosh \left (x\right )^{6} - 4470 \, \cosh \left (x\right )^{4} - 2244 \, \cosh \left (x\right )^{2} - 43\right )} \sinh \left (x\right )^{2} - 43 \, \cosh \left (x\right )^{2} + 2 \, {\left (25 \, \cosh \left (x\right )^{9} + 172 \, \cosh \left (x\right )^{7} - 894 \, \cosh \left (x\right )^{5} - 748 \, \cosh \left (x\right )^{3} - 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{720 \, {\left (a^{2} \cosh \left (x\right )^{5} + 5 \, a^{2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, a^{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, a^{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + a^{2} \sinh \left (x\right )^{5}\right )}} \]
-1/720*(672*sqrt(2)*(cosh(x)^5 + 5*cosh(x)^4*sinh(x) + 10*cosh(x)^3*sinh(x )^2 + 10*cosh(x)^2*sinh(x)^3 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5)*sqrt(a)*we ierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) - sqrt (2)*(5*cosh(x)^10 + 50*cosh(x)*sinh(x)^9 + 5*sinh(x)^10 + (225*cosh(x)^2 + 43)*sinh(x)^8 + 43*cosh(x)^8 + 8*(75*cosh(x)^3 + 43*cosh(x))*sinh(x)^7 + 2*(525*cosh(x)^4 + 602*cosh(x)^2 - 149)*sinh(x)^6 - 298*cosh(x)^6 + 4*(315 *cosh(x)^5 + 602*cosh(x)^3 - 447*cosh(x))*sinh(x)^5 + 2*(525*cosh(x)^6 + 1 505*cosh(x)^4 - 2235*cosh(x)^2 - 187)*sinh(x)^4 - 374*cosh(x)^4 + 8*(75*co sh(x)^7 + 301*cosh(x)^5 - 745*cosh(x)^3 - 187*cosh(x))*sinh(x)^3 + (225*co sh(x)^8 + 1204*cosh(x)^6 - 4470*cosh(x)^4 - 2244*cosh(x)^2 - 43)*sinh(x)^2 - 43*cosh(x)^2 + 2*(25*cosh(x)^9 + 172*cosh(x)^7 - 894*cosh(x)^5 - 748*co sh(x)^3 - 43*cosh(x))*sinh(x) - 5)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(a^2*cosh(x)^5 + 5*a^2*cosh(x)^4*s inh(x) + 10*a^2*cosh(x)^3*sinh(x)^2 + 10*a^2*cosh(x)^2*sinh(x)^3 + 5*a^2*c osh(x)*sinh(x)^4 + a^2*sinh(x)^5)
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{3/2}} \,d x \]