\(\int x^2 \sec ^2(a+b \log (c x^n)) \, dx\) [243]

Optimal result

Integrand size = 17, antiderivative size = 87 \[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {4 e^{2 i a} x^3 \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (2-\frac {3 i}{b n}\right ),\frac {1}{2} \left (4-\frac {3 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+2 i b n} \] Output:

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

Mathematica [A] (verified)

Time = 3.91 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.84 \[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^3 \left (3 e^{2 i a} \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (1,1-\frac {3 i}{2 b n},2-\frac {3 i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+(-3 i+2 b n) \left (-i \operatorname {Hypergeometric2F1}\left (1,-\frac {3 i}{2 b n},1-\frac {3 i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\tan \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b n (-3 i+2 b n)} \] Input:

Integrate[x^2*Sec[a + b*Log[c*x^n]]^2,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, 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.

\(\displaystyle \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 5020

\(\displaystyle \frac {x^3 \left (c x^n\right )^{-3/n} \int \left (c x^n\right )^{\frac {3}{n}-1} \sec ^2\left (a+b \log \left (c x^n\right )\right )d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 5016

\(\displaystyle \frac {4 e^{2 i a} x^3 \left (c x^n\right )^{-3/n} \int \frac {\left (c x^n\right )^{2 i b+\frac {3}{n}-1}}{\left (e^{2 i a} \left (c x^n\right )^{2 i b}+1\right )^2}d\left (c x^n\right )}{n}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {4 e^{2 i a} x^3 \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (2,\frac {1}{2} \left (2-\frac {3 i}{b n}\right ),\frac {1}{2} \left (4-\frac {3 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3+2 i b n}\)

Input:

Int[x^2*Sec[a + b*Log[c*x^n]]^2,x]
 

Output:

(4*E^((2*I)*a)*x^3*(c*x^n)^((2*I)*b)*Hypergeometric2F1[2, (2 - (3*I)/(b*n) 
)/2, (4 - (3*I)/(b*n))/2, -(E^((2*I)*a)*(c*x^n)^((2*I)*b))])/(3 + (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]

\[\int x^{2} {\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}d x\]

Input:

int(x^2*sec(a+b*ln(c*x^n))^2,x)
 

Output:

int(x^2*sec(a+b*ln(c*x^n))^2,x)
 

Fricas [F]

\[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2} \,d x } \] Input:

integrate(x^2*sec(a+b*log(c*x^n))^2,x, algorithm="fricas")
 

Output:

integral(x^2*sec(b*log(c*x^n) + a)^2, x)
 

Sympy [F]

\[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int x^{2} \sec ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate(x**2*sec(a+b*ln(c*x**n))**2,x)
 

Output:

Integral(x**2*sec(a + b*log(c*x**n))**2, x)
 

Maxima [F]

\[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2} \,d x } \] Input:

integrate(x^2*sec(a+b*log(c*x^n))^2,x, algorithm="maxima")
 

Output:

2*(x^3*cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + x^3*cos(2*b*log(c))*sin(2 
*b*log(x^n) + 2*a) - 3*(2*b^2*n^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) 
- 2*b^2*n^2*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b^2*cos(2*b*log(c)) 
^2 + b^2*sin(2*b*log(c))^2)*n^2*cos(2*b*log(x^n) + 2*a)^2 + (b^2*cos(2*b*l 
og(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*sin(2*b*log(x^n) + 2*a)^2 + b^2*n^2) 
*integrate((x^2*cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + x^2*cos(2*b*log( 
c))*sin(2*b*log(x^n) + 2*a))/(2*b^2*n^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 
 2*a) - 2*b^2*n^2*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b^2*cos(2*b*l 
og(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*cos(2*b*log(x^n) + 2*a)^2 + (b^2*cos 
(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*sin(2*b*log(x^n) + 2*a)^2 + b^ 
2*n^2), x))/(2*b*n*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + (b*cos(2*b*lo 
g(c))^2 + b*sin(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 - 2*b*n*sin(2*b 
*log(c))*sin(2*b*log(x^n) + 2*a) + (b*cos(2*b*log(c))^2 + b*sin(2*b*log(c) 
)^2)*n*sin(2*b*log(x^n) + 2*a)^2 + b*n)
 

