\(\int \frac {\sec ^3(a+b \log (c x^n))}{x^2} \, dx\) [252]

Optimal result

Integrand size = 17, antiderivative size = 87 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {8 e^{3 i a} \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {i}{b n}\right ),\frac {1}{2} \left (5+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-3 i b n) x} \]

[Out]

-8*exp(3*I*a)*(c*x^n)^(3*I*b)*hypergeom([3, 3/2+1/2*I/b/n],[5/2+1/2*I/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1-3*I
*b*n)/x

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4605, 4601, 371} \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {8 e^{3 i a} \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {i}{b n}\right ),\frac {1}{2} \left (5+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (1-3 i b n)} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^3/x^2,x]

[Out]

(-8*E^((3*I)*a)*(c*x^n)^((3*I)*b)*Hypergeometric2F1[3, (3 + I/(b*n))/2, (5 + I/(b*n))/2, -(E^((2*I)*a)*(c*x^n)
^((2*I)*b))])/((1 - (3*I)*b*n)*x)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 4601

Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[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 4605

Int[((e_.)*(x_))^(m_.)*Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = \frac {\left (c x^n\right )^{\frac {1}{n}} \text {Subst}\left (\int x^{-1-\frac {1}{n}} \sec ^3(a+b \log (x)) \, dx,x,c x^n\right )}{n x} \\ & = \frac {\left (8 e^{3 i a} \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+3 i b-\frac {1}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^3} \, dx,x,c x^n\right )}{n x} \\ & = -\frac {8 e^{3 i a} \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3+\frac {i}{b n}\right ),\frac {1}{2} \left (5+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-3 i b n) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.62 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.41 \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {-2 i e^{i a} (-i+b n) \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+\frac {i}{2 b n},\frac {3}{2}+\frac {i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec \left (a+b \log \left (c x^n\right )\right ) \left (1+b n \tan \left (a+b \log \left (c x^n\right )\right )\right )}{2 b^2 n^2 x} \]

[In]

Integrate[Sec[a + b*Log[c*x^n]]^3/x^2,x]

[Out]

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

Maple [F]

\[\int \frac {{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{x^{2}}d x\]

[In]

int(sec(a+b*ln(c*x^n))^3/x^2,x)

[Out]

int(sec(a+b*ln(c*x^n))^3/x^2,x)

Fricas [F]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^3/x^2,x, algorithm="fricas")

[Out]

integral(sec(b*log(c*x^n) + a)^3/x^2, x)

Sympy [F]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sec ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]

[In]

integrate(sec(a+b*ln(c*x**n))**3/x**2,x)

[Out]

Integral(sec(a + b*log(c*x**n))**3/x**2, x)

Maxima [F]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^3/x^2,x, algorithm="maxima")

[Out]

