\(\int \frac {\sec ^4(a+b \log (c x^n))}{x^2} \, dx\) [257]

Optimal result

Integrand size = 17, antiderivative size = 87 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {16 e^{4 i a} \left (c x^n\right )^{4 i b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4+\frac {i}{b n}\right ),\frac {1}{2} \left (6+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-4 i b n) x} \]

[Out]

-16*exp(4*I*a)*(c*x^n)^(4*I*b)*hypergeom([4, 2+1/2*I/b/n],[3+1/2*I/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1-4*I*b*
n)/x

Rubi [A] (verified)

Time = 0.08 (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} \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {16 e^{4 i a} \left (c x^n\right )^{4 i b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4+\frac {i}{b n}\right ),\frac {1}{2} \left (6+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (1-4 i b n)} \]

[In]

Int[Sec[a + b*Log[c*x^n]]^4/x^2,x]

[Out]

(-16*E^((4*I)*a)*(c*x^n)^((4*I)*b)*Hypergeometric2F1[4, (4 + I/(b*n))/2, (6 + I/(b*n))/2, -(E^((2*I)*a)*(c*x^n
)^((2*I)*b))])/((1 - (4*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 ^4(a+b \log (x)) \, dx,x,c x^n\right )}{n x} \\ & = \frac {\left (16 e^{4 i a} \left (c x^n\right )^{\frac {1}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+4 i b-\frac {1}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^4} \, dx,x,c x^n\right )}{n x} \\ & = -\frac {16 e^{4 i a} \left (c x^n\right )^{4 i b} \operatorname {Hypergeometric2F1}\left (4,\frac {1}{2} \left (4+\frac {i}{b n}\right ),\frac {1}{2} \left (6+\frac {i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-4 i b n) x} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(87)=174\).

Time = 7.00 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.47 \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {-2 e^{2 i a} (-i+2 b n) \left (c x^n\right )^{2 i b} \operatorname {Hypergeometric2F1}\left (1,1+\frac {i}{2 b n},2+\frac {i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-2 i \left (1+4 b^2 n^2\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {i}{2 b n},1+\frac {i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec ^2\left (a+b \log \left (c x^n\right )\right ) \left (2 b n+\left (1+8 b^2 n^2+\left (1+4 b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )\right )}{12 b^3 n^3 x} \]

[In]

Integrate[Sec[a + b*Log[c*x^n]]^4/x^2,x]

[Out]

(-2*E^((2*I)*a)*(-I + 2*b*n)*(c*x^n)^((2*I)*b)*Hypergeometric2F1[1, 1 + (I/2)/(b*n), 2 + (I/2)/(b*n), -E^((2*I
)*(a + b*Log[c*x^n]))] - (2*I)*(1 + 4*b^2*n^2)*Hypergeometric2F1[1, (I/2)/(b*n), 1 + (I/2)/(b*n), -E^((2*I)*(a
+ b*Log[c*x^n]))] + Sec[a + b*Log[c*x^n]]^2*(2*b*n + (1 + 8*b^2*n^2 + (1 + 4*b^2*n^2)*Cos[2*(a + b*Log[c*x^n]
)])*Tan[a + b*Log[c*x^n]]))/(12*b^3*n^3*x)

Maple [F]

\[\int \frac {{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{x^{2}}d x\]

[In]

int(sec(a+b*ln(c*x^n))^4/x^2,x)

[Out]

int(sec(a+b*ln(c*x^n))^4/x^2,x)

Fricas [F]

\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{2}} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^4/x^2,x, algorithm="fricas")

[Out]

integral(sec(b*log(c*x^n) + a)^4/x^2, x)

Sympy [F]

\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sec ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]

[In]

integrate(sec(a+b*ln(c*x**n))**4/x**2,x)

[Out]

Integral(sec(a + b*log(c*x**n))**4/x**2, x)

Maxima [F]

\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{2}} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^4/x^2,x, algorithm="maxima")

[Out]

