3.655 \(\int \sin (3 x) \sin (\cos (3 x)) \, dx\)

Optimal. Leaf size=9 \[ \frac{1}{3} \cos (\cos (3 x)) \]

[Out]

Cos[Cos[3*x]]/3

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Rubi [A]  time = 0.0107249, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4335, 2638} \[ \frac{1}{3} \cos (\cos (3 x)) \]

Antiderivative was successfully verified.

[In]

Int[Sin[3*x]*Sin[Cos[3*x]],x]

[Out]

Cos[Cos[3*x]]/3

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \sin (3 x) \sin (\cos (3 x)) \, dx &=-\left (\frac{1}{3} \operatorname{Subst}(\int \sin (x) \, dx,x,\cos (3 x))\right )\\ &=\frac{1}{3} \cos (\cos (3 x))\\ \end{align*}

Mathematica [A]  time = 2.75389, size = 9, normalized size = 1. \[ \frac{1}{3} \cos (\cos (3 x)) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[3*x]*Sin[Cos[3*x]],x]

[Out]

Cos[Cos[3*x]]/3

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Maple [A]  time = 0.006, size = 8, normalized size = 0.9 \begin{align*}{\frac{\cos \left ( \cos \left ( 3\,x \right ) \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(3*x)*sin(cos(3*x)),x)

[Out]

1/3*cos(cos(3*x))

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Maxima [A]  time = 0.960353, size = 9, normalized size = 1. \begin{align*} \frac{1}{3} \, \cos \left (\cos \left (3 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(cos(3*x)),x, algorithm="maxima")

[Out]

1/3*cos(cos(3*x))

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Fricas [B]  time = 1.99499, size = 65, normalized size = 7.22 \begin{align*} \frac{1}{3} \, \cos \left (\frac{\tan \left (\frac{3}{2} \, x\right )^{2} - 1}{\tan \left (\frac{3}{2} \, x\right )^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(cos(3*x)),x, algorithm="fricas")

[Out]

1/3*cos((tan(3/2*x)^2 - 1)/(tan(3/2*x)^2 + 1))

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Sympy [A]  time = 0.532767, size = 7, normalized size = 0.78 \begin{align*} \frac{\cos{\left (\cos{\left (3 x \right )} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(cos(3*x)),x)

[Out]

cos(cos(3*x))/3

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Giac [A]  time = 1.10921, size = 9, normalized size = 1. \begin{align*} \frac{1}{3} \, \cos \left (\cos \left (3 \, x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(cos(3*x)),x, algorithm="giac")

[Out]

1/3*cos(cos(3*x))