-((((b*cos(3*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(3*b*log(c)))*n - cos(4*b*log(c))*cos(3*b*log(c)
) - sin(4*b*log(c))*sin(3*b*log(c)))*cos(3*b*log(x^n) + 3*a) - ((b*cos(b*log(c))*sin(4*b*log(c)) - b*cos(4*b*l
og(c))*sin(b*log(c)))*n + cos(4*b*log(c))*cos(b*log(c)) + sin(4*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) -
((b*cos(4*b*log(c))*cos(3*b*log(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)))*n + cos(3*b*log(c))*sin(4*b*log(c))
- cos(4*b*log(c))*sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + ((b*cos(4*b*log(c))*cos(b*log(c)) + b*sin(4*b*log
(c))*sin(b*log(c)))*n - cos(b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*co
s(4*b*log(x^n) + 4*a) - (b*n*sin(3*b*log(c)) + 2*((b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2
*b*log(c)))*n + cos(3*b*log(c))*cos(2*b*log(c)) + sin(3*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 2
*((b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(3*b*log(c))*sin(2*b*log(c)))*n - cos(2*b*log(c))*sin(3*b*log(c))
+ cos(3*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + cos(3*b*log(c)))*cos(3*b*log(x^n) + 3*a) - 2*(((b
*cos(b*log(c))*sin(2*b*log(c)) - b*cos(2*b*log(c))*sin(b*log(c)))*n + cos(2*b*log(c))*cos(b*log(c)) + sin(2*b*
log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) - ((b*cos(2*b*log(c))*cos(b*log(c)) + b*sin(2*b*log(c))*sin(b*log(c
)))*n - cos(b*log(c))*sin(2*b*log(c)) + cos(2*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) +
2*a) + (b*n*sin(b*log(c)) - cos(b*log(c)))*cos(b*log(x^n) + a) - (((b^4*cos(4*b*log(c))^2*cos(b*log(c)) + b^4
*cos(b*log(c))*sin(4*b*log(c))^2)*n^4 + (b^2*cos(4*b*log(c))^2*cos(b*log(c)) + b^2*cos(b*log(c))*sin(4*b*log(c
))^2)*n^2)*x*cos(4*b*log(x^n) + 4*a)^2 + 4*((b^4*cos(2*b*log(c))^2*cos(b*log(c)) + b^4*cos(b*log(c))*sin(2*b*l
og(c))^2)*n^4 + (b^2*cos(2*b*log(c))^2*cos(b*log(c)) + b^2*cos(b*log(c))*sin(2*b*log(c))^2)*n^2)*x*cos(2*b*log
(x^n) + 2*a)^2 + ((b^4*cos(4*b*log(c))^2*cos(b*log(c)) + b^4*cos(b*log(c))*sin(4*b*log(c))^2)*n^4 + (b^2*cos(4
*b*log(c))^2*cos(b*log(c)) + b^2*cos(b*log(c))*sin(4*b*log(c))^2)*n^2)*x*sin(4*b*log(x^n) + 4*a)^2 + 4*((b^4*c
os(2*b*log(c))^2*cos(b*log(c)) + b^4*cos(b*log(c))*sin(2*b*log(c))^2)*n^4 + (b^2*cos(2*b*log(c))^2*cos(b*log(c
)) + b^2*cos(b*log(c))*sin(2*b*log(c))^2)*n^2)*x*sin(2*b*log(x^n) + 2*a)^2 + 4*(b^4*n^4*cos(2*b*log(c))*cos(b*
log(c)) + b^2*n^2*cos(2*b*log(c))*cos(b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - 4*(b^4*n^4*cos(b*log(c))*sin(2*b*
log(c)) + b^2*n^2*cos(b*log(c))*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) + (b^4*n^4*cos(b*log(c)) + b^2*n^2*
cos(b*log(c)))*x + 2*(2*((b^4*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b^4*cos(b*log(c))*sin(4*b*log(c)
)*sin(2*b*log(c)))*n^4 + (b^2*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b^2*cos(b*log(c))*sin(4*b*log(c)
)*sin(2*b*log(c)))*n^2)*x*cos(2*b*log(x^n) + 2*a) + 2*((b^4*cos(2*b*log(c))*cos(b*log(c))*sin(4*b*log(c)) - b^
4*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n^4 + (b^2*cos(2*b*log(c))*cos(b*log(c))*sin(4*b*log(c)) - b^
2*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n^2)*x*sin(2*b*log(x^n) + 2*a) + (b^4*n^4*cos(4*b*log(c))*cos
(b*log(c)) + b^2*n^2*cos(4*b*log(c))*cos(b*log(c)))*x)*cos(4*b*log(x^n) + 4*a) - 2*(2*((b^4*cos(2*b*log(c))*co
s(b*log(c))*sin(4*b*log(c)) - b^4*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n^4 + (b^2*cos(2*b*log(c))*co
