\(\int \frac {\cos ^3(a+b \log (c x^n))}{x^2} \, dx\) [100]

Optimal result

Integrand size = 17, antiderivative size = 158 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {6 b^2 n^2 \cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}-\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}+\frac {6 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}+\frac {3 b n \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x} \]

[Out]

-6*b^2*n^2*cos(a+b*ln(c*x^n))/(9*b^4*n^4+10*b^2*n^2+1)/x-cos(a+b*ln(c*x^n))^3/(9*b^2*n^2+1)/x+6*b^3*n^3*sin(a+
b*ln(c*x^n))/(9*b^4*n^4+10*b^2*n^2+1)/x+3*b*n*cos(a+b*ln(c*x^n))^2*sin(a+b*ln(c*x^n))/(9*b^2*n^2+1)/x

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {4576, 4574} \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}+\frac {3 b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^2 n^2+1\right )}-\frac {6 b^2 n^2 \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )}+\frac {6 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right )}{x \left (9 b^4 n^4+10 b^2 n^2+1\right )} \]

[In]

Int[Cos[a + b*Log[c*x^n]]^3/x^2,x]

[Out]

(-6*b^2*n^2*Cos[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x) - Cos[a + b*Log[c*x^n]]^3/((1 + 9*b^2*n^2)
*x) + (6*b^3*n^3*Sin[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x) + (3*b*n*Cos[a + b*Log[c*x^n]]^2*Sin[
a + b*Log[c*x^n]])/((1 + 9*b^2*n^2)*x)

Rule 4574

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(m + 1)*(e*x)^(m +
1)*(Cos[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)), x] + Simp[b*d*n*(e*x)^(m + 1)*(Sin[d*(a + b*Log[
c*x^n])]/(b^2*d^2*e*n^2 + e*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 + (m + 1)^2,
0]

Rule 4576

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(m + 1)*(e*x)^
(m + 1)*(Cos[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2)), x] + (Dist[b^2*d^2*n^2*p*((p - 1)/(b^
2*d^2*n^2*p^2 + (m + 1)^2)), Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[b*d*n*p*(e*x)^(m + 1
)*Sin[d*(a + b*Log[c*x^n])]*(Cos[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2)), x]) /; Free
Q[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}+\frac {3 b n \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}+\frac {\left (6 b^2 n^2\right ) \int \frac {\cos \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx}{1+9 b^2 n^2} \\ & = -\frac {6 b^2 n^2 \cos \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}-\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x}+\frac {6 b^3 n^3 \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+10 b^2 n^2+9 b^4 n^4\right ) x}+\frac {3 b n \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+9 b^2 n^2\right ) x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {3 \left (1+9 b^2 n^2\right ) \cos \left (a+b \log \left (c x^n\right )\right )+\left (1+b^2 n^2\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )-6 b n \left (1+5 b^2 n^2+\left (1+b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 \left (1+10 b^2 n^2+9 b^4 n^4\right ) x} \]

[In]

Integrate[Cos[a + b*Log[c*x^n]]^3/x^2,x]

[Out]

-1/4*(3*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + (1 + b^2*n^2)*Cos[3*(a + b*Log[c*x^n])] - 6*b*n*(1 + 5*b^2*n^2
+ (1 + b^2*n^2)*Cos[2*(a + b*Log[c*x^n])])*Sin[a + b*Log[c*x^n]])/((1 + 10*b^2*n^2 + 9*b^4*n^4)*x)

Maple [A] (verified)

Time = 7.09 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.49

method	result	size
parallelrisch	\(\frac {-1+\left (7 b^{2} n^{2}+1\right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{6}+6 \left (3 b^{3} n^{3}+b n \right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{5}+3 \left (b^{2} n^{2}-1\right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}+12 \left (b^{3} n^{3}-b n \right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+3 \left (-b^{2} n^{2}+1\right ) {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}+6 \left (3 b^{3} n^{3}+b n \right ) \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-7 b^{2} n^{2}}{9 \left (b^{2} n^{2}+\frac {1}{9}\right ) \left (b^{2} n^{2}+1\right ) x {\left (1+{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}\right )}^{3}}\)	\(235\)

