\(\int x^m \sec ^3(a+b \log (c x^n)) \, dx\) [278]

Optimal result

Integrand size = 17, antiderivative size = 102 \[ \int x^m \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {8 e^{3 i a} x^{1+m} \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,-\frac {i (1+m)-3 b n}{2 b n},-\frac {i (1+m)-5 b n}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+m+3 i b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 102, 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} \[ \int x^m \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {8 e^{3 i a} x^{m+1} \left (c x^n\right )^{3 i b} \operatorname {Hypergeometric2F1}\left (3,-\frac {i (m+1)-3 b n}{2 b n},-\frac {i (m+1)-5 b n}{2 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i b n+m+1} \]

[In]

Int[x^m*Sec[a + b*Log[c*x^n]]^3,x]

[Out]

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

Mathematica [A] (verified)

Time = 5.91 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.36 \[ \int x^m \sec ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{1+m} \left (4 e^{i a} (1+m-i b n) \left (c x^n\right )^{i b} \operatorname {Hypergeometric2F1}\left (1,\frac {-i-i m+b n}{2 b n},-\frac {i (1+m+3 i b n)}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )-2 \sec \left (a+b \log \left (c x^n\right )\right ) \left (1+m-b n \tan \left (a+b \log \left (c x^n\right )\right )\right )\right )}{4 b^2 n^2} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x^m/cos(a + b*log(c*x^n))^3,x)

[Out]

int(x^m/cos(a + b*log(c*x^n))^3, x)