\(\int \cos ^3(a+b \log (c x^n)) \, dx\) [98]

Optimal result

Integrand size = 13, antiderivative size = 149 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 b^2 n^2 x \cos \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac {x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac {6 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac {3 b n x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.79 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (3 \left (1+9 b^2 n^2\right ) \cos \left (a+b \log \left (c x^n\right )\right )+\left (1+b^2 n^2\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+6 b n \left (1+5 b^2 n^2+\left (1+b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4+40 b^2 n^2+36 b^4 n^4} \] Input:

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

Output:

(x*(3*(1 + 9*b^2*n^2)*Cos[a + b*Log[c*x^n]] + (1 + b^2*n^2)*Cos[3*(a + b*L 
og[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]]))/(4 + 40*b^2*n^2 + 36*b^4*n^4)
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4981, 4979}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 4981

\(\displaystyle \frac {6 b^2 n^2 \int \cos \left (a+b \log \left (c x^n\right )\right )dx}{9 b^2 n^2+1}+\frac {x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {3 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}\)

\(\Big \downarrow \) 4979

\(\displaystyle \frac {x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {3 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {6 b^2 n^2 \left (\frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}+\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}\right )}{9 b^2 n^2+1}\)

Input:

Int[Cos[a + b*Log[c*x^n]]^3,x]
 

Output:

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

Defintions of rubi rules used

rule 4979

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

rule 4981

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

Maple [A] (verified)

Time = 1.55 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.28

method	result	size
default	\(\frac {3 \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 n^{2} \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {3 b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 n \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{4 n^{2} \left (\frac {1}{n^{2}}+9 b^{2}\right )}+\frac {3 b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{4 n \left (\frac {1}{n^{2}}+9 b^{2}\right )}\)	\(190\)
parallelrisch	\(\frac {x \left (1+12 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3} b^{3} n^{3}+18 b^{3} n^{3} \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+6 b n {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{5}+18 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{5} b^{3} n^{3}-7 b^{2} n^{2} {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{6}-12 b n {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}-3 b^{2} n^{2} {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}+3 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2} b^{2} n^{2}+6 b n \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+3 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}-3 {\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}-{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{6}+7 b^{2} n^{2}\right )}{\left (b^{2} n^{2}+1\right ) \left (9 b^{2} n^{2}+1\right ) {\left (1+{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}\right )}^{3}}\)	\(323\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.80 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 \, b^{2} n^{2} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (b^{2} n^{2} + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, {\left (2 \, b^{3} n^{3} x + {\left (b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1} \] Input:

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

Output:

(6*b^2*n^2*x*cos(b*n*log(x) + b*log(c) + a) + (b^2*n^2 + 1)*x*cos(b*n*log( 
x) + b*log(c) + a)^3 + 3*(2*b^3*n^3*x + (b^3*n^3 + b*n)*x*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)
 

Sympy [F]

\[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \cos ^{3}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int \cos ^{3}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = - \frac {i}{3 n} \\\int \cos ^{3}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {i}{3 n} \\\int \cos ^{3}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\\frac {6 b^{3} n^{3} x \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {9 b^{3} n^{3} x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {6 b^{2} n^{2} x \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {7 b^{2} n^{2} x \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {3 b n x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {x \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((Integral(cos(a - I*log(c*x**n)/n)**3, x), Eq(b, -I/n)), (Integr 
al(cos(a - I*log(c*x**n)/(3*n))**3, x), Eq(b, -I/(3*n))), (Integral(cos(a 
+ I*log(c*x**n)/(3*n))**3, x), Eq(b, I/(3*n))), (Integral(cos(a + I*log(c* 
x**n)/n)**3, x), Eq(b, I/n)), (6*b**3*n**3*x*sin(a + b*log(c*x**n))**3/(9* 
b**4*n**4 + 10*b**2*n**2 + 1) + 9*b**3*n**3*x*sin(a + b*log(c*x**n))*cos(a 
 + b*log(c*x**n))**2/(9*b**4*n**4 + 10*b**2*n**2 + 1) + 6*b**2*n**2*x*sin( 
a + b*log(c*x**n))**2*cos(a + b*log(c*x**n))/(9*b**4*n**4 + 10*b**2*n**2 + 
 1) + 7*b**2*n**2*x*cos(a + b*log(c*x**n))**3/(9*b**4*n**4 + 10*b**2*n**2 
+ 1) + 3*b*n*x*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**2/(9*b**4*n* 
*4 + 10*b**2*n**2 + 1) + x*cos(a + b*log(c*x**n))**3/(9*b**4*n**4 + 10*b** 
2*n**2 + 1), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (149) = 298\).