[In]

int(cos(a+b*ln(c*x^n))^3/x^2,x,method=_RETURNVERBOSE)

[Out]

1/9*(-1+(7*b^2*n^2+1)*tan(1/2*a+b*ln((c*x^n)^(1/2)))^6+6*(3*b^3*n^3+b*n)*tan(1/2*a+b*ln((c*x^n)^(1/2)))^5+3*(b
^2*n^2-1)*tan(1/2*a+b*ln((c*x^n)^(1/2)))^4+12*(b^3*n^3-b*n)*tan(1/2*a+b*ln((c*x^n)^(1/2)))^3+3*(-b^2*n^2+1)*ta
n(1/2*a+b*ln((c*x^n)^(1/2)))^2+6*(3*b^3*n^3+b*n)*tan(1/2*a+b*ln((c*x^n)^(1/2)))-7*b^2*n^2)/(b^2*n^2+1/9)/(b^2*
n^2+1)/x/(1+tan(1/2*a+b*ln((c*x^n)^(1/2)))^2)^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.75 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {6 \, b^{2} n^{2} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (b^{2} n^{2} + 1\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, {\left (2 \, b^{3} n^{3} + {\left (b^{3} n^{3} + b n\right )} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{{\left (9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1\right )} x} \]

[In]

integrate(cos(a+b*log(c*x^n))^3/x^2,x, algorithm="fricas")

[Out]

-(6*b^2*n^2*cos(b*n*log(x) + b*log(c) + a) + (b^2*n^2 + 1)*cos(b*n*log(x) + b*log(c) + a)^3 - 3*(2*b^3*n^3 + (
b^3*n^3 + b*n)*cos(b*n*log(x) + b*log(c) + a)^2)*sin(b*n*log(x) + b*log(c) + a))/((9*b^4*n^4 + 10*b^2*n^2 + 1)
*x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 21.93 (sec) , antiderivative size = 774, normalized size of antiderivative = 4.90 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\begin {cases} \frac {3 i \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 x} + \frac {3 i \sin {\left (3 a - \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}}{32 x} + \frac {\cos {\left (3 a - \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}}{32 x} + \frac {3 i \log {\left (c x^{n} \right )} \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} + \frac {3 \log {\left (c x^{n} \right )} \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} & \text {for}\: b = - \frac {i}{n} \\- \frac {9 i \sin {\left (a - \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}}{32 x} + \frac {i \sin {\left (3 a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 x} - \frac {27 \cos {\left (a - \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}}{32 x} + \frac {i \log {\left (c x^{n} \right )} \sin {\left (3 a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} + \frac {\log {\left (c x^{n} \right )} \cos {\left (3 a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} & \text {for}\: b = - \frac {i}{3 n} \\\frac {9 i \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}}{32 x} - \frac {27 \cos {\left (a + \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}}{32 x} - \frac {\cos {\left (3 a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 x} - \frac {i \log {\left (c x^{n} \right )} \sin {\left (3 a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} + \frac {\log {\left (c x^{n} \right )} \cos {\left (3 a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} & \text {for}\: b = \frac {i}{3 n} \\- \frac {3 i \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 x} - \frac {3 i \sin {\left (3 a + \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}}{32 x} + \frac {\cos {\left (3 a + \frac {3 i \log {\left (c x^{n} \right )}}{n} \right )}}{32 x} - \frac {3 i \log {\left (c x^{n} \right )} \sin {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} + \frac {3 \log {\left (c x^{n} \right )} \cos {\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}}{8 n x} & \text {for}\: b = \frac {i}{n} \\\frac {6 b^{3} n^{3} \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} x + 10 b^{2} n^{2} x + x} + \frac {9 b^{3} n^{3} \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} x + 10 b^{2} n^{2} x + x} - \frac {6 b^{2} n^{2} \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} x + 10 b^{2} n^{2} x + x} - \frac {7 b^{2} n^{2} \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} x + 10 b^{2} n^{2} x + x} + \frac {3 b n \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} x + 10 b^{2} n^{2} x + x} - \frac {\cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} x + 10 b^{2} n^{2} x + x} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(a+b*ln(c*x**n))**3/x**2,x)

