Integrand size = 10, antiderivative size = 121 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=-\frac {26 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}} \] Output:
-26/77*I*InverseJacobiAM(1/2*I*x,2^(1/2))/a^2/cosh(x)^(3/2)/(a*sech(x)^3)^ (1/2)+78/385*cosh(x)*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+26/165*cosh(x)^3*sinh (x)/a^2/(a*sech(x)^3)^(1/2)+2/15*cosh(x)^5*sinh(x)/a^2/(a*sech(x)^3)^(1/2) +26/77*tanh(x)/a^2/(a*sech(x)^3)^(1/2)
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\frac {\cosh (x) \sqrt {a \text {sech}^3(x)} \left (-24960 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+19122 \sinh (2 x)+4406 \sinh (4 x)+826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \] Input:
Integrate[(a*Sech[x]^3)^(-5/2),x]
Output:
(Cosh[x]*Sqrt[a*Sech[x]^3]*((-24960*I)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2] + 19122*Sinh[2*x] + 4406*Sinh[4*x] + 826*Sinh[6*x] + 77*Sinh[8*x]))/(7392 0*a^3)
Time = 0.64 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {3042, 4611, 3042, 4256, 3042, 4256, 3042, 4256, 3042, 4256, 3042, 4258, 3042, 3120}
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.
| \(\displaystyle \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \sec (i x)^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4611 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {15}{2}}(x)}dx}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{15/2}}dx}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {13}{15} \int \frac {1}{\text {sech}^{\frac {11}{2}}(x)}dx+\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}+\frac {13}{15} \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{11/2}}dx\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {13}{15} \left (\frac {9}{11} \int \frac {1}{\text {sech}^{\frac {7}{2}}(x)}dx+\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}+\frac {13}{15} \left (\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}+\frac {9}{11} \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{7/2}}dx\right )\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)}dx+\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}\right )+\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}+\frac {13}{15} \left (\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}+\frac {9}{11} \left (\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}+\frac {5}{7} \int \frac {1}{\csc \left (i x+\frac {\pi }{2}\right )^{3/2}}dx\right )\right )\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \int \sqrt {\text {sech}(x)}dx+\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}\right )+\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}\right )+\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}+\frac {13}{15} \left (\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}+\frac {9}{11} \left (\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}+\frac {5}{7} \left (\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \int \sqrt {\csc \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\right )\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {13}{15} \left (\frac {9}{11} \left (\frac {5}{7} \left (\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\cosh (x)}}dx+\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}\right )+\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}\right )+\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}\right )+\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}+\frac {13}{15} \left (\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}+\frac {9}{11} \left (\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}+\frac {5}{7} \left (\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\right )\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{15 \text {sech}^{\frac {13}{2}}(x)}+\frac {13}{15} \left (\frac {2 \sinh (x)}{11 \text {sech}^{\frac {9}{2}}(x)}+\frac {9}{11} \left (\frac {2 \sinh (x)}{7 \text {sech}^{\frac {5}{2}}(x)}+\frac {5}{7} \left (\frac {2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {2}{3} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )\right )\right )\right )\right )}{a^2 \sqrt {a \text {sech}^3(x)}}\) |
Input:
Int[(a*Sech[x]^3)^(-5/2),x]
Output:
(Sech[x]^(3/2)*((2*Sinh[x])/(15*Sech[x]^(13/2)) + (13*((2*Sinh[x])/(11*Sec h[x]^(9/2)) + (9*((2*Sinh[x])/(7*Sech[x]^(5/2)) + (5*(((-2*I)/3)*Sqrt[Cosh [x]]*EllipticF[(I/2)*x, 2]*Sqrt[Sech[x]] + (2*Sinh[x])/(3*Sqrt[Sech[x]]))) /7))/11))/15))/(a^2*Sqrt[a*Sech[x]^3])
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(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]
\[\int \frac {1}{\left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]
Input:
int(1/(a*sech(x)^3)^(5/2),x)
Output:
int(1/(a*sech(x)^3)^(5/2),x)
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (95) = 190\).
Time = 0.12 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.93 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="fricas")
Output:
1/147840*(49920*sqrt(2)*(cosh(x)^8 + 8*cosh(x)^7*sinh(x) + 28*cosh(x)^6*si nh(x)^2 + 56*cosh(x)^5*sinh(x)^3 + 70*cosh(x)^4*sinh(x)^4 + 56*cosh(x)^3*s inh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8)*sqrt( a)*weierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + sqrt(2)*(77*cosh(x)^16 + 1232*cosh(x)*sinh(x)^15 + 77*sinh(x)^16 + 14*(660*cosh(x)^2 + 59)*sinh(x )^14 + 826*cosh(x)^14 + 196*(220*cosh(x)^3 + 59*cosh(x))*sinh(x)^13 + 2*(7 0070*cosh(x)^4 + 37583*cosh(x)^2 + 2203)*sinh(x)^12 + 4406*cosh(x)^12 + 8* (42042*cosh(x)^5 + 37583*cosh(x)^3 + 6609*cosh(x))*sinh(x)^11 + 2*(308308* cosh(x)^6 + 413413*cosh(x)^4 + 145398*cosh(x)^2 + 9561)*sinh(x)^10 + 19122 *cosh(x)^10 + 4*(220220*cosh(x)^7 + 413413*cosh(x)^5 + 242330*cosh(x)^3 + 47805*cosh(x))*sinh(x)^9 + 6*(165165*cosh(x)^8 + 413413*cosh(x)^6 + 363495 *cosh(x)^4 + 143415*cosh(x)^2)*sinh(x)^8 + 16*(55055*cosh(x)^9 + 177177*co sh(x)^7 + 218097*cosh(x)^5 + 143415*cosh(x)^3)*sinh(x)^7 + 2*(308308*cosh( x)^10 + 1240239*cosh(x)^8 + 2035572*cosh(x)^6 + 2007810*cosh(x)^4 - 9561)* sinh(x)^6 - 19122*cosh(x)^6 + 4*(84084*cosh(x)^11 + 413413*cosh(x)^9 + 872 388*cosh(x)^7 + 1204686*cosh(x)^5 - 28683*cosh(x))*sinh(x)^5 + 2*(70070*co sh(x)^12 + 413413*cosh(x)^10 + 1090485*cosh(x)^8 + 2007810*cosh(x)^6 - 143 415*cosh(x)^2 - 2203)*sinh(x)^4 - 4406*cosh(x)^4 + 8*(5390*cosh(x)^13 + 37 583*cosh(x)^11 + 121165*cosh(x)^9 + 286830*cosh(x)^7 - 47805*cosh(x)^3 - 2 203*cosh(x))*sinh(x)^3 + 2*(4620*cosh(x)^14 + 37583*cosh(x)^12 + 145398...
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(a*sech(x)**3)**(5/2),x)
Output:
Integral((a*sech(x)**3)**(-5/2), x)
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="maxima")
Output:
integrate((a*sech(x)^3)^(-5/2), x)
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="giac")
Output:
integrate((a*sech(x)^3)^(-5/2), x)
Timed out. \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{5/2}} \,d x \] Input:
int(1/(a/cosh(x)^3)^(5/2),x)
Output:
int(1/(a/cosh(x)^3)^(5/2), x)
\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\mathrm {sech}\left (x \right )}}{\mathrm {sech}\left (x \right )^{8}}d x \right )}{a^{3}} \] Input:
int(1/(a*sech(x)^3)^(5/2),x)
Output:
(sqrt(a)*int(sqrt(sech(x))/sech(x)**8,x))/a**3