\(\int \frac {1}{x \text {csch}^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [175]

Optimal result

Integrand size = 19, antiderivative size = 109 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{4} \left (2 i a-\pi +2 i b \log \left (c x^n\right )\right )\right |2\right )}{5 b n \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}} \] Output:

2/5*cosh(a+b*ln(c*x^n))/b/n/csch(a+b*ln(c*x^n))^(3/2)-6/5*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 = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.87 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (\cosh \left (a+b \log \left (c x^n\right )\right )-3 \text {csch}^2\left (a+b \log \left (c x^n\right )\right ) E\left (\left .\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right )}{5 b n \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \] Input:

Integrate[1/(x*Csch[a + b*Log[c*x^n]]^(5/2)),x]
 

Output:

(2*(Cosh[a + b*Log[c*x^n]] - 3*Csch[a + b*Log[c*x^n]]^2*EllipticE[((-2*I)* 
a + Pi - (2*I)*b*Log[c*x^n])/4, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))/(5*b*n 
*Csch[a + b*Log[c*x^n]]^(3/2))
 

Rubi [A] (verified)

Time = 0.43 (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, 4256, 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.

\(\displaystyle \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \frac {1}{\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\left (i \csc \left (i a+i b \log \left (c x^n\right )\right )\right )^{5/2}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3}{5} \int \frac {1}{\sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3}{5} \int \frac {1}{\sqrt {i \csc \left (i a+i b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3 \int \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{5 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {3 \int \sqrt {\sin \left (i a+i b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{5 \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}}{n}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{5 b \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}}{n}\)

Input:

Int[1/(x*Csch[a + b*Log[c*x^n]]^(5/2)),x]
 

Output:

((2*Cosh[a + b*Log[c*x^n]])/(5*b*Csch[a + b*Log[c*x^n]]^(3/2)) + (((6*I)/5 
)*EllipticE[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2])/(b*Sqrt[Csch[a + b*Log[c* 
x^n]]]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))/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 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 226 vs. \(2 (94 ) = 188\).

Time = 0.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 2.08

method	result	size
derivativedivides	\(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(227\)
default	\(\frac {-\frac {6 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{5}-\frac {2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{5}}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(227\)

Input:

int(1/x/csch(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/n*(-6/5*(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)*EllipticE((1-I*sinh(a+b*ln(c*x^n)))^ 
(1/2),1/2*2^(1/2))+3/5*(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))+2/5*cosh(a+b*ln(c*x^n))^4-2/5*cosh(a+b*ln( 
c*x^n))^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. 602 vs. \(2 (90) = 180\).

Time = 0.10 (sec) , antiderivative size = 602, normalized size of antiderivative = 5.52 \[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")
 

Output:

1/20*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^6 + 6*cosh(b*n*log(x) + b*l 
og(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^5 + sinh(b*n*log(x) + b*log(c) 
+ a)^6 + (15*cosh(b*n*log(x) + b*log(c) + a)^2 + 11)*sinh(b*n*log(x) + b*l 
og(c) + a)^4 + 11*cosh(b*n*log(x) + b*log(c) + a)^4 + 4*(5*cosh(b*n*log(x) 
 + b*log(c) + a)^3 + 11*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + 
 b*log(c) + a)^3 + (15*cosh(b*n*log(x) + b*log(c) + a)^4 + 66*cosh(b*n*log 
(x) + b*log(c) + a)^2 - 13)*sinh(b*n*log(x) + b*log(c) + a)^2 - 13*cosh(b* 
n*log(x) + b*log(c) + a)^2 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^5 + 22*c 
osh(b*n*log(x) + b*log(c) + a)^3 - 13*cosh(b*n*log(x) + b*log(c) + a))*sin 
h(b*n*log(x) + b*log(c) + a) + 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*co 
sh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*l 
og(x) + b*log(c) + a)^2 - 1)) + 24*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a 
)^3 + 3*sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log( 
c) + a) + 3*sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*lo 
g(c) + a)^2 + sqrt(2)*sinh(b*n*log(x) + b*log(c) + a)^3)*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)^3 + 3*b*n*c 
osh(b*n*log(x) + b*log(c) + a)^2*sinh(b*n*log(x) + b*log(c) + a) + 3*b*n*c 
osh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^2 + b*n*...
 

Sympy [F]

\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \operatorname {csch}^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \] Input:

integrate(1/x/csch(a+b*ln(c*x**n))**(5/2),x)
 

Output:

Integral(1/(x*csch(a + b*log(c*x**n))**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-1)]

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

integrate(1/x/csch(a+b*log(c*x^n))^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{x \text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{{\mathrm {csch}\left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} x}d x \] Input:

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

Output:

int(sqrt(csch(log(x**n*c)*b + a))/(csch(log(x**n*c)*b + a)**3*x),x)