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

Optimal. Leaf size=43 \[ \frac{\cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0317094, antiderivative size = 43, 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} \[ \frac{\cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^3/x,x]

[Out]

-(Cos[a + b*Log[c*x^n]]/(b*n)) + Cos[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{\sin ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sin ^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,\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{b n}+\frac{\cos ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n}\\ \end{align*}

Mathematica [A]  time = 0.0591127, size = 45, normalized size = 1.05 \[ \frac{\cos \left (3 \left (a+b \log \left (c x^n\right )\right )\right )}{12 b n}-\frac{3 \cos \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Cos[a + b*Log[c*x^n]])/(4*b*n) + Cos[3*(a + b*Log[c*x^n])]/(12*b*n)

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Maple [A]  time = 0.025, size = 35, normalized size = 0.8 \begin{align*} -{\frac{ \left ( 2+ \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \cos \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(sin(a+b*ln(c*x^n))^3/x,x)

[Out]

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

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Maxima [B]  time = 1.10253, size = 315, normalized size = 7.33 \begin{align*} \frac{{\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 )} \cos \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 )} \cos \left (b \log \left (x^{n}\right ) + a\right ) -{\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 )} \sin \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 )} \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(sin(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.491843, size = 109, normalized size = 2.53 \begin{align*} \frac{\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 )}{3 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 55.5855, size = 83, normalized size = 1.93 \begin{align*} \begin{cases} \log{\left (x \right )} \sin ^{3}{\left (a \right )} & \text{for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log{\left (x \right )} \sin ^{3}{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\- \frac{\sin ^{2}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )} \cos{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{b n} - \frac{2 \cos ^{3}{\left (a + b n \log{\left (x \right )} + b \log{\left (c \right )} \right )}}{3 b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \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(sin(a+b*log(c*x^n))^3/x,x, algorithm="giac")

[Out]

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