Integrand size = 13, antiderivative size = 191 \[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {24 b^4 n^4 x}{1+20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac {x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac {24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac {4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2} \]
24*b^4*n^4*x/(64*b^4*n^4+20*b^2*n^2+1)+12*b^2*n^2*x*cos(a+b*ln(c*x^n))^2/( 64*b^4*n^4+20*b^2*n^2+1)+x*cos(a+b*ln(c*x^n))^4/(16*b^2*n^2+1)+24*b^3*n^3* x*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/(64*b^4*n^4+20*b^2*n^2+1)+4*b*n*x* cos(a+b*ln(c*x^n))^3*sin(a+b*ln(c*x^n))/(16*b^2*n^2+1)
Time = 0.39 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (3+60 b^2 n^2+192 b^4 n^4+\left (4+64 b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (1+4 b^2 n^2\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 \left (1+20 b^2 n^2+64 b^4 n^4\right )} \]
(x*(3 + 60*b^2*n^2 + 192*b^4*n^4 + (4 + 64*b^2*n^2)*Cos[2*(a + b*Log[c*x^n ])] + (1 + 4*b^2*n^2)*Cos[4*(a + b*Log[c*x^n])] + 8*b*n*Sin[2*(a + b*Log[c *x^n])] + 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 4*b*n*Sin[4*(a + b*Log[c *x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(1 + 20*b^2*n^2 + 64*b ^4*n^4))
Time = 0.32 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4981, 4981, 24}
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.
| \(\displaystyle \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 4981 |
| \(\displaystyle \frac {12 b^2 n^2 \int \cos ^2\left (a+b \log \left (c x^n\right )\right )dx}{16 b^2 n^2+1}+\frac {x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac {4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}\) |
| \(\Big \downarrow \) 4981 |
| \(\displaystyle \frac {12 b^2 n^2 \left (\frac {2 b^2 n^2 \int 1dx}{4 b^2 n^2+1}+\frac {x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}+\frac {2 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}\right )}{16 b^2 n^2+1}+\frac {x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac {4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac {4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac {12 b^2 n^2 \left (\frac {x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}+\frac {2 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+1}+\frac {2 b^2 n^2 x}{4 b^2 n^2+1}\right )}{16 b^2 n^2+1}\) |
(x*Cos[a + b*Log[c*x^n]]^4)/(1 + 16*b^2*n^2) + (4*b*n*x*Cos[a + b*Log[c*x^ n]]^3*Sin[a + b*Log[c*x^n]])/(1 + 16*b^2*n^2) + (12*b^2*n^2*((2*b^2*n^2*x) /(1 + 4*b^2*n^2) + (x*Cos[a + b*Log[c*x^n]]^2)/(1 + 4*b^2*n^2) + (2*b*n*x* Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(1 + 4*b^2*n^2)))/(1 + 16*b^2 *n^2)
3.2.1.3.1 Defintions of rubi rules used
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 = 7.80 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.69
| method | result | size |
| parallelrisch | \(\frac {128 \left (\left (\frac {1}{8} b^{3} n^{3}+\frac {1}{32} b n \right ) \sin \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+\left (\frac {b^{2} n^{2}}{32}+\frac {1}{128}\right ) \cos \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+\left (\frac {3 b^{2} n^{2}}{2}+b n \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+\frac {\cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{2}+\frac {3}{8}\right ) \left (b^{2} n^{2}+\frac {1}{16}\right )\right ) x}{512 b^{4} n^{4}+160 b^{2} n^{2}+8}\) | \(131\) |
| default | \(\frac {3 x}{8}+\frac {{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{2 n^{2} \left (\frac {1}{n^{2}}+4 b^{2}\right )}+\frac {b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{n \left (\frac {1}{n^{2}}+4 b^{2}\right )}+\frac {{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \cos \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )}{8 n^{2} \left (\frac {1}{n^{2}}+16 b^{2}\right )}+\frac {b \,{\mathrm e}^{\frac {\ln \left (c \,x^{n}\right )}{n}-\frac {\ln \left (c \right )}{n}} \sin \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )}{2 n \left (\frac {1}{n^{2}}+16 b^{2}\right )}\) | \(202\) |
128*((1/8*b^3*n^3+1/32*b*n)*sin(4*b*ln(c*x^n)+4*a)+(1/32*b^2*n^2+1/128)*co s(4*b*ln(c*x^n)+4*a)+(3/2*b^2*n^2+b*n*sin(2*b*ln(c*x^n)+2*a)+1/2*cos(2*b*l n(c*x^n)+2*a)+3/8)*(b^2*n^2+1/16))*x/(512*b^4*n^4+160*b^2*n^2+8)
Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.75 \[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {24 \, b^{4} n^{4} x + 12 \, b^{2} n^{2} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + {\left (4 \, b^{2} n^{2} + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, {\left (6 \, b^{3} n^{3} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (4 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{64 \, b^{4} n^{4} + 20 \, b^{2} n^{2} + 1} \]
(24*b^4*n^4*x + 12*b^2*n^2*x*cos(b*n*log(x) + b*log(c) + a)^2 + (4*b^2*n^2 + 1)*x*cos(b*n*log(x) + b*log(c) + a)^4 + 4*(6*b^3*n^3*x*cos(b*n*log(x) + b*log(c) + a) + (4*b^3*n^3 + b*n)*x*cos(b*n*log(x) + b*log(c) + a)^3)*sin (b*n*log(x) + b*log(c) + a))/(64*b^4*n^4 + 20*b^2*n^2 + 1)
\[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \cos ^{4}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = - \frac {i}{2 n} \\\int \cos ^{4}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = - \frac {i}{4 n} \\\int \cos ^{4}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = \frac {i}{4 n} \\\int \cos ^{4}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {i}{2 n} \\\frac {24 b^{4} n^{4} x \sin ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {48 b^{4} n^{4} x \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {24 b^{4} n^{4} x \cos ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {24 b^{3} n^{3} x \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos {\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {40 b^{3} n^{3} x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {12 b^{2} n^{2} x \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {16 b^{2} n^{2} x \cos ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {4 b n x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {x \cos ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \]
