\(\int x \sec (a+b \log (c x^n)) \, dx\) [238]

Optimal result

Integrand size = 13, antiderivative size = 87 \[ \int x \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right ),\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n} \] Output:

2*exp(I*a)*x^2*(c*x^n)^(I*b)*hypergeom([1, 1/2-I/b/n],[3/2-I/b/n],-exp(2*I 
*a)*(c*x^n)^(2*I*b))/(2+I*b*n)
 

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94 \[ \int x \sec \left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 i e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {i}{b n},\frac {3}{2}-\frac {i}{b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-2 i+b n} \] Input:

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

Output:

((-2*I)*E^(I*a)*x^2*(c*x^n)^(I*b)*Hypergeometric2F1[1, 1/2 - I/(b*n), 3/2 
- I/(b*n), -E^((2*I)*(a + b*Log[c*x^n]))])/(-2*I + b*n)
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5020, 5016, 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.

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

\(\Big \downarrow \) 5020

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

\(\Big \downarrow \) 5016

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

\(\Big \downarrow \) 888

\(\displaystyle \frac {2 e^{i a} x^2 \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {2 i}{b n}\right ),\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n}\)

Input:

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

Output:

(2*E^(I*a)*x^2*(c*x^n)^(I*b)*Hypergeometric2F1[1, (1 - (2*I)/(b*n))/2, (3 
- (2*I)/(b*n))/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/(2 + I*b*n)
 

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 5016

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

rule 5020

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(16*cos(log(x**n*c)*b + a)*int((tan((log(x**n*c)*b + a)/2)*x)/(tan((log(x* 
*n*c)*b + a)/2)**4*b**2*n**2 - 2*tan((log(x**n*c)*b + a)/2)**4 - 2*tan((lo 
g(x**n*c)*b + a)/2)**2*b**2*n**2 + 4*tan((log(x**n*c)*b + a)/2)**2 + b**2* 
n**2 - 2),x)*b**3*n**3 - 32*cos(log(x**n*c)*b + a)*int((tan((log(x**n*c)*b 
 + a)/2)*x)/(tan((log(x**n*c)*b + a)/2)**4*b**2*n**2 - 2*tan((log(x**n*c)* 
b + a)/2)**4 - 2*tan((log(x**n*c)*b + a)/2)**2*b**2*n**2 + 4*tan((log(x**n 
*c)*b + a)/2)**2 + b**2*n**2 - 2),x)*b*n + 8*cos(log(x**n*c)*b + a)*int(x/ 
(tan((log(x**n*c)*b + a)/2)**4*b**2*n**2 - 2*tan((log(x**n*c)*b + a)/2)**4 
 - 2*tan((log(x**n*c)*b + a)/2)**2*b**2*n**2 + 4*tan((log(x**n*c)*b + a)/2 
)**2 + b**2*n**2 - 2),x)*b**4*n**4 - 16*cos(log(x**n*c)*b + a)*int(x/(tan( 
(log(x**n*c)*b + a)/2)**4*b**2*n**2 - 2*tan((log(x**n*c)*b + a)/2)**4 - 2* 
tan((log(x**n*c)*b + a)/2)**2*b**2*n**2 + 4*tan((log(x**n*c)*b + a)/2)**2 
+ b**2*n**2 - 2),x)*b**2*n**2 + 2*cos(log(x**n*c)*b + a)*sec(log(x**n*c)*b 
 + a)*b**2*n**2*x**2 - 4*cos(log(x**n*c)*b + a)*sec(log(x**n*c)*b + a)*x** 
2 - cos(log(x**n*c)*b + a)*b**2*n**2*x**2 - 2*sin(log(x**n*c)*b + a)*b*n*x 
**2 - 2*b**2*n**2*x**2)/(4*cos(log(x**n*c)*b + a)*(b**2*n**2 - 2))