\(\int x \sec ^3(a+b \log (c x^n)) \, dx\) [249]

Optimal result

Integrand size = 15, antiderivative size = 87 \[ \int x \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {8 e^{3 i a} x^2 \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),\frac {1}{2} \left (5-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+3 i b n} \]

[Out]

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

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4605, 4601, 371} \[ \int x \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {8 e^{3 i a} x^2 \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} \left (3-\frac {2 i}{b n}\right ),\frac {1}{2} \left (5-\frac {2 i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+3 i b n} \]

[In]

Int[x*Sec[a + b*Log[c*x^n]]^3,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 4.96 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.36 \[ \int x \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^2 \left (2 e^{i a} (2-i b n) \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}-\frac {i}{b n},\frac {3}{2}-\frac {i}{b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec \left (a+b \log \left (c x^n\right )\right ) \left (-2+b n \tan \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 b^2 n^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x/cos(a + b*log(c*x^n))^3,x)

[Out]

int(x/cos(a + b*log(c*x^n))^3, x)