[Out]

Piecewise((3*I*sin(a - I*log(c*x**n)/n)/(8*x) + 3*I*sin(3*a - 3*I*log(c*x**n)/n)/(32*x) + cos(3*a - 3*I*log(c*
x**n)/n)/(32*x) + 3*I*log(c*x**n)*sin(a - I*log(c*x**n)/n)/(8*n*x) + 3*log(c*x**n)*cos(a - I*log(c*x**n)/n)/(8
*n*x), Eq(b, -I/n)), (-9*I*sin(a - I*log(c*x**n)/(3*n))/(32*x) + I*sin(3*a - I*log(c*x**n)/n)/(8*x) - 27*cos(a
- I*log(c*x**n)/(3*n))/(32*x) + I*log(c*x**n)*sin(3*a - I*log(c*x**n)/n)/(8*n*x) + log(c*x**n)*cos(3*a - I*lo
g(c*x**n)/n)/(8*n*x), Eq(b, -I/(3*n))), (9*I*sin(a + I*log(c*x**n)/(3*n))/(32*x) - 27*cos(a + I*log(c*x**n)/(3
*n))/(32*x) - cos(3*a + I*log(c*x**n)/n)/(8*x) - I*log(c*x**n)*sin(3*a + I*log(c*x**n)/n)/(8*n*x) + log(c*x**n
)*cos(3*a + I*log(c*x**n)/n)/(8*n*x), Eq(b, I/(3*n))), (-3*I*sin(a + I*log(c*x**n)/n)/(8*x) - 3*I*sin(3*a + 3*
I*log(c*x**n)/n)/(32*x) + cos(3*a + 3*I*log(c*x**n)/n)/(32*x) - 3*I*log(c*x**n)*sin(a + I*log(c*x**n)/n)/(8*n*
x) + 3*log(c*x**n)*cos(a + I*log(c*x**n)/n)/(8*n*x), Eq(b, I/n)), (6*b**3*n**3*sin(a + b*log(c*x**n))**3/(9*b*
*4*n**4*x + 10*b**2*n**2*x + x) + 9*b**3*n**3*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**2/(9*b**4*n**4*x
+ 10*b**2*n**2*x + x) - 6*b**2*n**2*sin(a + b*log(c*x**n))**2*cos(a + b*log(c*x**n))/(9*b**4*n**4*x + 10*b**2*
n**2*x + x) - 7*b**2*n**2*cos(a + b*log(c*x**n))**3/(9*b**4*n**4*x + 10*b**2*n**2*x + x) + 3*b*n*sin(a + b*log
(c*x**n))*cos(a + b*log(c*x**n))**2/(9*b**4*n**4*x + 10*b**2*n**2*x + x) - cos(a + b*log(c*x**n))**3/(9*b**4*n
**4*x + 10*b**2*n**2*x + x), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (158) = 316\).

