\(\int \frac {1}{x \sqrt {\text {csch}(a+b \log (c x^n))}} \, dx\) [173]

Optimal result

Integrand size = 19, antiderivative size = 70 \[ \int \frac {1}{x \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i E\left (\left .\frac {1}{4} \left (2 i a-\pi +2 i b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \] Output:

2*I*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))/b/n/csch(a+b* 
ln(c*x^n))^(1/2)/(I*sinh(a+b*ln(c*x^n)))^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\frac {2 \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (\frac {\pi }{2}-i \left (a+b \log \left (c x^n\right )\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{b n} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.03, 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.

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

\(\Big \downarrow \) 3039

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}\)

\(\Big \downarrow \) 3119

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

Input:

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

Output:

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

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (61 ) = 122\).

Time = 0.44 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.09

method	result	size
derivativedivides	\(\frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(146\)
default	\(\frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(146\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (57) = 114\).

Time = 0.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.54 \[ \int \frac {1}{x \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1}} + 2 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \] Input:

integrate(1/x/csch(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*l 
og(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))))/(b*n*cosh(b*n*l 
og(x) + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c) + a))
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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