3.255 \(\int \sec ^4(a+b \log (c x^n)) \, dx\)
Optimal. Leaf size=85 \[ \frac{16 e^{4 i a} x \left (c x^n\right )^{4 i b} \text{Hypergeometric2F1}\left (4,\frac{1}{2} \left (4-\frac{i}{b n}\right ),\frac{1}{2} \left (6-\frac{i}{b n}\right ),-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \]
[Out]
(16*E^((4*I)*a)*x*(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)
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Rubi [A] time = 0.0625623, antiderivative size = 85, normalized size of antiderivative = 1.,
number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used =
{4503, 4505, 364} \[ \frac{16 e^{4 i a} x \left (c x^n\right )^{4 i b} \, _2F_1\left (4,\frac{1}{2} \left (4-\frac{i}{b n}\right );\frac{1}{2} \left (6-\frac{i}{b n}\right );-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+4 i b n} \]
Antiderivative was successfully verified.
[In]
Int[Sec[a + b*Log[c*x^n]]^4,x]
[Out]
(16*E^((4*I)*a)*x*(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)
Rule 4503
Int[Sec[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sec[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])
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 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])
Rubi steps
\begin{align*} \int \sec ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sec ^4(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (16 e^{4 i a} x \left (c x^n\right )^{-1/n}\right ) \operatorname{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}\\ &=\frac{16 e^{4 i a} x \left (c x^n\right )^{4 i b} \, _2F_1\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}\\ \end{align*}
Mathematica [B] time = 10.7975, size = 213, normalized size = 2.51 \[ \frac{x \left (-2 i \left (4 b^2 n^2+1\right ) \text{Hypergeometric2F1}\left (1,-\frac{i}{2 b n},1-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+2 e^{2 i a} (2 b n+i) \left (c x^n\right )^{2 i b} \text{Hypergeometric2F1}\left (1,1-\frac{i}{2 b n},2-\frac{i}{2 b n},-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )+\sec ^2\left (a+b \log \left (c x^n\right )\right ) \left (\tan \left (a+b \log \left (c x^n\right )\right ) \left (\left (4 b^2 n^2+1\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+8 b^2 n^2+1\right )-2 b n\right )\right )}{12 b^3 n^3} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[Sec[a + b*Log[c*x^n]]^4,x]
[Out]
(x*(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)
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Maple [F] time = 1.325, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sec(a+b*ln(c*x^n))^4,x)
[Out]
int(sec(a+b*ln(c*x^n))^4,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(a+b*log(c*x^n))^4,x, algorithm="maxima")
[Out]
-1/3*(6*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*x*cos(4*b*log(x^n) + 4*a)^2 + 6*(b*cos(2*b*log(c))^2 + b
*sin(2*b*log(c))^2)*n*x*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*x*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*x*sin(2*b*log(x^n) + 2*a)^2 + (2*b*n*cos(2
*b*log(c)) - sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) - (2*b*n*sin(2*b*log(c)) + cos(2*b*log(c)))*x*sin(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)))*x*cos(4*b*log(x^n) + 4*a) - 2*(6*(b^2*cos(2*b*log(c))*s
in(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)))*x*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)) + sin(6*b*log(c))*sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) + 2*(6*(b^2*cos(6*b*log(c))*cos(2*b*l
og(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)))*x*sin(2*b*log(x^n) + 2*a)
- (4*b^2*n^2*sin(6*b*log(c)) + sin(6*b*log(c)))*x)*cos(6*b*log(x^n) + 6*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)))*x*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)))*x*sin(2*b*log(x^n) + 2*a) + 2*(6*b^2*n^2*sin(4*b*log(c)) - b*n*cos(4*b*log(c)) + sin(4*
b*log(c)))*x)*cos(4*b*log(x^n) + 4*a) + 9*(4*b^8*n^8 + b^6*n^6 + (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)*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)*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*l
og(c))^2)*n^6)*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*log(c))^2 + b^6*sin(6*b*log(c))^2)*n^6)*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)*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)*sin(2*b
*log(x^n) + 2*a)^2 + 2*(4*b^8*n^8*cos(6*b*log(c)) + b^6*n^6*cos(6*b*log(c)) + 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))*cos(4*b*log(c)) + b^6*sin(6*b*log(
c))*sin(4*b*log(c)))*n^6)*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*lo
g(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)*
