\(\int (a \text {sech}^3(x))^{5/2} \, dx\) [39]

Optimal result

Integrand size = 10, antiderivative size = 121 \[ \int \left (a \text {sech}^3(x)\right )^{5/2} \, dx=\frac {154}{195} i a^2 \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+\frac {154}{195} a^2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {154}{585} a^2 \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {22}{117} a^2 \text {sech}^2(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {2}{13} a^2 \text {sech}^4(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \] Output:

154/195*I*a^2*cosh(x)^(3/2)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*sech(x)^3) 
^(1/2)+154/195*a^2*cosh(x)*(a*sech(x)^3)^(1/2)*sinh(x)+154/585*a^2*(a*sech 
(x)^3)^(1/2)*tanh(x)+22/117*a^2*sech(x)^2*(a*sech(x)^3)^(1/2)*tanh(x)+2/13 
*a^2*sech(x)^4*(a*sech(x)^3)^(1/2)*tanh(x)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52 \[ \int \left (a \text {sech}^3(x)\right )^{5/2} \, dx=\frac {2}{585} a \text {sech}(x) \left (a \text {sech}^3(x)\right )^{3/2} \left (231 i \cosh ^{\frac {11}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+55 \cosh (x) \sinh (x)+77 \cosh ^3(x) \sinh (x)+231 \cosh ^5(x) \sinh (x)+45 \tanh (x)\right ) \] Input:

Integrate[(a*Sech[x]^3)^(5/2),x]
 

Output:

(2*a*Sech[x]*(a*Sech[x]^3)^(3/2)*((231*I)*Cosh[x]^(11/2)*EllipticE[(I/2)*x 
, 2] + 55*Cosh[x]*Sinh[x] + 77*Cosh[x]^3*Sinh[x] + 231*Cosh[x]^5*Sinh[x] + 
 45*Tanh[x]))/585
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.88, 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, 4255, 3042, 4255, 3042, 4255, 3042, 4255, 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.

\(\displaystyle \int \left (a \text {sech}^3(x)\right )^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \left (a \sec (i x)^3\right )^{5/2}dx\)

\(\Big \downarrow \) 4611

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \int \text {sech}^{\frac {15}{2}}(x)dx}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \int \csc \left (i x+\frac {\pi }{2}\right )^{15/2}dx}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {11}{13} \int \text {sech}^{\frac {11}{2}}(x)dx+\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)+\frac {11}{13} \int \csc \left (i x+\frac {\pi }{2}\right )^{11/2}dx\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \int \text {sech}^{\frac {7}{2}}(x)dx+\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)\right )+\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)+\frac {11}{13} \left (\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)+\frac {7}{9} \int \csc \left (i x+\frac {\pi }{2}\right )^{7/2}dx\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \int \text {sech}^{\frac {3}{2}}(x)dx+\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)\right )+\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)\right )+\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)+\frac {11}{13} \left (\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)+\frac {7}{9} \left (\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)+\frac {3}{5} \int \csc \left (i x+\frac {\pi }{2}\right )^{3/2}dx\right )\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\int \frac {1}{\sqrt {\text {sech}(x)}}dx\right )+\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)\right )+\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)\right )+\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)+\frac {11}{13} \left (\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)+\frac {7}{9} \left (\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)+\frac {3}{5} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\int \frac {1}{\sqrt {\csc \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {11}{13} \left (\frac {7}{9} \left (\frac {3}{5} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \sqrt {\cosh (x)}dx\right )+\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)\right )+\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)\right )+\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)+\frac {11}{13} \left (\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)+\frac {7}{9} \left (\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)+\frac {3}{5} \left (2 \sinh (x) \sqrt {\text {sech}(x)}-\sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \sqrt {\sin \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {a^2 \sqrt {a \text {sech}^3(x)} \left (\frac {2}{13} \sinh (x) \text {sech}^{\frac {13}{2}}(x)+\frac {11}{13} \left (\frac {2}{9} \sinh (x) \text {sech}^{\frac {9}{2}}(x)+\frac {7}{9} \left (\frac {2}{5} \sinh (x) \text {sech}^{\frac {5}{2}}(x)+\frac {3}{5} \left (2 \sinh (x) \sqrt {\text {sech}(x)}+2 i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} E\left (\left .\frac {i x}{2}\right |2\right )\right )\right )\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

Input:

Int[(a*Sech[x]^3)^(5/2),x]
 

Output:

(a^2*Sqrt[a*Sech[x]^3]*((2*Sech[x]^(13/2)*Sinh[x])/13 + (11*((2*Sech[x]^(9 
/2)*Sinh[x])/9 + (7*((2*Sech[x]^(5/2)*Sinh[x])/5 + (3*((2*I)*Sqrt[Cosh[x]] 
*EllipticE[(I/2)*x, 2]*Sqrt[Sech[x]] + 2*Sqrt[Sech[x]]*Sinh[x]))/5))/9))/1 
3))/Sech[x]^(3/2)
 

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 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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]

\[\int \left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {5}{2}}d x\]

Input:

int((a*sech(x)^3)^(5/2),x)
 

Output:

int((a*sech(x)^3)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1382 vs. \(2 (98) = 196\).