Time = 0.08 (sec) , antiderivative size = 989, normalized size of antiderivative = 6.64 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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*c 
os(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)))*x*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*cos(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*l 
og(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*log(c)) + sin(3*b*log(c))*sin(2*b*log(c)))*x*c 
os(b*log(x^n) + a) + (3*(b^3*cos(6*b*log(c))*cos(3*b*log(c)) + b^3*sin(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)))*x*sin(3*b*log(x^n) + 3*a) ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17458 vs. \(2 (149) = 298\).

Time = 0.59 (sec) , antiderivative size = 17458, normalized size of antiderivative = 117.17 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(54*b^3*n^3*x*e^(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1 
/2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(ab 
s(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2*tan(1/2*a) + 54*b^3*n^3*x*e^(-1/ 
2*pi*b*n*sgn(x) + 1/2*pi*b*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(3/2*b*n*log 
(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c) 
))^2*tan(3/2*a)^2*tan(1/2*a) + 6*b^3*n^3*x*e^(3/2*pi*b*n*sgn(x) - 3/2*pi*b 
*n + 3/2*pi*b*sgn(c) - 3/2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c 
)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)*tan(1/2*a) 
^2 + 6*b^3*n^3*x*e^(-3/2*pi*b*n*sgn(x) + 3/2*pi*b*n - 3/2*pi*b*sgn(c) + 3/ 
2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(abs 
(x)) + 1/2*b*log(abs(c)))^2*tan(3/2*a)*tan(1/2*a)^2 + 54*b^3*n^3*x*e^(1/2* 
pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*tan(3/2*b*n*log(a 
bs(x)) + 3/2*b*log(abs(c)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c))) 
*tan(3/2*a)^2*tan(1/2*a)^2 + 54*b^3*n^3*x*e^(-1/2*pi*b*n*sgn(x) + 1/2*pi*b 
*n - 1/2*pi*b*sgn(c) + 1/2*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c 
)))^2*tan(1/2*b*n*log(abs(x)) + 1/2*b*log(abs(c)))*tan(3/2*a)^2*tan(1/2*a) 
^2 + 6*b^3*n^3*x*e^(3/2*pi*b*n*sgn(x) - 3/2*pi*b*n + 3/2*pi*b*sgn(c) - 3/2 
*pi*b)*tan(3/2*b*n*log(abs(x)) + 3/2*b*log(abs(c)))*tan(1/2*b*n*log(abs(x) 
) + 1/2*b*log(abs(c)))^2*tan(3/2*a)^2*tan(1/2*a)^2 + 6*b^3*n^3*x*e^(-3/2*p 
i*b*n*sgn(x) + 3/2*pi*b*n - 3/2*pi*b*sgn(c) + 3/2*pi*b)*tan(3/2*b*n*log...
 

Mupad [B] (verification not implemented)

Time = 21.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}\,3{}\mathrm {i}}{8\,b\,n+8{}\mathrm {i}}+\frac {3\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{8+b\,n\,8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{24\,b\,n+8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}{8+b\,n\,24{}\mathrm {i}} \] Input:

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

Output:

(x*exp(-a*1i)/(c*x^n)^(b*1i)*3i)/(8*b*n + 8i) + (3*x*exp(a*1i)*(c*x^n)^(b* 
1i))/(b*n*8i + 8) + (x*exp(-a*3i)/(c*x^n)^(b*3i)*1i)/(24*b*n + 8i) + (x*ex 
p(a*3i)*(c*x^n)^(b*3i))/(b*n*24i + 8)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.23 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (-\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2} b^{2} n^{2}-\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}+7 \cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) b^{2} n^{2}+\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )-3 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} b^{3} n^{3}-3 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} b n +9 \sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) b^{3} n^{3}+3 \sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) b n \right )}{9 b^{4} n^{4}+10 b^{2} n^{2}+1} \] Input:

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

Output:

(x*( - cos(log(x**n*c)*b + a)*sin(log(x**n*c)*b + a)**2*b**2*n**2 - cos(lo 
g(x**n*c)*b + a)*sin(log(x**n*c)*b + a)**2 + 7*cos(log(x**n*c)*b + a)*b**2 
*n**2 + cos(log(x**n*c)*b + a) - 3*sin(log(x**n*c)*b + a)**3*b**3*n**3 - 3 
*sin(log(x**n*c)*b + a)**3*b*n + 9*sin(log(x**n*c)*b + a)*b**3*n**3 + 3*si 
n(log(x**n*c)*b + a)*b*n))/(9*b**4*n**4 + 10*b**2*n**2 + 1)