s(b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*cos(b*log(c))*sin(2*b*log(c)))*n^2)*x*cos(2*b*log(x^n) + 2*a
) - 2*((b^4*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b^4*cos(b*log(c))*sin(4*b*log(c))*sin(2*b*log(c)))
*n^4 + (b^2*cos(4*b*log(c))*cos(2*b*log(c))*cos(b*log(c)) + b^2*cos(b*log(c))*sin(4*b*log(c))*sin(2*b*log(c)))
*n^2)*x*sin(2*b*log(x^n) + 2*a) + (b^4*n^4*cos(b*log(c))*sin(4*b*log(c)) + b^2*n^2*cos(b*log(c))*sin(4*b*log(c
)))*x)*sin(4*b*log(x^n) + 4*a))*integrate(((cos(2*b*log(c))*cos(b*log(x^n) + a) + sin(2*b*log(c))*sin(b*log(x^
n) + a))*cos(2*b*log(x^n) + 2*a) - (cos(b*log(x^n) + a)*sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(x^n) + a))
*sin(2*b*log(x^n) + 2*a) + cos(b*log(x^n) + a))/(2*b^2*n^2*x^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*b^2
*n^2*x^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*x^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*x^2*sin(2*b*log(x^n) + 2*a)^2 + b
^2*n^2*x^2), x) - (((b^4*cos(4*b*log(c))^2*sin(b*log(c)) + b^4*sin(4*b*log(c))^2*sin(b*log(c)))*n^4 + (b^2*cos
(4*b*log(c))^2*sin(b*log(c)) + b^2*sin(4*b*log(c))^2*sin(b*log(c)))*n^2)*x*cos(4*b*log(x^n) + 4*a)^2 + 4*((b^4
*cos(2*b*log(c))^2*sin(b*log(c)) + b^4*sin(2*b*log(c))^2*sin(b*log(c)))*n^4 + (b^2*cos(2*b*log(c))^2*sin(b*log
(c)) + b^2*sin(2*b*log(c))^2*sin(b*log(c)))*n^2)*x*cos(2*b*log(x^n) + 2*a)^2 + ((b^4*cos(4*b*log(c))^2*sin(b*l
og(c)) + b^4*sin(4*b*log(c))^2*sin(b*log(c)))*n^4 + (b^2*cos(4*b*log(c))^2*sin(b*log(c)) + b^2*sin(4*b*log(c))
^2*sin(b*log(c)))*n^2)*x*sin(4*b*log(x^n) + 4*a)^2 + 4*((b^4*cos(2*b*log(c))^2*sin(b*log(c)) + b^4*sin(2*b*log
(c))^2*sin(b*log(c)))*n^4 + (b^2*cos(2*b*log(c))^2*sin(b*log(c)) + b^2*sin(2*b*log(c))^2*sin(b*log(c)))*n^2)*x
*sin(2*b*log(x^n) + 2*a)^2 + 4*(b^4*n^4*cos(2*b*log(c))*sin(b*log(c)) + b^2*n^2*cos(2*b*log(c))*sin(b*log(c)))
*x*cos(2*b*log(x^n) + 2*a) - 4*(b^4*n^4*sin(2*b*log(c))*sin(b*log(c)) + b^2*n^2*sin(2*b*log(c))*sin(b*log(c)))
*x*sin(2*b*log(x^n) + 2*a) + (b^4*n^4*sin(b*log(c)) + b^2*n^2*sin(b*log(c)))*x + 2*(2*((b^4*cos(4*b*log(c))*co
s(2*b*log(c))*sin(b*log(c)) + b^4*sin(4*b*log(c))*sin(2*b*log(c))*sin(b*log(c)))*n^4 + (b^2*cos(4*b*log(c))*co
s(2*b*log(c))*sin(b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c))*sin(b*log(c)))*n^2)*x*cos(2*b*log(x^n) + 2*a
) + 2*((b^4*cos(2*b*log(c))*sin(4*b*log(c))*sin(b*log(c)) - b^4*cos(4*b*log(c))*sin(2*b*log(c))*sin(b*log(c)))
*n^4 + (b^2*cos(2*b*log(c))*sin(4*b*log(c))*sin(b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c))*sin(b*log(c)))
*n^2)*x*sin(2*b*log(x^n) + 2*a) + (b^4*n^4*cos(4*b*log(c))*sin(b*log(c)) + b^2*n^2*cos(4*b*log(c))*sin(b*log(c
)))*x)*cos(4*b*log(x^n) + 4*a) - 2*(2*((b^4*cos(2*b*log(c))*sin(4*b*log(c))*sin(b*log(c)) - b^4*cos(4*b*log(c)
)*sin(2*b*log(c))*sin(b*log(c)))*n^4 + (b^2*cos(2*b*log(c))*sin(4*b*log(c))*sin(b*log(c)) - b^2*cos(4*b*log(c)
)*sin(2*b*log(c))*sin(b*log(c)))*n^2)*x*cos(2*b*log(x^n) + 2*a) - 2*((b^4*cos(4*b*log(c))*cos(2*b*log(c))*sin(
b*log(c)) + b^4*sin(4*b*log(c))*sin(2*b*log(c))*sin(b*log(c)))*n^4 + (b^2*cos(4*b*log(c))*cos(2*b*log(c))*sin(
b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c))*sin(b*log(c)))*n^2)*x*sin(2*b*log(x^n) + 2*a) + (b^4*n^4*sin(4
*b*log(c))*sin(b*log(c)) + b^2*n^2*sin(4*b*log(c))*sin(b*log(c)))*x)*sin(4*b*log(x^n) + 4*a))*integrate(((cos(
