3.258 \(\int \frac {\sec ^4(a+b \log (c x^n))}{x^3} \, dx\)

Optimal. Leaf size=79 \[ -\frac {8 e^{4 i a} \left (c x^n\right )^{4 i b} \, _2F_1\left (4,2+\frac {i}{b n};3+\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (1-2 i b n)} \]

[Out]

-8*exp(4*I*a)*(c*x^n)^(4*I*b)*hypergeom([4, 2+I/b/n],[3+I/b/n],-exp(2*I*a)*(c*x^n)^(2*I*b))/(1-2*I*b*n)/x^2

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {4509, 4505, 364} \[ -\frac {8 e^{4 i a} \left (c x^n\right )^{4 i b} \, _2F_1\left (4,2+\frac {i}{b n};3+\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x^2 (1-2 i b n)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*Log[c*x^n]]^4/x^3,x]

[Out]

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

Rule 364

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

Rule 4505

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 4509

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*} \int \frac {\sec ^4\left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int x^{-1-\frac {2}{n}} \sec ^4(a+b \log (x)) \, dx,x,c x^n\right )}{n x^2}\\ &=\frac {\left (16 e^{4 i a} \left (c x^n\right )^{2/n}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+4 i b-\frac {2}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^4} \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac {8 e^{4 i a} \left (c x^n\right )^{4 i b} \, _2F_1\left (4,2+\frac {i}{b n};3+\frac {i}{b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(1-2 i b n) x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 9.24, size = 203, normalized size = 2.57 \[ \frac {-2 i \left (b^2 n^2+1\right ) \, _2F_1\left (1,\frac {i}{b n};1+\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec ^2\left (a+b \log \left (c x^n\right )\right ) \left (\tan \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 b^2 n^2+1\right )+b n\right )-2 e^{2 i a} (b n-i) \left (c x^n\right )^{2 i b} \, _2F_1\left (1,1+\frac {i}{b n};2+\frac {i}{b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{3 b^3 n^3 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[a + b*Log[c*x^n]]^4/x^3,x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(a+b*log(c*x^n))^4/x^3,x, algorithm="fricas")

[Out]

integral(sec(b*log(c*x^n) + a)^4/x^3, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(a+b*log(c*x^n))^4/x^3,x, algorithm="giac")

[Out]

integrate(sec(b*log(c*x^n) + a)^4/x^3, x)

________________________________________________________________________________________

maple [F]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(a+b*ln(c*x^n))^4/x^3,x)

[Out]

int(sec(a+b*ln(c*x^n))^4/x^3,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(a+b*log(c*x^n))^4/x^3,x, algorithm="maxima")

[Out]

