\(\int \frac {\cos ^4(a+b \log (c x^n))}{x} \, dx\) [102]

Optimal result

Integrand size = 17, antiderivative size = 73 \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \log (x)}{8}+\frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

[Out]

3/8*ln(x)+3/8*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/b/n+1/4*cos(a+b*ln(c*x^n))^3*sin(a+b*ln(c*x^n))/b/n

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2715, 8} \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {3 \log (x)}{8} \]

[In]

Int[Cos[a + b*Log[c*x^n]]^4/x,x]

[Out]

(3*Log[x])/8 + (3*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]])/(8*b*n) + (Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Lo
g[c*x^n]])/(4*b*n)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \cos ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int \cos ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n} \\ & = \frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n} \\ & = \frac {3 \log (x)}{8}+\frac {3 \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{4 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \left (a+b \log \left (c x^n\right )\right )+8 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]

[In]

Integrate[Cos[a + b*Log[c*x^n]]^4/x,x]

[Out]

(12*(a + b*Log[c*x^n]) + 8*Sin[2*(a + b*Log[c*x^n])] + Sin[4*(a + b*Log[c*x^n])])/(32*b*n)

Maple [A] (verified)

Time = 12.88 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63

method	result	size
parallelrisch	\(\frac {12 \ln \left (x \right ) b n +\sin \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+8 \sin \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )}{32 b n}\)	\(46\)
derivativedivides	\(\frac {\frac {\left ({\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}+\frac {3 \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\)	\(61\)
default	\(\frac {\frac {\left ({\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}+\frac {3 \cos \left (a +b \ln \left (c \,x^{n}\right )\right )}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{4}+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\)	\(61\)

[In]

int(cos(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)

[Out]

1/32*(12*ln(x)*b*n+sin(4*b*ln(c*x^n)+4*a)+8*sin(2*b*ln(c*x^n)+2*a))/b/n

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.81 \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \, b n \log \left (x\right ) + {\left (2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \]

[In]

integrate(cos(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

1/8*(3*b*n*log(x) + (2*cos(b*n*log(x) + b*log(c) + a)^3 + 3*cos(b*n*log(x) + b*log(c) + a))*sin(b*n*log(x) + b
*log(c) + a))/(b*n)

Sympy [A] (verification not implemented)

Time = 7.54 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.37 \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\begin {cases} \log {\left (x \right )} \cos {\left (2 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (2 a + 2 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (2 a + 2 b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases}}{2} + \frac {\begin {cases} \log {\left (x \right )} \cos {\left (4 a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos {\left (4 a + 4 b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\sin {\left (4 a + 4 b \log {\left (c x^{n} \right )} \right )}}{4 b n} & \text {otherwise} \end {cases}}{8} + \frac {3 \log {\left (x \right )}}{8} \]

[In]

integrate(cos(a+b*ln(c*x**n))**4/x,x)

[Out]

Piecewise((log(x)*cos(2*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*cos(2*a + 2*b*log(c)), Eq(n, 0)), (sin(
2*a + 2*b*log(c*x**n))/(2*b*n), True))/2 + Piecewise((log(x)*cos(4*a), Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log
(x)*cos(4*a + 4*b*log(c)), Eq(n, 0)), (sin(4*a + 4*b*log(c*x**n))/(4*b*n), True))/8 + 3*log(x)/8

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \, b n \log \left (x\right ) + \cos \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) \sin \left (4 \, b \log \left (c\right )\right ) + 8 \, \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right ) + 8 \, \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{32 \, b n} \]

[In]

integrate(cos(a+b*log(c*x^n))^4/x,x, algorithm="maxima")

[Out]

1/32*(12*b*n*log(x) + cos(4*b*log(x^n) + 4*a)*sin(4*b*log(c)) + 8*cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + co
s(4*b*log(c))*sin(4*b*log(x^n) + 4*a) + 8*cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(b*n)

Giac [F]

\[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\cos \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x} \,d x } \]

[In]

integrate(cos(a+b*log(c*x^n))^4/x,x, algorithm="giac")

[Out]

integrate(cos(b*log(c*x^n) + a)^4/x, x)

Mupad [B] (verification not implemented)

Time = 27.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3\,\ln \left (x^n\right )}{8\,n}+\frac {\frac {\sin \left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}+\frac {\sin \left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \]

[In]

int(cos(a + b*log(c*x^n))^4/x,x)

[Out]

(3*log(x^n))/(8*n) + (sin(2*a + 2*b*log(c*x^n))/4 + sin(4*a + 4*b*log(c*x^n))/32)/(b*n)