Piecewise((Integral(cos(a - I*log(c*x**n)/(2*n))**4, x), Eq(b, -I/(2*n))), (Integral(cos(a - I*log(c*x**n)/(4*n))**4, x), Eq(b, -I/(4*n))), (Integra l(cos(a + I*log(c*x**n)/(4*n))**4, x), Eq(b, I/(4*n))), (Integral(cos(a + I*log(c*x**n)/(2*n))**4, x), Eq(b, I/(2*n))), (24*b**4*n**4*x*sin(a + b*lo g(c*x**n))**4/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 48*b**4*n**4*x*sin(a + b *log(c*x**n))**2*cos(a + b*log(c*x**n))**2/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 24*b**4*n**4*x*cos(a + b*log(c*x**n))**4/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 24*b**3*n**3*x*sin(a + b*log(c*x**n))**3*cos(a + b*log(c*x**n))/(6 4*b**4*n**4 + 20*b**2*n**2 + 1) + 40*b**3*n**3*x*sin(a + b*log(c*x**n))*co s(a + b*log(c*x**n))**3/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 12*b**2*n**2*x *sin(a + b*log(c*x**n))**2*cos(a + b*log(c*x**n))**2/(64*b**4*n**4 + 20*b* *2*n**2 + 1) + 16*b**2*n**2*x*cos(a + b*log(c*x**n))**4/(64*b**4*n**4 + 20 *b**2*n**2 + 1) + 4*b*n*x*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**3 /(64*b**4*n**4 + 20*b**2*n**2 + 1) + x*cos(a + b*log(c*x**n))**4/(64*b**4* n**4 + 20*b**2*n**2 + 1), True))
Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (191) = 382\).
Time = 0.27 (sec) , antiderivative size = 1078, normalized size of antiderivative = 5.64 \[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
1/16*((16*(b^3*cos(4*b*log(c))*sin(8*b*log(c)) - b^3*cos(8*b*log(c))*sin(4 *b*log(c)) + b^3*sin(4*b*log(c)))*n^3 + 4*(b^2*cos(8*b*log(c))*cos(4*b*log (c)) + b^2*sin(8*b*log(c))*sin(4*b*log(c)) + b^2*cos(4*b*log(c)))*n^2 + 4* (b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b *sin(4*b*log(c)))*n + cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*si n(4*b*log(c)) + cos(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*a) + 4*(32*(b^3*co s(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)) + 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 + 16*(b^2*cos(6*b*log(c))*cos(4*b*log(c)) + b^2*cos(4*b*log(c))*cos(2*b*l og(c)) + b^2*sin(6*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b *log(c)))*n^2 + 2*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*s in(4*b*log(c)) + b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin (2*b*log(c)))*n + cos(6*b*log(c))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2* b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)) + sin(4*b*log(c))*sin(2*b*log( c)))*x*cos(2*b*log(x^n) + 2*a) + (16*(b^3*cos(8*b*log(c))*cos(4*b*log(c)) + b^3*sin(8*b*log(c))*sin(4*b*log(c)) + b^3*cos(4*b*log(c)))*n^3 - 4*(b^2* cos(4*b*log(c))*sin(8*b*log(c)) - b^2*cos(8*b*log(c))*sin(4*b*log(c)) + b^ 2*sin(4*b*log(c)))*n^2 + 4*(b*cos(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b* log(c))*sin(4*b*log(c)) + b*cos(4*b*log(c)))*n - cos(4*b*log(c))*sin(8*b*l og(c)) + cos(8*b*log(c))*sin(4*b*log(c)) - sin(4*b*log(c)))*x*sin(4*b*l...
Leaf count of result is larger than twice the leaf count of optimal. 16422 vs. \(2 (191) = 382\).
Time = 0.76 (sec) , antiderivative size = 16422, normalized size of antiderivative = 85.98 \[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\text {Too large to display} \]
3/8*x - 1/16*(256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b )*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(a bs(c)))^2*tan(2*a)^2*tan(a) + 256*b^3*n^3*x*e^(-pi*b*n*sgn(x) + pi*b*n - p i*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log( abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a) + 32*b^3*n^3*x*e^(2*pi*b*n*sg n(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log( abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)*tan(a)^2 + 32*b ^3*n^3*x*e^(-2*pi*b*n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b* n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2* tan(2*a)*tan(a)^2 + 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b *log(abs(c)))*tan(2*a)^2*tan(a)^2 + 256*b^3*n^3*x*e^(-pi*b*n*sgn(x) + pi*b *n - pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b* n*log(abs(x)) + b*log(abs(c)))*tan(2*a)^2*tan(a)^2 + 32*b^3*n^3*x*e^(2*pi* b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*n*log(abs(x)) + 2* b*log(abs(c)))*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 + 32*b^3*n^3*x*e^(-2*pi*b*n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*ta n(2*b*n*log(abs(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b*log(abs(c)) )^2*tan(2*a)^2*tan(a)^2 - 4*b^2*n^2*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi *b*sgn(c) - 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*...
Time = 27.22 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.61 \[ \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3\,x}{8}+\frac {x\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,b\,n+4{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{4+b\,n\,8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{-a\,4{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}\,1{}\mathrm {i}}{64\,b\,n+16{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}{16+b\,n\,64{}\mathrm {i}} \]