Integrand size = 13, antiderivative size = 149 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 b^2 n^2 x \cos \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac {x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2}+\frac {6 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right )}{1+10 b^2 n^2+9 b^4 n^4}+\frac {3 b n x \cos ^2\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+9 b^2 n^2} \]
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)
Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.79 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (3 \left (1+9 b^2 n^2\right ) \cos \left (a+b \log \left (c x^n\right )\right )+\left (1+b^2 n^2\right ) \cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+6 b n \left (1+5 b^2 n^2+\left (1+b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )\right )}{4+40 b^2 n^2+36 b^4 n^4} \]
(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)
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 |
\(\displaystyle \frac {x \cos ^3\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {3 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^2\left (a+b \log \left (c x^n\right )\right )}{9 b^2 n^2+1}+\frac {6 b^2 n^2 \left (\frac {b n x \sin \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}+\frac {x \cos \left (a+b \log \left (c x^n\right )\right )}{b^2 n^2+1}\right )}{9 b^2 n^2+1}\) |
(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)
3.1.98.3.1 Defintions of rubi rules used
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]
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]
Time = 3.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.76
method | result | size |
parallelrisch | \(\frac {27 \left (\frac {\left (b^{2} n^{2}+1\right ) \cos \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{27}+\frac {b n \left (b^{2} n^{2}+1\right ) \sin \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{9}+\left (b^{2} n^{2}+\frac {1}{9}\right ) \left (\sin \left (a +b \ln \left (c \,x^{n}\right )\right ) b n +\cos \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right ) x}{4 \left (9 b^{4} n^{4}+10 b^{2} n^{2}+1\right )}\) | \(113\) |
default | \(\frac {3 \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 n^{2} \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {3 b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{4 n \left (\frac {1}{n^{2}}+b^{2}\right )}+\frac {{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{4 n^{2} \left (\frac {1}{n^{2}}+9 b^{2}\right )}+\frac {3 b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )}{4 n \left (\frac {1}{n^{2}}+9 b^{2}\right )}\) | \(190\) |
27/4*(1/27*(b^2*n^2+1)*cos(3*b*ln(c*x^n)+3*a)+1/9*b*n*(b^2*n^2+1)*sin(3*b* ln(c*x^n)+3*a)+(b^2*n^2+1/9)*(sin(a+b*ln(c*x^n))*b*n+cos(a+b*ln(c*x^n))))* x/(9*b^4*n^4+10*b^2*n^2+1)
Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.80 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {6 \, b^{2} n^{2} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (b^{2} n^{2} + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, {\left (2 \, b^{3} n^{3} x + {\left (b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{9 \, b^{4} n^{4} + 10 \, b^{2} n^{2} + 1} \]
(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)
\[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \cos ^{3}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = - \frac {i}{n} \\\int \cos ^{3}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = - \frac {i}{3 n} \\\int \cos ^{3}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {i}{3 n} \\\int \cos ^{3}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {i}{n} \\\frac {6 b^{3} n^{3} x \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {9 b^{3} n^{3} x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {6 b^{2} n^{2} x \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {7 b^{2} n^{2} x \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {3 b n x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} + \frac {x \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} + 10 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]
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))
Leaf count of result is larger than twice the leaf count of optimal. 989 vs. \(2 (149) = 298\).
Time = 0.26 (sec) , antiderivative size = 989, normalized size of antiderivative = 6.64 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
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) ...
Leaf count of result is larger than twice the leaf count of optimal. 17458 vs. \(2 (149) = 298\).
Time = 0.80 (sec) , antiderivative size = 17458, normalized size of antiderivative = 117.17 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
-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...
Time = 27.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.77 \[ \int \cos ^3\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}\,3{}\mathrm {i}}{8\,b\,n+8{}\mathrm {i}}+\frac {3\,x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{8+b\,n\,8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}\,1{}\mathrm {i}}{24\,b\,n+8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,3{}\mathrm {i}}}{8+b\,n\,24{}\mathrm {i}} \]