\(\int \frac {1}{(a \text {sech}^3(x))^{3/2}} \, dx\) [43]

Optimal result

Integrand size = 10, antiderivative size = 77 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=-\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}} \] Output:

-14/15*I*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)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\frac {-\frac {84 i E\left (\left .\frac {i x}{2}\right |2\right )}{\cosh ^{\frac {3}{2}}(x)}+33 \sinh (x)+5 \sinh (3 x)}{90 a \sqrt {a \text {sech}^3(x)}} \] Input:

Integrate[(a*Sech[x]^3)^(-3/2),x]
 

Output:

(((-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])
 

Rubi [A] (verified)

Time = 0.45 (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

\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \left (\frac {2 \sinh (x)}{9 \text {sech}^{\frac {7}{2}}(x)}+\frac {7}{9} \left (\frac {2 \sinh (x)}{5 \text {sech}^{\frac {3}{2}}(x)}-\frac {6}{5} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )\right )}{a \sqrt {a \text {sech}^3(x)}}\)

Input:

Int[(a*Sech[x]^3)^(-3/2),x]
 

Output:

(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])
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 4256

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]
 

rule 4258

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]
 

rule 4611

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]
 

Maple [F]

\[\int \frac {1}{\left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {3}{2}}}d x\]

Input:

int(1/(a*sech(x)^3)^(3/2),x)
 

Output:

int(1/(a*sech(x)^3)^(3/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 407 vs. \(2 (62) = 124\).

Time = 0.10 (sec) , antiderivative size = 407, normalized size of antiderivative = 5.29 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

integrate(1/(a*sech(x)^3)^(3/2),x, algorithm="fricas")
 

Output:

-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)
 

Sympy [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(1/(a*sech(x)**3)**(3/2),x)
 

Output:

Integral((a*sech(x)**3)**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a*sech(x)^3)^(3/2),x, algorithm="maxima")
 

Output:

integrate((a*sech(x)^3)^(-3/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate(1/(a*sech(x)^3)^(3/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((a*sech(x)^3)^(-3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{3/2}} \,d x \] Input:

int(1/(a/cosh(x)^3)^(3/2),x)
 

Output:

int(1/(a/cosh(x)^3)^(3/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\int \frac {\sqrt {\mathrm {sech}\left (x \right )}}{\mathrm {sech}\left (x \right )^{5}}d x \right )}{a^{2}} \] Input:

int(1/(a*sech(x)^3)^(3/2),x)
 

Output:

(sqrt(a)*int(sqrt(sech(x))/sech(x)**5,x))/a**2