\(\int \frac {1}{x \sqrt {\text {sech}(a+b \log (c x^n))}} \, dx\) [199]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b n} \] Output:

-2*I*cosh(a+b*ln(c*x^n))^(1/2)*EllipticE(I*sinh(1/2*a+1/2*b*ln(c*x^n)),2^( 
1/2))*sech(a+b*ln(c*x^n))^(1/2)/b/n
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \] Input:

Integrate[1/(x*Sqrt[Sech[a + b*Log[c*x^n]]]),x]
 

Output:

((-2*I)*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2])/(b*n*Sqrt[Cosh[a + b*Log[c 
*x^n]]]*Sqrt[Sech[a + b*Log[c*x^n]]])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3039, 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.

\(\displaystyle \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \frac {1}{\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\sqrt {\csc \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n}\)

Input:

Int[1/(x*Sqrt[Sech[a + b*Log[c*x^n]]]),x]
 

Output:

((-2*I)*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2 
]*Sqrt[Sech[a + b*Log[c*x^n]]])/(b*n)
 

Defintions of rubi rules used

rule 3039

Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst 
[[3]]   Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /;  !FalseQ[lst]] /; 
NonsumQ[u]
 

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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(56)=112\).

Time = 0.96 (sec) , antiderivative size = 183, normalized size of antiderivative = 3.16

method	result	size
derivativedivides	\(-\frac {2 \sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\)	\(183\)
default	\(-\frac {2 \sqrt {\left (2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}-1}\, b}\)	\(183\)

Input:

int(1/x/sech(a+b*ln(c*x^n))^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-2/n*((2*cosh(1/2*a+1/2*b*ln(c*x^n))^2-1)*sinh(1/2*a+1/2*b*ln(c*x^n))^2)^( 
1/2)*(-sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-2*cosh(1/2*a+1/2*b*ln(c*x^n) 
)^2+1)^(1/2)*EllipticE(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/(2*sinh(1/2*a+ 
1/2*b*ln(c*x^n))^4+sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/sinh(1/2*a+1/2*b*l 
n(c*x^n))/(2*cosh(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (54) = 108\).

Time = 0.24 (sec) , antiderivative size = 248, normalized size of antiderivative = 4.28 \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1}} + 2 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \] Input:

integrate(1/x/sech(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")
 

Output:

-(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c 
) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 
 + 1)*sqrt((cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + 
 a))/(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a 
)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 + 1) 
) + 2*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*log(x) + 
 b*log(c) + a))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(b*n 
*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))))/(b*n*cosh(b*n 
*log(x) + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c) + a))
 

Sympy [F]

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

integrate(1/x/sech(a+b*ln(c*x**n))**(1/2),x)
 

Output:

Integral(1/(x*sqrt(sech(a + b*log(c*x**n)))), x)
 

Maxima [F]

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

integrate(1/x/sech(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(x*sqrt(sech(b*log(c*x^n) + a))), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Timed out} \] Input:

integrate(1/x/sech(a+b*log(c*x^n))^(1/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

int(1/(x*(1/cosh(a + b*log(c*x^n)))^(1/2)),x)
 

Output:

int(1/(x*(1/cosh(a + b*log(c*x^n)))^(1/2)), x)
 

Reduce [F]

\[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{\mathrm {sech}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) x}d x \] Input:

int(1/x/sech(a+b*log(c*x^n))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(sqrt(sech(log(x**n*c)*b + a))/(sech(log(x**n*c)*b + a)*x),x)