4/3*(3*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 + 3*(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 + 3*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n
) + 4*a)^2 + 3*(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^2*n^2*sin(6*b*log(
c)) + ((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) + (3*(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 + 2*cos(2*b*log(c))*sin(6*b*log(c)) - 2*cos(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + ((b*co
s(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)))*sin(4*b*log(x^n) + 4*a) - (3*(b^2*cos(6*b*log(c))*cos(2*b*log(c)) + b^2*sin(6*b*lo
g(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 + 2*cos
(6*b*log(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))*c
os(6*b*log(x^n) + 6*a) + (3*b^2*n^2*sin(4*b*log(c)) + b*n*cos(4*b*log(c)) + 3*(3*(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 + 2*(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)))*cos(2*b*log(x^n) + 2*a
) - 3*(3*(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 - 2*(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))*s
in(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) + (b*n*cos(2*b*log(c)) +
sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 18*(((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^2*cos(6*b*log(x^n) + 6*a)^2 + 9*((b^8*cos(4*b*log(c))^2 + b^8*s
in(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^2*cos(4*b*log(x^n) + 4*a)^2 + 9
*((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
^2*cos(2*b*log(x^n) + 2*a)^2 + ((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^2*sin(6*b*log(x^n) + 6*a)^2 + 9*((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^2*sin(4*b*log(x^n) + 4*a)^2 + 9*((b^8*cos(2*b*l
og(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^2*sin(2*b*log(x
^n) + 2*a)^2 + 6*(b^8*n^8*cos(2*b*log(c)) + b^6*n^6*cos(2*b*log(c)))*x^2*cos(2*b*log(x^n) + 2*a) - 6*(b^8*n^8*
sin(2*b*log(c)) + b^6*n^6*sin(2*b*log(c)))*x^2*sin(2*b*log(x^n) + 2*a) + (b^8*n^8 + b^6*n^6)*x^2 + 2*(3*((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))*cos(4*b*log(
c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*x^2*cos(4*b*log(x^n) + 4*a) + 3*((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^2*cos(2*b*log(x^n) + 2*a) + 3*((b^8*cos(4*b*log(c))*sin(6*b*log(c)) - b^8*cos(6*b*lo
g(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^2*sin(4*b*log(x^n) + 4*a) + 3*((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^2*sin(2*b*log(x^n) + 2
*a) + (b^8*n^8*cos(6*b*log(c)) + b^6*n^6*cos(6*b*log(c)))*x^2)*cos(6*b*log(x^n) + 6*a) + 6*(3*((b^8*cos(4*b*lo
g(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^2*cos(2*b*log(x^n) + 2*a) + 3*((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^2*sin(2*b*log(x^n) + 2*a) + (b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(c)))*x^2)*cos(4*b*l
og(x^n) + 4*a) - 2*(3*((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^2*cos(4*b*log(x^n) + 4*a) + 3*((
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^2*cos(2*b*log(x^n) + 2*a) - 3*((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))*cos(4*b*log(c)) + b^6*sin(6*b*lo
g(c))*sin(4*b*log(c)))*n^6)*x^2*sin(4*b*log(x^n) + 4*a) - 3*((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^2*sin(2*b*log(x^n) + 2*a) + (b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*log(c)))*x^2)*sin(6*b*log(x^n) +
6*a) - 6*(3*((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*lo
g(c))*sin(4*b*log(c)) - b^6*cos(4*b*log(c))*sin(2*b*log(c)))*n^6)*x^2*cos(2*b*log(x^n) + 2*a) - 3*((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^2*sin(2*b*log(x^n) + 2*a) + (b^8*n^8*sin(4*b*log(c)) + b^6*n^6*sin
(4*b*log(c)))*x^2)*sin(4*b*log(x^n) + 4*a))*integrate(1/9*(cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + cos(2*b*l
og(c))*sin(2*b*log(x^n) + 2*a))/(2*b^6*n^6*x^3*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 2*b^6*n^6*x^3*sin(2*b
*log(c))*sin(2*b*log(x^n) + 2*a) + b^6*n^6*x^3 + (b^6*cos(2*b*log(c))^2 + b^6*sin(2*b*log(c))^2)*n^6*x^3*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^3*sin(2*b*log(x^n) + 2*a)^2), x)
+ (b^2*n^2*cos(6*b*log(c)) - ((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) + (3*(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 + 2*cos(6*b*log(c))*cos(2*b*log(c)) + 2*sin(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*l
og(x^n) + 2*a) + ((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) + (3*(b^2*cos(2*b*log(c))*sin(6*b*l
og(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 + 2*cos(2*b*log(c))*sin(6*b*log(c)) - 2*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) + (3*b^2*n^2*cos(4*b*log(c)) - b*n*sin(4*b*log(c)) + 3*(3*(b^2*c
os(4*b*log(c))*cos(2*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 - 2*(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))*sin(2*b*log(c)))
*cos(2*b*log(x^n) + 2*a) + 3*(3*(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 + 2*(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) -
(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^2*cos(2*b*log(c))*cos(2*b*log(x^
n) + 2*a) - 6*b^3*n^3*x^2*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + b^3*n^3*x^2 + (b^3*cos(6*b*log(c))^2 + b^3
*sin(6*b*log(c))^2)*n^3*x^2*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^2*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^2*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^2*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^2*sin(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^2*sin(2*b*log(x^n) + 2*a)^2 + 2*(b^3*n^3*x^2*cos(6*b*log(c)) + 3*(b^3*cos(6*b*log(c))*cos(4*b*lo
g(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)))*n^3*x^2*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^2*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^2*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^2*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) +
6*(b^3*n^3*x^2*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^2*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^2*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 2*(b^3*n^3*x^2*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^2*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^2*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^2*sin(4*b*log(x^n) + 4*a) - 3*(b^3*co
s(6*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(2*b*log(c)))*n^3*x^2*sin(2*b*log(x^n) + 2*a))*sin(6*b*
log(x^n) + 6*a) - 6*(b^3*n^3*x^2*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^2*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^2*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*cos(a + b*log(c*x^n))^4),x)

[Out]

int(1/(x^3*cos(a + b*log(c*x^n))^4), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(a+b*ln(c*x**n))**4/x**3,x)

[Out]

Integral(sec(a + b*log(c*x**n))**4/x**3, x)

________________________________________________________________________________________