b*log(x^n) + a)*sin(2*b*log(c)) - cos(2*b*log(c))*sin(b*log(x^n) + a))*cos(2*b*log(x^n) + 2*a) + (cos(2*b*log(
c))*cos(b*log(x^n) + a) + sin(2*b*log(c))*sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) - sin(b*log(x^n) + a))/
(2*b^2*n^2*x^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*b^2*n^2*x^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*x^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*x^2*sin(2*b*log(x^n) + 2*a)^2 + b^2*n^2*x^2), x) + (((b*cos(4*b*log(c))*cos(3*b*l
og(c)) + b*sin(4*b*log(c))*sin(3*b*log(c)))*n + cos(3*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(3*b*log(
c)))*cos(3*b*log(x^n) + 3*a) - ((b*cos(4*b*log(c))*cos(b*log(c)) + b*sin(4*b*log(c))*sin(b*log(c)))*n - cos(b*
log(c))*sin(4*b*log(c)) + cos(4*b*log(c))*sin(b*log(c)))*cos(b*log(x^n) + a) + ((b*cos(3*b*log(c))*sin(4*b*log
(c)) - b*cos(4*b*log(c))*sin(3*b*log(c)))*n - cos(4*b*log(c))*cos(3*b*log(c)) - sin(4*b*log(c))*sin(3*b*log(c)
))*sin(3*b*log(x^n) + 3*a) - ((b*cos(b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(b*log(c)))*n + cos(4*b*
log(c))*cos(b*log(c)) + sin(4*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*sin(4*b*log(x^n) + 4*a) - (b*n*cos
(3*b*log(c)) + 2*((b*cos(3*b*log(c))*cos(2*b*log(c)) + b*sin(3*b*log(c))*sin(2*b*log(c)))*n - cos(2*b*log(c))*
sin(3*b*log(c)) + cos(3*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*((b*cos(2*b*log(c))*sin(3*b*log
(c)) - b*cos(3*b*log(c))*sin(2*b*log(c)))*n + cos(3*b*log(c))*cos(2*b*log(c)) + sin(3*b*log(c))*sin(2*b*log(c)
))*sin(2*b*log(x^n) + 2*a) - sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) - 2*(((b*cos(2*b*log(c))*cos(b*log(c)) +
b*sin(2*b*log(c))*sin(b*log(c)))*n - cos(b*log(c))*sin(2*b*log(c)) + cos(2*b*log(c))*sin(b*log(c)))*cos(b*log
(x^n) + a) + ((b*cos(b*log(c))*sin(2*b*log(c)) - b*cos(2*b*log(c))*sin(b*log(c)))*n + cos(2*b*log(c))*cos(b*lo
g(c)) + sin(2*b*log(c))*sin(b*log(c)))*sin(b*log(x^n) + a))*sin(2*b*log(x^n) + 2*a) + (b*n*cos(b*log(c)) + sin
(b*log(c)))*sin(b*log(x^n) + a))/(4*b^2*n^2*x*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 4*b^2*n^2*x*sin(2*b*lo
g(c))*sin(2*b*log(x^n) + 2*a) + (b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2*x*cos(4*b*log(x^n) + 4*a)^
2 + 4*(b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*n^2*x*cos(2*b*log(x^n) + 2*a)^2 + (b^2*cos(4*b*log(c))^2
+ b^2*sin(4*b*log(c))^2)*n^2*x*sin(4*b*log(x^n) + 4*a)^2 + 4*(b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*
n^2*x*sin(2*b*log(x^n) + 2*a)^2 + b^2*n^2*x + 2*(b^2*n^2*x*cos(4*b*log(c)) + 2*(b^2*cos(4*b*log(c))*cos(2*b*lo
g(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2*x*cos(2*b*log(x^n) + 2*a) + 2*(b^2*cos(2*b*log(c))*sin(4*b*lo
g(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2*x*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 2*(b^2*n
^2*x*sin(4*b*log(c)) + 2*(b^2*cos(2*b*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2*x*cos
(2*b*log(x^n) + 2*a) - 2*(b^2*cos(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2*x*sin
(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

Giac [F]

\[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^3/x^2,x, algorithm="giac")

[Out]

integrate(sec(b*log(c*x^n) + a)^3/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {1}{x^2\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^3} \,d x \]

[In]

int(1/(x^2*cos(a + b*log(c*x^n))^3),x)

[Out]

int(1/(x^2*cos(a + b*log(c*x^n))^3), x)