Time = 0.12 (sec) , antiderivative size = 1382, normalized size of antiderivative = 11.42 \[ \int \left (a \text {sech}^3(x)\right )^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a*sech(x)^3)^(5/2),x, algorithm="fricas")
 

Output:

2/585*(231*sqrt(2)*(a^2*cosh(x)^12 + 12*a^2*cosh(x)*sinh(x)^11 + a^2*sinh( 
x)^12 + 6*a^2*cosh(x)^10 + 6*(11*a^2*cosh(x)^2 + a^2)*sinh(x)^10 + 15*a^2* 
cosh(x)^8 + 20*(11*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^9 + 15*(33*a^2*c 
osh(x)^4 + 18*a^2*cosh(x)^2 + a^2)*sinh(x)^8 + 20*a^2*cosh(x)^6 + 24*(33*a 
^2*cosh(x)^5 + 30*a^2*cosh(x)^3 + 5*a^2*cosh(x))*sinh(x)^7 + 4*(231*a^2*co 
sh(x)^6 + 315*a^2*cosh(x)^4 + 105*a^2*cosh(x)^2 + 5*a^2)*sinh(x)^6 + 15*a^ 
2*cosh(x)^4 + 24*(33*a^2*cosh(x)^7 + 63*a^2*cosh(x)^5 + 35*a^2*cosh(x)^3 + 
 5*a^2*cosh(x))*sinh(x)^5 + 15*(33*a^2*cosh(x)^8 + 84*a^2*cosh(x)^6 + 70*a 
^2*cosh(x)^4 + 20*a^2*cosh(x)^2 + a^2)*sinh(x)^4 + 6*a^2*cosh(x)^2 + 20*(1 
1*a^2*cosh(x)^9 + 36*a^2*cosh(x)^7 + 42*a^2*cosh(x)^5 + 20*a^2*cosh(x)^3 + 
 3*a^2*cosh(x))*sinh(x)^3 + 6*(11*a^2*cosh(x)^10 + 45*a^2*cosh(x)^8 + 70*a 
^2*cosh(x)^6 + 50*a^2*cosh(x)^4 + 15*a^2*cosh(x)^2 + a^2)*sinh(x)^2 + a^2 
+ 12*(a^2*cosh(x)^11 + 5*a^2*cosh(x)^9 + 10*a^2*cosh(x)^7 + 10*a^2*cosh(x) 
^5 + 5*a^2*cosh(x)^3 + a^2*cosh(x))*sinh(x))*sqrt(a)*weierstrassZeta(-4, 0 
, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) + sqrt(2)*(231*a^2*cosh(x 
)^13 + 3003*a^2*cosh(x)*sinh(x)^12 + 231*a^2*sinh(x)^13 + 1540*a^2*cosh(x) 
^11 + 154*(117*a^2*cosh(x)^2 + 10*a^2)*sinh(x)^11 + 4367*a^2*cosh(x)^9 + 1 
694*(39*a^2*cosh(x)^3 + 10*a^2*cosh(x))*sinh(x)^10 + 11*(15015*a^2*cosh(x) 
^4 + 7700*a^2*cosh(x)^2 + 397*a^2)*sinh(x)^9 + 6808*a^2*cosh(x)^7 + 33*(90 
09*a^2*cosh(x)^5 + 7700*a^2*cosh(x)^3 + 1191*a^2*cosh(x))*sinh(x)^8 + 4...
 

Sympy [F]

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

integrate((a*sech(x)**3)**(5/2),x)
 

Output:

Integral((a*sech(x)**3)**(5/2), x)
 

Maxima [F]

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

integrate((a*sech(x)^3)^(5/2),x, algorithm="maxima")
 

Output:

integrate((a*sech(x)^3)^(5/2), x)
 

Giac [F]

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

integrate((a*sech(x)^3)^(5/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((a*sech(x)^3)^(5/2), x)
 

Mupad [F(-1)]

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

int((a/cosh(x)^3)^(5/2),x)
 

Output:

int((a/cosh(x)^3)^(5/2), x)
 

Reduce [F]

\[ \int \left (a \text {sech}^3(x)\right )^{5/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\mathrm {sech}\left (x \right )}\, \mathrm {sech}\left (x \right )^{7}d x \right ) a^{2} \] Input:

int((a*sech(x)^3)^(5/2),x)
 

Output:

sqrt(a)*int(sqrt(sech(x))*sech(x)**7,x)*a**2