Giac [F]

\[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { x^{2} \sec \left (b \log \left (c x^{n}\right ) + a\right )^{2} \,d x } \] Input:

integrate(x^2*sec(a+b*log(c*x^n))^2,x, algorithm="giac")
 

Output:

integrate(x^2*sec(b*log(c*x^n) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {x^2}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2} \,d x \] Input:

int(x^2/cos(a + b*log(c*x^n))^2,x)
 

Output:

int(x^2/cos(a + b*log(c*x^n))^2, x)
 

Reduce [F]

\[ \int x^2 \sec ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {too large to display} \] Input:

int(x^2*sec(a+b*log(c*x^n))^2,x)
 

Output:

( - 42*cos(log(x**n*c)*b + a)*sin(log(x**n*c)*b + a)*tan((log(x**n*c)*b + 
a)/2)**4*b**3*n**3*x**3 - 54*cos(log(x**n*c)*b + a)*sin(log(x**n*c)*b + a) 
*tan((log(x**n*c)*b + a)/2)**4*b*n*x**3 + 84*cos(log(x**n*c)*b + a)*sin(lo 
g(x**n*c)*b + a)*tan((log(x**n*c)*b + a)/2)**2*b**3*n**3*x**3 + 108*cos(lo 
g(x**n*c)*b + a)*sin(log(x**n*c)*b + a)*tan((log(x**n*c)*b + a)/2)**2*b*n* 
x**3 - 42*cos(log(x**n*c)*b + a)*sin(log(x**n*c)*b + a)*b**3*n**3*x**3 - 5 
4*cos(log(x**n*c)*b + a)*sin(log(x**n*c)*b + a)*b*n*x**3 - 252*cos(log(x** 
n*c)*b + a)*tan((log(x**n*c)*b + a)/2)**4*b**2*n**2*x**3 + 504*cos(log(x** 
n*c)*b + a)*tan((log(x**n*c)*b + a)/2)**2*b**2*n**2*x**3 - 252*cos(log(x** 
n*c)*b + a)*b**2*n**2*x**3 - 16128*int((tan((log(x**n*c)*b + a)/2)**3*x**2 
)/(14*tan((log(x**n*c)*b + a)/2)**6*b**4*n**4 - 135*tan((log(x**n*c)*b + a 
)/2)**6*b**2*n**2 + 81*tan((log(x**n*c)*b + a)/2)**6 - 42*tan((log(x**n*c) 
*b + a)/2)**4*b**4*n**4 + 405*tan((log(x**n*c)*b + a)/2)**4*b**2*n**2 - 24 
3*tan((log(x**n*c)*b + a)/2)**4 + 42*tan((log(x**n*c)*b + a)/2)**2*b**4*n* 
*4 - 405*tan((log(x**n*c)*b + a)/2)**2*b**2*n**2 + 243*tan((log(x**n*c)*b 
+ a)/2)**2 - 14*b**4*n**4 + 135*b**2*n**2 - 81),x)*sin(log(x**n*c)*b + a)* 
*2*tan((log(x**n*c)*b + a)/2)**4*b**7*n**7 + 155520*int((tan((log(x**n*c)* 
b + a)/2)**3*x**2)/(14*tan((log(x**n*c)*b + a)/2)**6*b**4*n**4 - 135*tan(( 
log(x**n*c)*b + a)/2)**6*b**2*n**2 + 81*tan((log(x**n*c)*b + a)/2)**6 - 42 
*tan((log(x**n*c)*b + a)/2)**4*b**4*n**4 + 405*tan((log(x**n*c)*b + a)/...