\(\int (a \text {sech}^3(x))^{3/2} \, dx\) [40]

Optimal result

Integrand size = 10, antiderivative size = 69 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {a \text {sech}^3(x)}+\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \] Output:

-10/21*I*a*cosh(x)^(3/2)*InverseJacobiAM(1/2*I*x,2^(1/2))*(a*sech(x)^3)^(1 
/2)+10/21*a*(a*sech(x)^3)^(1/2)*sinh(x)+2/7*a*sech(x)*(a*sech(x)^3)^(1/2)* 
tanh(x)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\frac {2}{21} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \left (-5 i \cosh ^{\frac {5}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+5 \cosh (x) \sinh (x)+3 \tanh (x)\right ) \] Input:

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

Output:

(2*a*Sech[x]*Sqrt[a*Sech[x]^3]*((-5*I)*Cosh[x]^(5/2)*EllipticF[(I/2)*x, 2] 
 + 5*Cosh[x]*Sinh[x] + 3*Tanh[x]))/21
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09, 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, 4255, 3042, 4255, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4611

\(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \int \text {sech}^{\frac {9}{2}}(x)dx}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)+\frac {1}{3} \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right )}}dx\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {a \sqrt {a \text {sech}^3(x)} \left (\frac {2}{7} \sinh (x) \text {sech}^{\frac {7}{2}}(x)+\frac {5}{7} \left (\frac {2}{3} \sinh (x) \text {sech}^{\frac {3}{2}}(x)-\frac {2}{3} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )\right )\right )}{\text {sech}^{\frac {3}{2}}(x)}\)

Input:

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

Output:

(a*Sqrt[a*Sech[x]^3]*((2*Sech[x]^(7/2)*Sinh[x])/7 + (5*(((-2*I)/3)*Sqrt[Co 
sh[x]]*EllipticF[(I/2)*x, 2]*Sqrt[Sech[x]] + (2*Sech[x]^(3/2)*Sinh[x])/3)) 
/7))/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 3120

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]
 

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]

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

Input:

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

Output:

int((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. 391 vs. \(2 (51) = 102\).

Time = 0.10 (sec) , antiderivative size = 391, normalized size of antiderivative = 5.67 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx =\text {Too large to display} \] Input:

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

Output:

2/21*(5*sqrt(2)*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 + 3*a*c 
osh(x)^4 + 3*(5*a*cosh(x)^2 + a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 + 3*a*cosh(x 
))*sinh(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 + 6*a*cosh(x)^2 + a)*sinh( 
x)^2 + 6*(a*cosh(x)^5 + 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)*sqrt(a)*we 
ierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + sqrt(2)*(5*a*cosh(x)^6 + 30* 
a*cosh(x)*sinh(x)^5 + 5*a*sinh(x)^6 + 17*a*cosh(x)^4 + (75*a*cosh(x)^2 + 1 
7*a)*sinh(x)^4 + 4*(25*a*cosh(x)^3 + 17*a*cosh(x))*sinh(x)^3 - 17*a*cosh(x 
)^2 + (75*a*cosh(x)^4 + 102*a*cosh(x)^2 - 17*a)*sinh(x)^2 + 2*(15*a*cosh(x 
)^5 + 34*a*cosh(x)^3 - 17*a*cosh(x))*sinh(x) - 5*a)*sqrt((a*cosh(x) + a*si 
nh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(cosh(x)^6 + 6*co 
sh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 
+ 4*(5*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1 
)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(x)^3 + cosh(x))*sinh(x) 
+ 1)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\sqrt {a}\, \left (\int \sqrt {\mathrm {sech}\left (x \right )}\, \mathrm {sech}\left (x \right )^{4}d x \right ) a \] Input:

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

Output:

sqrt(a)*int(sqrt(sech(x))*sech(x)**4,x)*a