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)*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))*si
n(6*b*log(c)) - b^6*cos(6*b*log(c))*sin(2*b*log(c)))*n^6)*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6
*(4*b^8*n^8*cos(4*b*log(c)) + b^6*n^6*cos(4*b*log(c)) + 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)*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)*sin(2*b*log(x^n) + 2*a
))*cos(4*b*log(x^n) + 4*a) + 6*(4*b^8*n^8*cos(2*b*log(c)) + b^6*n^6*cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) -
2*(4*b^8*n^8*sin(6*b*log(c)) + b^6*n^6*sin(6*b*log(c)) + 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)*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)*cos(2*b*log(x^n) + 2
*a) - 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))*cos(4*b*log(c)) + b^6*sin(6*b*log(c))*sin(4*b*log(c)))*n^6)*sin(4*b*log(x^n) + 4*a) - 3*(4*(b^8*cos(6*b*lo
g(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)*sin(2*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 6*(4*b^8*n^8*sin(4*b*
log(c)) + b^6*n^6*sin(4*b*log(c)) + 3*(4*(b^8*cos(2*b*log(c))*sin(4*b*log(c)) - b^8*cos(4*b*log(c))*sin(2*b*lo
g(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)*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)*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n)
+ 4*a) - 6*(4*b^8*n^8*sin(2*b*log(c)) + b^6*n^6*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*integrate(1/9*(cos(2
*b*log(x^n) + 2*a)*sin(2*b*log(c)) + cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(2*b^6*n^6*cos(2*b*log(c))*cos(2
*b*log(x^n) + 2*a) - 2*b^6*n^6*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + b^6*n^6 + (b^6*cos(2*b*log(c))^2 + b^
6*sin(2*b*log(c))^2)*n^6*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*sin(2
*b*log(x^n) + 2*a)^2), x) - ((2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n + co
s(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*x*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*c
os(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)))*x*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*lo
g(c))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)))*x*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*l
og(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)))*x*sin(2*b*log(x
^n) + 2*a) + (4*b^2*n^2*cos(6*b*log(c)) + cos(6*b*log(c)))*x)*sin(6*b*log(x^n) + 6*a) - (3*(12*(b^2*cos(4*b*lo
g(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*co
s(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)))*x*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)))*x*sin(2*b*log(x^n) + 2*a) + 2*(6*b^2*n^2*cos(4*b*log(c)) + b*n*sin(4*b*log(c)
) + cos(4*b*log(c)))*x)*sin(4*b*log(x^n) + 4*a))/(6*b^3*n^3*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) - 6*b^3*n^
3*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + b^3*n^3 + (b^3*cos(6*b*log(c))^2 + b^3*sin(6*b*log(c))^2)*n^3*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*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*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*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*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*sin(2*b*log(x^n) + 2*a)^2 + 2*(b^3*n
^3*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*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*cos(2*b*l
og(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*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*sin(2*b*log(x
^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 6*(b^3*n^3*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*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*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 2*(b^3*n^3*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*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*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*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*sin(2*b*log(x^n) + 2*a))*si
n(6*b*log(x^n) + 6*a) - 6*(b^3*n^3*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*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*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(a+b*log(c*x^n))^4,x, algorithm="fricas")
[Out]
integral(sec(b*log(c*x^n) + a)^4, x)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec ^{4}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(a+b*ln(c*x**n))**4,x)
[Out]
Integral(sec(a + b*log(c*x**n))**4, x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec \left (b \log \left (c x^{n}\right ) + a\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sec(a+b*log(c*x^n))^4,x, algorithm="giac")
[Out]
integrate(sec(b*log(c*x^n) + a)^4, x)