1/3*(6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b*si
n(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 + 6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n
) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(2*b*log(x^n) + 2*a)^2 + (4*b^2*n^2*sin(6*b*lo
g(c)) + (2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n + cos(4*b*log(c))*sin(6*b
*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 2*(6*(b^2*cos(2*b*log(c))*sin(6*b*log(c)
) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*
log(c)))*n + cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + (2*(
b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n - cos(6*b*log(c))*cos(4*b*log(c)) - s
in(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 2*(6*(b^2*cos(6*b*log(c))*cos(2*b*log(c)) + b^2*sin(
6*b*log(c))*sin(2*b*log(c)))*n^2 - (b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n +
cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))
*cos(6*b*log(x^n) + 6*a) + (12*b^2*n^2*sin(4*b*log(c)) + 2*b*n*cos(4*b*log(c)) + 3*(12*(b^2*cos(2*b*log(c))*si
n(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 + 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*lo
g(c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n)
+ 2*a) - 3*(12*(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 - 4*(b*cos(2*b
*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*lo
g(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + 2*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + (2*b*n*cos(2*b*l
og(c)) + sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 9*((4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8
+ (b^6*cos(6*b*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*x*cos(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))
^2 + b^8*sin(4*b*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*x*cos(4*b*log(x^n) + 4*
a)^2 + 9*(4*(b^8*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))
^2)*n^6)*x*cos(2*b*log(x^n) + 2*a)^2 + (4*(b^8*cos(6*b*log(c))^2 + b^8*sin(6*b*log(c))^2)*n^8 + (b^6*cos(6*b*l
og(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*x*sin(6*b*log(x^n) + 6*a)^2 + 9*(4*(b^8*cos(4*b*log(c))^2 + b^8*sin(4*b
*log(c))^2)*n^8 + (b^6*cos(4*b*log(c))^2 + b^6*sin(4*b*log(c))^2)*n^6)*x*sin(4*b*log(x^n) + 4*a)^2 + 9*(4*(b^8
*cos(2*b*log(c))^2 + b^8*sin(2*b*log(c))^2)*n^8 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6)*x*sin(2
*b*log(x^n) + 2*a)^2 + 6*(4*b^8*n^8*cos(2*b*log(c)) + b^6*n^6*cos(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - 6*(
4*b^8*n^8*sin(2*b*log(c)) + b^6*n^6*sin(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) + (4*b^8*n^8 + b^6*n^6)*x + 2*(
3*(4*(b^8*cos(6*b*log(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*co
s(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*x*cos(4*b*log(x^n) + 4*a) + 3*(4*(b^8*cos(6*b*log(c)
)*cos(2*b*log(c)) + b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(
6*b*log(c))*sin(2*b*log(c)))*n^6)*x*cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*
cos(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(
c)))*n^6)*x*sin(4*b*log(x^n) + 4*a) + 3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*
log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*x*sin(2*b*log(
x^n) + 2*a) + (4*b^8*n^8*cos(6*b*log(c)) + b^6*n^6*cos(6*b*log(c)))*x)*cos(6*b*log(x^n) + 6*a) + 6*(3*(4*(b^8*
cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*log(
c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*x*cos(2*b*log(x^n) + 2*a) + 3*(4*(b^8*cos(2*b*log(c))*sin(4*b*
log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c)
)*sin(2*b*log(c)))*n^6)*x*sin(2*b*log(x^n) + 2*a) + (4*b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(c)))*x)*c
os(4*b*log(x^n) + 4*a) - 2*(3*(4*(b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(4*b*log(c)))*n
^8 + (b^6*cos(4*b*log(c))*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(4*b*log(c)))*n^6)*x*cos(4*b*log(x^n) + 4*a
) + 3*(4*(b^8*cos(2*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(2*b*log(c)
)*sin(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*x*cos(2*b*log(x^n) + 2*a) - 3*(4*(b^8*cos(6*b*lo
g(c))*cos(4*b*log(c)) + b^8*sin(6*b*log(c))*sin(4*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(4*b*log(c)) + b^6*
sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*x*sin(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(6*b*log(c))*cos(2*b*log(c)) +
b^8*sin(6*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(6*b*log(c))*cos(2*b*log(c)) + b^6*sin(6*b*log(c))*sin(2*b*
log(c)))*n^6)*x*sin(2*b*log(x^n) + 2*a) + (4*b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*log(c)))*x)*sin(6*b*log
(x^n) + 6*a) - 6*(3*(4*(b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*
cos(2*b*log(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*x*cos(2*b*log(x^n) + 2*a) - 3*(4*(