Time = 0.26 (sec) , antiderivative size = 994, normalized size of antiderivative = 6.29 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\text {Too large to display} \]

[In]

integrate(cos(a+b*log(c*x^n))^3/x^2,x, algorithm="maxima")

[Out]

1/8*((3*(b^3*cos(3*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(3*b*log(c)) + b^3*sin(3*b*log(c)))*n^3
- (b^2*cos(6*b*log(c))*cos(3*b*log(c)) + b^2*sin(6*b*log(c))*sin(3*b*log(c)) + b^2*cos(3*b*log(c)))*n^2 + 3*(b
*cos(3*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(3*b*log(c)) + b*sin(3*b*log(c)))*n - cos(6*b*log(c))*
cos(3*b*log(c)) - sin(6*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c)))*cos(3*b*log(x^n) + 3*a) + 3*(9*(b^3*cos(3
*b*log(c))*sin(4*b*log(c)) - b^3*cos(4*b*log(c))*sin(3*b*log(c)) + b^3*cos(2*b*log(c))*sin(3*b*log(c)) - b^3*c
os(3*b*log(c))*sin(2*b*log(c)))*n^3 - 9*(b^2*cos(4*b*log(c))*cos(3*b*log(c)) + b^2*cos(3*b*log(c))*cos(2*b*log
(c)) + b^2*sin(4*b*log(c))*sin(3*b*log(c)) + b^2*sin(3*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(3*b*log(c))*sin
(4*b*log(c)) - b*cos(4*b*log(c))*sin(3*b*log(c)) + b*cos(2*b*log(c))*sin(3*b*log(c)) - b*cos(3*b*log(c))*sin(2
*b*log(c)))*n - cos(4*b*log(c))*cos(3*b*log(c)) - cos(3*b*log(c))*cos(2*b*log(c)) - sin(4*b*log(c))*sin(3*b*lo
g(c)) - sin(3*b*log(c))*sin(2*b*log(c)))*cos(b*log(x^n) + a) + (3*(b^3*cos(6*b*log(c))*cos(3*b*log(c)) + b^3*s
in(6*b*log(c))*sin(3*b*log(c)) + b^3*cos(3*b*log(c)))*n^3 + (b^2*cos(3*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b
*log(c))*sin(3*b*log(c)) + b^2*sin(3*b*log(c)))*n^2 + 3*(b*cos(6*b*log(c))*cos(3*b*log(c)) + b*sin(6*b*log(c))
*sin(3*b*log(c)) + b*cos(3*b*log(c)))*n + cos(3*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(3*b*log(c)) +
sin(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + 3*(9*(b^3*cos(4*b*log(c))*cos(3*b*log(c)) + b^3*cos(3*b*log(c))*cos
(2*b*log(c)) + b^3*sin(4*b*log(c))*sin(3*b*log(c)) + b^3*sin(3*b*log(c))*sin(2*b*log(c)))*n^3 + 9*(b^2*cos(3*b
*log(c))*sin(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(3*b*log(c)) + b^2*cos(2*b*log(c))*sin(3*b*log(c)) - b^2*cos
(3*b*log(c))*sin(2*b*log(c)))*n^2 + (b*cos(4*b*log(c))*cos(3*b*log(c)) + b*cos(3*b*log(c))*cos(2*b*log(c)) + b
*sin(4*b*log(c))*sin(3*b*log(c)) + b*sin(3*b*log(c))*sin(2*b*log(c)))*n + cos(3*b*log(c))*sin(4*b*log(c)) - co
s(4*b*log(c))*sin(3*b*log(c)) + cos(2*b*log(c))*sin(3*b*log(c)) - cos(3*b*log(c))*sin(2*b*log(c)))*sin(b*log(x
^n) + a))/((9*(b^4*cos(3*b*log(c))^2 + b^4*sin(3*b*log(c))^2)*n^4 + 10*(b^2*cos(3*b*log(c))^2 + b^2*sin(3*b*lo
g(c))^2)*n^2 + cos(3*b*log(c))^2 + sin(3*b*log(c))^2)*x)

Giac [F]

\[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}} \,d x } \]

[In]

integrate(cos(a+b*log(c*x^n))^3/x^2,x, algorithm="giac")

[Out]

integrate(cos(b*log(c*x^n) + a)^3/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]

[In]

int(cos(a + b*log(c*x^n))^3/x^2,x)

[Out]

int(cos(a + b*log(c*x^n))^3/x^2, x)