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

Optimal. Leaf size=42 \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

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

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Rubi [A]  time = 0.032968, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2633} \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[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 2633

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*} \int \frac{\cos ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cos ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\operatorname{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]  time = 0.0561742, size = 42, normalized size = 1. \[ \frac{\sin \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.033, size = 35, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{3\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.11751, size = 313, normalized size = 7.45 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 0.489189, size = 109, normalized size = 2.6 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 27.5633, size = 82, normalized size = 1.95 \begin{align*} \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 n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{3 b n} + \frac{\sin{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )} \cos ^{2}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*n*log(x) + b*log(c))**3/(3*b*n) + sin(a + b*n*log(x) + b*log(c))*cos(a + b*n*log(x) + b*log(c))**2/(b*
n), True))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)