\(\int \text {csch}^p(a-\frac {\log (c x^n)}{n (-2+p)}) \, dx\) [164]

Optimal result

Integrand size = 21, antiderivative size = 66 \[ \int \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {(2-p) x \left (1-e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \] Output:

(2-p)*x*(1-1/exp(2*a)/((c*x^n)^(2/n/(2-p))))*csch(a+ln(c*x^n)/n/(2-p))^p/( 
2-2*p)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(140\) vs. \(2(66)=132\).

Time = 0.85 (sec) , antiderivative size = 140, normalized size of antiderivative = 2.12 \[ \int \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {2^{-1+p} e^{-\frac {2 a p}{-2+p}} (-2+p) x \left (e^{\frac {2 a p}{-2+p}}-e^{\frac {4 a}{-2+p}} \left (c x^n\right )^{\frac {2}{n (-2+p)}}\right ) \left (-\frac {e^{\frac {a (2+p)}{-2+p}} \left (c x^n\right )^{\frac {1}{n (-2+p)}}}{-e^{\frac {2 a p}{-2+p}}+e^{\frac {4 a}{-2+p}} \left (c x^n\right )^{\frac {2}{n (-2+p)}}}\right )^p}{-1+p} \] Input:

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

Output:

(2^(-1 + p)*(-2 + p)*x*(E^((2*a*p)/(-2 + p)) - E^((4*a)/(-2 + p))*(c*x^n)^ 
(2/(n*(-2 + p))))*(-((E^((a*(2 + p))/(-2 + p))*(c*x^n)^(1/(n*(-2 + p))))/( 
-E^((2*a*p)/(-2 + p)) + E^((4*a)/(-2 + p))*(c*x^n)^(2/(n*(-2 + p))))))^p)/ 
(E^((2*a*p)/(-2 + p))*(-1 + p))
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.62, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6080, 6084, 796}

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.

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

\(\Big \downarrow \) 6080

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

\(\Big \downarrow \) 6084

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

\(\Big \downarrow \) 796

\(\displaystyle \frac {(2-p) x \left (c x^n\right )^{\frac {2 (1-p)}{n (2-p)}+\frac {p}{n (2-p)}-\frac {1}{n}} \left (1-e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\)

Input:

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

Output:

((2 - p)*x*(c*x^n)^(-n^(-1) + (2*(1 - p))/(n*(2 - p)) + p/(n*(2 - p)))*(1 
- 1/(E^(2*a)*(c*x^n)^(2/(n*(2 - p)))))*Csch[a + Log[c*x^n]/(n*(2 - p))]^p) 
/(2*(1 - p))
 

Defintions of rubi rules used

rule 796

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* 
x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, 
 p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
 

rule 6080

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> S 
imp[x/(n*(c*x^n)^(1/n))   Subst[Int[x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x 
], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1] 
)
 

rule 6084

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[Csch[d*(a + b*Log[x])]^p*((1 - 1/(E^(2*a*d)*x^(2*b*d)))^p/x^((-b)* 
d*p))   Int[(e*x)^m*(1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p)), x], x] 
 /; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]
 

Maple [F]

\[\int {\operatorname {csch}\left (a -\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )}^{p}d x\]

Input:

int(csch(a-ln(c*x^n)/n/(-2+p))^p,x)
 

Output:

int(csch(a-ln(c*x^n)/n/(-2+p))^p,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (55) = 110\).

Time = 0.10 (sec) , antiderivative size = 539, normalized size of antiderivative = 8.17 \[ \int \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=-\frac {{\left (p - 2\right )} x \cosh \left (p \log \left (-\frac {2 \, {\left (\cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) + {\left (p - 2\right )} x \sinh \left (p \log \left (-\frac {2 \, {\left (\cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )\right )}}{\cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )^{2} + 2 \, \cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) + \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )^{2} - 1}\right )\right ) \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )}{{\left (p - 1\right )} \cosh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right ) - {\left (p - 1\right )} \sinh \left (-\frac {a n p - 2 \, a n - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )} \] Input:

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

Output:

-((p - 2)*x*cosh(p*log(-2*(cosh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p 
- 2*n)) + sinh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)))/(cosh(-( 
a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n))^2 + 2*cosh(-(a*n*p - 2*a*n 
 - n*log(x) - log(c))/(n*p - 2*n))*sinh(-(a*n*p - 2*a*n - n*log(x) - log(c 
))/(n*p - 2*n)) + sinh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n))^2 
 - 1)))*sinh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)) + (p - 2)*x 
*sinh(p*log(-2*(cosh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)) + s 
inh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)))/(cosh(-(a*n*p - 2*a 
*n - n*log(x) - log(c))/(n*p - 2*n))^2 + 2*cosh(-(a*n*p - 2*a*n - n*log(x) 
 - log(c))/(n*p - 2*n))*sinh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2 
*n)) + sinh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n))^2 - 1)))*sin 
h(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)))/((p - 1)*cosh(-(a*n*p 
 - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)) - (p - 1)*sinh(-(a*n*p - 2*a*n 
- n*log(x) - log(c))/(n*p - 2*n)))
 

Sympy [F]

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

integrate(csch(a-ln(c*x**n)/n/(-2+p))**p,x)
 

Output:

Integral(csch(a - log(c*x**n)/(n*(p - 2)))**p, x)
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

integrate((-csch(-a + log(c*x^n)/(n*(p - 2))))^p, x)
 

Giac [F]

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

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

Output:

integrate(csch(a - log(c*x^n)/(n*(p - 2)))^p, x)
 

Mupad [F(-1)]

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

int((1/sinh(a - log(c*x^n)/(n*(p - 2))))^p,x)
 

Output:

int((1/sinh(a - log(c*x^n)/(n*(p - 2))))^p, x)
 

Reduce [F]

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

int(csch(a-log(c*x^n)/n/(-2+p))^p,x)
 

Output:

( - 1)**p*int(csch((log(x**n*c) - a*n*p + 2*a*n)/(n*p - 2*n))**p,x)