3.2.90 \(\int \text {sech}^p(a-\frac {\log (c x^n)}{n (-2+p)}) \, dx\) [190]

3.2.90.1 Optimal result

Integrand size = 21, antiderivative size = 65 \[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {(2-p) x \left (1+e^{-2 a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \text {sech}^p\left (a+\frac {\log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

output

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

3.2.90.2 Mathematica [A] (warning: unable to verify)

Time = 1.42 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.66 \[ \int \text {sech}^p\left (a-\frac {\log \left (c x^n\right )}{n (-2+p)}\right ) \, dx=\frac {2^{-1+p} e^{-a} (-2+p) x \left (c x^n\right )^{\frac {1}{n (-2+p)}} \left (\frac {e^{\frac {a (2+p)}{-2+p}} \left (c x^n\right )^{\frac {1}{n (-2+p)}}}{e^{\frac {2 a p}{-2+p}}+e^{\frac {4 a}{-2+p}} \left (c x^n\right )^{\frac {2}{n (-2+p)}}}\right )^{-1+p}}{-1+p} \]

input

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

output

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

3.2.90.3 Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.63, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6079, 6083, 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.

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 796

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

input

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

output

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

3.2.90.3.1 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 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]
 

3.2.90.4 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)
 

3.2.90.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (55) = 110\).

Time = 0.27 (sec) , antiderivative size = 538, normalized size of antiderivative = 8.28 \[ \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*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)))*cosh(-(a*n*p - 2*a*n - n*log(x) - log(c))/(n*p - 2*n)) + (p - 2)*x*c 
osh(-(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)))
 

3.2.90.6 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)
 

3.2.90.7 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)
 

3.2.90.8 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)
 

3.2.90.9 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)