b^8*cos(4*b*log(c))*cos(2*b*log(c)) + b^8*sin(4*b*log(c))*sin(2*b*log(c)))*n^8 + (b^6*cos(4*b*log(c))*cos(2*b*
log(c)) + b^6*sin(4*b*log(c))*sin(2*b*log(c)))*n^6)*x*sin(2*b*log(x^n) + 2*a) + (4*b^8*n^8*sin(4*b*log(c)) + b
^6*n^6*sin(4*b*log(c)))*x)*sin(4*b*log(x^n) + 4*a))*integrate(1/9*(cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + c
os(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(2*b^6*n^6*x^2*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*b^6*n^6*x^2
*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + b^6*n^6*x^2 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6*x
^2*cos(2*b*log(x^n) + 2*a)^2 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6*x^2*sin(2*b*log(x^n) + 2*a)
^2), x) + (4*b^2*n^2*cos(6*b*log(c)) - (2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c
)))*n - cos(6*b*log(c))*cos(4*b*log(c)) - sin(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 2*(6*(b^2
*cos(6*b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(2*b*log(c)))*n^2 - (b*cos(2*b*log(c))*sin(6*b*log(c
)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n + cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))
*cos(2*b*log(x^n) + 2*a) + (2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n + cos(
4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 2*(6*(b^2*cos(2*b*log
(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6
*b*log(c))*sin(2*b*log(c)))*n + cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log
(x^n) + 2*a) + cos(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + (12*b^2*n^2*cos(4*b*log(c)) - 2*b*n*sin(4*b*log(c))
+ 3*(12*(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 - 4*(b*cos(2*b*log(c))
*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n + cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*si
n(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 3*(12*(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 + 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n + cos(2*b*log(c
))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + 2*cos(4*b*log(c)))*sin(4*b*log
(x^n) + 4*a) - (2*b*n*sin(2*b*log(c)) - cos(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))/(6*b^3*n^3*x*cos(2*b*log(c))
*cos(2*b*log(x^n) + 2*a) - 6*b^3*n^3*x*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + b^3*n^3*x + (b^3*cos(6*b*log(
c))^2 + b^3*sin(6*b*log(c))^2)*n^3*x*cos(6*b*log(x^n) + 6*a)^2 + 9*(b^3*cos(4*b*log(c))^2 + b^3*sin(4*b*log(c)
)^2)*n^3*x*cos(4*b*log(x^n) + 4*a)^2 + 9*(b^3*cos(2*b*log(c))^2 + b^3*sin(2*b*log(c))^2)*n^3*x*cos(2*b*log(x^n
) + 2*a)^2 + (b^3*cos(6*b*log(c))^2 + b^3*sin(6*b*log(c))^2)*n^3*x*sin(6*b*log(x^n) + 6*a)^2 + 9*(b^3*cos(4*b*
log(c))^2 + b^3*sin(4*b*log(c))^2)*n^3*x*sin(4*b*log(x^n) + 4*a)^2 + 9*(b^3*cos(2*b*log(c))^2 + b^3*sin(2*b*lo
g(c))^2)*n^3*x*sin(2*b*log(x^n) + 2*a)^2 + 2*(b^3*n^3*x*cos(6*b*log(c)) + 3*(b^3*cos(6*b*log(c))*cos(4*b*log(c
)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)))*n^3*x*cos(4*b*log(x^n) + 4*a) + 3*(b^3*cos(6*b*log(c))*cos(2*b*log(c
)) + b^3*sin(6*b*log(c))*sin(2*b*log(c)))*n^3*x*cos(2*b*log(x^n) + 2*a) + 3*(b^3*cos(4*b*log(c))*sin(6*b*log(c
)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)))*n^3*x*sin(4*b*log(x^n) + 4*a) + 3*(b^3*cos(2*b*log(c))*sin(6*b*log(c
)) - b^3*cos(6*b*log(c))*sin(2*b*log(c)))*n^3*x*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(b^3*n^3*
x*cos(4*b*log(c)) + 3*(b^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3*x*cos(2*
b*log(x^n) + 2*a) + 3*(b^3*cos(2*b*log(c))*sin(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3*x*sin(2*
b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 2*(b^3*n^3*x*sin(6*b*log(c)) + 3*(b^3*cos(4*b*log(c))*sin(6*b*log
(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)))*n^3*x*cos(4*b*log(x^n) + 4*a) + 3*(b^3*cos(2*b*log(c))*sin(6*b*log
(c)) - b^3*cos(6*b*log(c))*sin(2*b*log(c)))*n^3*x*cos(2*b*log(x^n) + 2*a) - 3*(b^3*cos(6*b*log(c))*cos(4*b*log
(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)))*n^3*x*sin(4*b*log(x^n) + 4*a) - 3*(b^3*cos(6*b*log(c))*cos(2*b*log
(c)) + b^3*sin(6*b*log(c))*sin(2*b*log(c)))*n^3*x*sin(2*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(b^3*n^
3*x*sin(4*b*log(c)) + 3*(b^3*cos(2*b*log(c))*sin(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3*x*cos(
2*b*log(x^n) + 2*a) - 3*(b^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c)))*n^3*x*sin(
2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

Giac [F]

\[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{2}} \,d x } \]

[In]

integrate(sec(a+b*log(c*x^n))^4/x^2,x, algorithm="giac")

[Out]

integrate(sec(b*log(c*x^n) + a)^4/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {1}{x^2\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \]

[In]

int(1/(x^2*cos(a + b*log(c*x^n))^4),x)

[Out]

int(1/(x^2*cos(a + b*log(c*x^n))^4), x)