\(\int \frac {\text {csch}^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\) [170]

Optimal result

Integrand size = 19, antiderivative size = 109 \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 i \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{4} \left (2 i a-\pi +2 i b \log \left (c x^n\right )\right ),2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \] Output:

-2/3*cosh(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))^(3/2)/b/n+2/3*I*csch(a+b*ln(c 
*x^n))^(1/2)*InverseJacobiAM(1/2*I*a-1/4*Pi+1/2*I*b*ln(c*x^n),2^(1/2))*(I* 
sinh(a+b*ln(c*x^n)))^(1/2)/b/n
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.77 \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \left (\coth \left (a+b \log \left (c x^n\right )\right )+i \operatorname {EllipticF}\left (\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right ),2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \] Input:

Integrate[Csch[a + b*Log[c*x^n]]^(5/2)/x,x]
 

Output:

(-2*Sqrt[Csch[a + b*Log[c*x^n]]]*(Coth[a + b*Log[c*x^n]] + I*EllipticF[((- 
2*I)*a + Pi - (2*I)*b*Log[c*x^n])/4, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))/( 
3*b*n)
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3039, 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.

\(\displaystyle \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \left (i \csc \left (i a+i b \log \left (c x^n\right )\right )\right )^{5/2}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {-\frac {1}{3} \int \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}-\frac {1}{3} \int \sqrt {i \csc \left (i a+i b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}-\frac {1}{3} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \int \frac {1}{\sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}-\frac {1}{3} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \int \frac {1}{\sqrt {\sin \left (i a+i b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b}+\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b}}{n}\)

Input:

Int[Csch[a + b*Log[c*x^n]]^(5/2)/x,x]
 

Output:

((-2*Cosh[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^(3/2))/(3*b) + (((2*I)/ 
3)*Sqrt[Csch[a + b*Log[c*x^n]]]*EllipticF[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 
 2]*Sqrt[I*Sinh[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 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]
 

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.32

method	result	size
derivativedivides	\(-\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\)	\(144\)
default	\(-\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\)	\(144\)

Input:

int(csch(a+b*ln(c*x^n))^(5/2)/x,x,method=_RETURNVERBOSE)
 

Output:

-1/3/n/sinh(a+b*ln(c*x^n))^(3/2)*(I*(1-I*sinh(a+b*ln(c*x^n)))^(1/2)*2^(1/2 
)*(1+I*sinh(a+b*ln(c*x^n)))^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/2)*EllipticF( 
(1-I*sinh(a+b*ln(c*x^n)))^(1/2),1/2*2^(1/2))*sinh(a+b*ln(c*x^n))+2*cosh(a+ 
b*ln(c*x^n))^2)/cosh(a+b*ln(c*x^n))/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (89) = 178\).

Time = 0.10 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.92 \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1}} + {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )}} \] Input:

integrate(csch(a+b*log(c*x^n))^(5/2)/x,x, algorithm="fricas")
 

Output:

-2/3*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*l 
og(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)) + (sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*sqrt(2)*cosh(b*n*lo 
g(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*lo 
g(x) + b*log(c) + a)^2 - sqrt(2))*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)^2 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) 
 + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c) + a)^2 - b*n)
 

Sympy [F]

\[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:

integrate(csch(a+b*ln(c*x**n))**(5/2)/x,x)
 

Output:

Integral(csch(a + b*log(c*x**n))**(5/2)/x, x)
 

Maxima [F]

\[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \] Input:

integrate(csch(a+b*log(c*x^n))^(5/2)/x,x, algorithm="maxima")
 

Output:

integrate(csch(b*log(c*x^n) + a)^(5/2)/x, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \] Input:

integrate(csch(a+b*log(c*x^n))^(5/2)/x,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \] Input:

int((1/sinh(a + b*log(c*x^n)))^(5/2)/x,x)
 

Output:

int((1/sinh(a + b*log(c*x^n)))^(5/2)/x, x)
 

Reduce [F]

\[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}\, {\mathrm {csch}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}}{x}d x \] Input:

int(csch(a+b*log(c*x^n))^(5/2)/x,x)
 

Output:

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