### 3.43 $$\int \sin (2 x) \sin (4 x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x)$

[Out]

Sin[2*x]/4 - Sin[6*x]/12

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Rubi [A]  time = 0.0080002, 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} $\frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Sin[2*x]*Sin[4*x],x]

[Out]

Sin[2*x]/4 - Sin[6*x]/12

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 (2 x) \sin (4 x) \, dx &=\frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x)\\ \end{align*}

Mathematica [A]  time = 0.0061412, size = 17, normalized size = 1. $\frac{1}{4} \sin (2 x)-\frac{1}{12} \sin (6 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sin[2*x]*Sin[4*x],x]

[Out]

Sin[2*x]/4 - Sin[6*x]/12

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sin(2*x)*sin(4*x),x)

[Out]

1/3*sin(2*x)^3

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Maxima [A]  time = 0.932666, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{12} \, \sin \left (6 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(4*x),x, algorithm="maxima")

[Out]

-1/12*sin(6*x) + 1/4*sin(2*x)

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Fricas [A]  time = 2.41495, size = 43, normalized size = 2.53 \begin{align*} -\frac{1}{3} \,{\left (\cos \left (2 \, x\right )^{2} - 1\right )} \sin \left (2 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(4*x),x, algorithm="fricas")

[Out]

-1/3*(cos(2*x)^2 - 1)*sin(2*x)

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Sympy [A]  time = 0.558186, size = 22, normalized size = 1.29 \begin{align*} - \frac{\sin{\left (2 x \right )} \cos{\left (4 x \right )}}{3} + \frac{\sin{\left (4 x \right )} \cos{\left (2 x \right )}}{6} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(4*x),x)

[Out]

-sin(2*x)*cos(4*x)/3 + sin(4*x)*cos(2*x)/6

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Giac [A]  time = 1.10083, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{12} \, \sin \left (6 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(2*x)*sin(4*x),x, algorithm="giac")

[Out]

-1/12*sin(6*x) + 1/4*sin(2*x)