Integrand size = 20, antiderivative size = 90 \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=-\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
-1/2*exp(2*a)*(2-p)*x*(1-(c*x^n)^(2/n/(2-p))/exp(2*a))*csch(a-ln(c*x^n)/n/ (2-p))^p/(1-p)/((c*x^n)^(2/n/(2-p)))
Time = 1.54 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.28 \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {2^{-1+p} (-2+p) x \left (\frac {e^a \left (c x^n\right )^{\frac {1}{n (-2+p)}}}{-1+e^{2 a} \left (c x^n\right )^{\frac {2}{n (-2+p)}}}\right )^p \left (1+e^{2 a} \left (c x^n\right )^{\frac {2}{n (-2+p)}} \left (-1+\left (1-e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p\right )\right )}{-1+p} \]
(2^(-1 + p)*(-2 + p)*x*((E^a*(c*x^n)^(1/(n*(-2 + p))))/(-1 + E^(2*a)*(c*x^ n)^(2/(n*(-2 + p)))))^p*(1 + E^(2*a)*(c*x^n)^(2/(n*(-2 + p)))*(-1 + (1 - 1 /(E^(2*a)*(c*x^n)^(2/(n*(-2 + p)))))^p)))/(-1 + p)
Time = 0.36 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6080, 6084, 793}
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 |
| \(\displaystyle \frac {x \left (c x^n\right )^{-\frac {p}{n (2-p)}-\frac {1}{n}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right )^p \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right ) \int \left (c x^n\right )^{\frac {p}{2 n-n p}+\frac {1}{n}-1} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right )^{-p}d\left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 793 |
| \(\displaystyle -\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {p}{n (2-p)}-\frac {1}{n}} \left (1-e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {csch}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\) |
-1/2*(E^(2*a)*(2 - p)*x*(c*x^n)^(-n^(-1) - p/(n*(2 - p)))*(1 - (c*x^n)^(2/ (n*(2 - p)))/E^(2*a))*Csch[a - Log[c*x^n]/(n*(2 - p))]^p)/(1 - p)
3.2.63.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
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] )
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]
\[\int {\operatorname {csch}\left (a +\frac {\ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )}^{p}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (76) = 152\).
Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 5.28 \[ \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 )} \]
-((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)) + 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 - 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)))
\[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int \operatorname {csch}^{p}{\left (a + \frac {\log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]
\[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int { \operatorname {csch}\left (a + \frac {\log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p} \,d x } \]
Timed out. \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \text {csch}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\int {\left (\frac {1}{\mathrm {sinh}\left (a+\frac {\ln \left (c\,x^n\right )}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]