\(\int \text {sech}^p(a+\frac {\log (c x^n)}{n (-2+p)}) \, dx\) [189]

Optimal result

Integrand size = 20, antiderivative size = 89 \[ \int \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {e^{2 a} (2-p) x \left (c x^n\right )^{-\frac {2}{n (2-p)}} \left (1+e^{-2 a} \left (c x^n\right )^{\frac {2}{n (2-p)}}\right ) \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \] Output:

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

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.28 \[ \int \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=-\frac {2^{-1+p} (-2+p) x \left (\frac {e^a \left (c x^n\right )^{\frac {1}{2 n-n p}}}{e^{2 a}+\left (c x^n\right )^{-\frac {2}{n (-2+p)}}}\right )^p \left (-1+e^{2 a} \left (c x^n\right )^{\frac {2}{n (-2+p)}} \left (-1+\left (1+e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (-2+p)}}\right )^p\right )\right )}{-1+p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 96, 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 = {6079, 6083, 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.

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

\(\Big \downarrow \) 6079

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

\(\Big \downarrow \) 6083

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

\(\Big \downarrow \) 793

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 793

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]
 

rule 6079

Int[Sech[((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)*Sech[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 6083

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] 
 :> Simp[Sech[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]

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

Input:

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

Output:

int(sech(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. 474 vs. \(2 (76) = 152\).

Time = 0.25 (sec) , antiderivative size = 474, normalized size of antiderivative = 5.33 \[ \int \text {sech}^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 ) \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 - 2\right )} x \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 (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 )}{{\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(sech(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*l 
og(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)))* 
cosh((a*n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n)) + (p - 2)*x*cosh((a* 
n*p - 2*a*n + n*log(x) + log(c))/(n*p - 2*n))*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))))/((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]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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