\(\int \frac {\cos ^3(a+b \log (c x^n))}{x} \, dx\) [99]

Optimal result

Integrand size = 17, antiderivative size = 42 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

sin(a+b*ln(c*x^n))/b/n-1/3*sin(a+b*ln(c*x^n))^3/b/n

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2713} \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[In]

Int[Cos[a + b*Log[c*x^n]]^3/x,x]

[Out]

Sin[a + b*Log[c*x^n]]/(b*n) - Sin[a + b*Log[c*x^n]]^3/(3*b*n)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[In]

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

[Out]

Sin[a + b*Log[c*x^n]]/(b*n) - Sin[a + b*Log[c*x^n]]^3/(3*b*n)

Maple [A] (verified)

Time = 4.70 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\left (2+{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 n b}\)	\(35\)
default	\(\frac {\left (2+{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{3 n b}\)	\(35\)
parallelrisch	\(\frac {\sin \left (3 b \ln \left (c \,x^{n}\right )+3 a \right )+9 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )}{12 b n}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, b n} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (32) = 64\).

Time = 1.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.69 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \cos ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \cos ^{3}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {2 \sin ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{3 b n} + \frac {\sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{b n} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (40) = 80\).

Time = 0.23 (sec) , antiderivative size = 232, normalized size of antiderivative = 5.52 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (\cos \left (3 \, b \log \left (c\right )\right ) \sin \left (6 \, b \log \left (c\right )\right ) - \cos \left (6 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \sin \left (3 \, b \log \left (c\right )\right )\right )} \cos \left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right ) + 9 \, {\left (\cos \left (3 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) - \cos \left (3 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right )\right )} \cos \left (b \log \left (x^{n}\right ) + a\right ) + {\left (\cos \left (6 \, b \log \left (c\right )\right ) \cos \left (3 \, b \log \left (c\right )\right ) + \sin \left (6 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \cos \left (3 \, b \log \left (c\right )\right )\right )} \sin \left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right ) + 9 \, {\left (\cos \left (4 \, b \log \left (c\right )\right ) \cos \left (3 \, b \log \left (c\right )\right ) + \cos \left (3 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (3 \, b \log \left (c\right )\right ) + \sin \left (3 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right )\right )} \sin \left (b \log \left (x^{n}\right ) + a\right )}{24 \, b n} \]

[In]

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

[Out]

1/24*((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)))*cos(3*b*log(x^n) +
3*a) + 9*(cos(3*b*log(c))*sin(4*b*log(c)) - cos(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)))*cos(b*log(x^n) + a) + (cos(6*b*log(c))*cos(3*b*log(c)) + sin(6*b*log(c))*s
in(3*b*log(c)) + cos(3*b*log(c)))*sin(3*b*log(x^n) + 3*a) + 9*(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)))*sin(b*log(x^n) + a))/(
b*n)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 27.89 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3\,\sin \left (a+b\,\ln \left (c\,x^n\right )\right )-{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{3\,b\,n} \]

[In]

int(cos(a + b*log(c*x^n))^3/x,x)

[Out]

(3*sin(a + b*log(c*x^n)) - sin(a + b*log(c*x^n))^3)/(3*b*n)