\(\int \sec ^p(a+b \log (c x^n)) \, dx\) [288]

Optimal result

Integrand size = 13, antiderivative size = 107 \[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,-\frac {i-b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}+p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{1+i b n p} \] Output:

x*(1+exp(2*I*a)*(c*x^n)^(2*I*b))^p*hypergeom([p, -1/2*(I-b*n*p)/b/n],[1-1/ 
2*I/b/n+1/2*p],-exp(2*I*a)*(c*x^n)^(2*I*b))*sec(a+b*ln(c*x^n))^p/(1+I*b*n* 
p)
 

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.33 \[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {i 2^p x \left (\frac {e^{i a} \left (c x^n\right )^{i b}}{1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {-i+b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}+p\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-i+b n p} \] Input:

Integrate[Sec[a + b*Log[c*x^n]]^p,x]
 

Output:

((-I)*2^p*x*((E^(I*a)*(c*x^n)^(I*b))/(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)))^ 
p*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*Hypergeometric2F1[p, (-I + b*n*p)/ 
(2*b*n), (2 - I/(b*n) + p)/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/(-I + b*n 
*p)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5014, 5018, 888}

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.

\(\displaystyle \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 5014

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

\(\Big \downarrow \) 5018

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {x \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,-\frac {i-b n p}{2 b n},\frac {1}{2} \left (p-\frac {i}{b n}+2\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sec ^p\left (a+b \log \left (c x^n\right )\right )}{n \left (\frac {1}{n}+i b p\right )}\)

Input:

Int[Sec[a + b*Log[c*x^n]]^p,x]
 

Output:

(x*(1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*Hypergeometric2F1[p, -1/2*(I - b* 
n*p)/(b*n), (2 - I/(b*n) + p)/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Sec[a + 
 b*Log[c*x^n]]^p)/(n*(n^(-1) + I*b*p))
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 5014

Int[Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Si 
mp[x/(n*(c*x^n)^(1/n))   Subst[Int[x^(1/n - 1)*Sec[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 5018

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

Maple [F]

\[\int {\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]

Input:

int(sec(a+b*ln(c*x^n))^p,x)
 

Output:

int(sec(a+b*ln(c*x^n))^p,x)
 

Fricas [F]

\[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sec \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \] Input:

integrate(sec(a+b*log(c*x^n))^p,x, algorithm="fricas")
 

Output:

integral(sec(b*log(c*x^n) + a)^p, x)
 

Sympy [F]

\[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \sec ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate(sec(a+b*ln(c*x**n))**p,x)
 

Output:

Integral(sec(a + b*log(c*x**n))**p, x)
 

Maxima [F]

\[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sec \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \] Input:

integrate(sec(a+b*log(c*x^n))^p,x, algorithm="maxima")
 

Output:

integrate(sec(b*log(c*x^n) + a)^p, x)
 

Giac [F]

\[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \sec \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \] Input:

integrate(sec(a+b*log(c*x^n))^p,x, algorithm="giac")
 

Output:

integrate(sec(b*log(c*x^n) + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^p \,d x \] Input:

int((1/cos(a + b*log(c*x^n)))^p,x)
 

Output:

int((1/cos(a + b*log(c*x^n)))^p, x)
 

Reduce [F]

\[ \int \sec ^p\left (a+b \log \left (c x^n\right )\right ) \, dx={\sec \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} x -\left (\int {\sec \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{p} \tan \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )d x \right ) b n p \] Input:

int(sec(a+b*log(c*x^n))^p,x)
 

Output:

sec(log(x**n*c)*b + a)**p*x - int(sec(log(x**n*c)*b + a)**p*tan(log(x**n*c 
)*b + a),x)*b*n*p