\(\int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx\) [42]

Optimal result

Integrand size = 10, antiderivative size = 48 \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \tanh (x)}{3 \sqrt {a \text {sech}^3(x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\frac {-\frac {2 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{\cosh ^{\frac {3}{2}}(x)}+2 \tanh (x)}{3 \sqrt {a \text {sech}^3(x)}} \] Input:

Integrate[1/Sqrt[a*Sech[x]^3],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4611, 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.

\(\displaystyle \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\sqrt {a \sec (i x)^3}}dx\)

\(\Big \downarrow \) 4611

\(\displaystyle \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)}dx}{\sqrt {a \text {sech}^3(x)}}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4256

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

Input:

Int[1/Sqrt[a*Sech[x]^3],x]
 

Output:

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

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (34) = 68\).

Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.62 \[ \int \frac {1}{\sqrt {a \text {sech}^3(x)}} \, dx=\frac {4 \, \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right )^{3} \sinh \left (x\right ) + 6 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} - 1\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{6 \, {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2}\right )}} \] Input:

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

Output:

1/6*(4*sqrt(2)*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2)*sqrt(a)*weierst 
rassPInverse(-4, 0, cosh(x) + sinh(x)) + sqrt(2)*(cosh(x)^4 + 4*cosh(x)^3* 
sinh(x) + 6*cosh(x)^2*sinh(x)^2 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 - 1)*sqr 
t((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)) 
)/(a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2)
 

Sympy [F]

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

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

Output:

Integral(1/sqrt(a*sech(x)**3), x)
 

Maxima [F]

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

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

Output:

integrate(1/sqrt(a*sech(x)^3), x)
 

Giac [F]

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

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

Output:

integrate(1/sqrt(a*sech(x)^3), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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