### 3.109 $$\int \sin (3 x) \sin (6 x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x)$

[Out]

Sin[3*x]/6 - Sin[9*x]/18

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Rubi [A]  time = 0.0082184, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {4282} $\frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sin[3*x]*Sin[6*x],x]

[Out]

Sin[3*x]/6 - Sin[9*x]/18

Rule 4282

Int[sin[(a_.) + (b_.)*(x_)]*sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int \sin (3 x) \sin (6 x) \, dx &=\frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x)\\ \end{align*}

Mathematica [A]  time = 0.0064867, size = 17, normalized size = 1. $\frac{1}{6} \sin (3 x)-\frac{1}{18} \sin (9 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sin[3*x]*Sin[6*x],x]

[Out]

Sin[3*x]/6 - Sin[9*x]/18

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Maple [A]  time = 0.01, size = 9, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( \sin \left ( 3\,x \right ) \right ) ^{3}}{9}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(3*x)*sin(6*x),x)

[Out]

2/9*sin(3*x)^3

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Maxima [A]  time = 0.938837, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{18} \, \sin \left (9 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(6*x),x, algorithm="maxima")

[Out]

-1/18*sin(9*x) + 1/6*sin(3*x)

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Fricas [A]  time = 2.10254, size = 43, normalized size = 2.53 \begin{align*} -\frac{2}{9} \,{\left (\cos \left (3 \, x\right )^{2} - 1\right )} \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(6*x),x, algorithm="fricas")

[Out]

-2/9*(cos(3*x)^2 - 1)*sin(3*x)

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Sympy [A]  time = 0.53335, size = 24, normalized size = 1.41 \begin{align*} - \frac{2 \sin{\left (3 x \right )} \cos{\left (6 x \right )}}{9} + \frac{\sin{\left (6 x \right )} \cos{\left (3 x \right )}}{9} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(6*x),x)

[Out]

-2*sin(3*x)*cos(6*x)/9 + sin(6*x)*cos(3*x)/9

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Giac [A]  time = 1.05213, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{18} \, \sin \left (9 \, x\right ) + \frac{1}{6} \, \sin \left (3 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(3*x)*sin(6*x),x, algorithm="giac")

[Out]

-1/18*sin(9*x) + 1/6*sin(3*x)