3.66 \(\int \sin (x) \sin (2 x) \, dx\)

Optimal. Leaf size=15 \[ \frac{\sin (x)}{2}-\frac{1}{6} \sin (3 x) \]

[Out]

Sin[x]/2 - Sin[3*x]/6

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Rubi [A]  time = 0.0088231, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4282} \[ \frac{\sin (x)}{2}-\frac{1}{6} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 - Sin[3*x]/6

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 (x) \sin (2 x) \, dx &=\frac{\sin (x)}{2}-\frac{1}{6} \sin (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0052901, size = 15, normalized size = 1. \[ \frac{\sin (x)}{2}-\frac{1}{6} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 - Sin[3*x]/6

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Maple [A]  time = 0.01, size = 7, normalized size = 0.5 \begin{align*}{\frac{2\, \left ( \sin \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(x)^3

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Maxima [A]  time = 0.960009, size = 15, normalized size = 1. \begin{align*} -\frac{1}{6} \, \sin \left (3 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sin(3*x) + 1/2*sin(x)

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Fricas [A]  time = 2.30363, size = 38, normalized size = 2.53 \begin{align*} -\frac{2}{3} \,{\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(cos(x)^2 - 1)*sin(x)

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Sympy [A]  time = 0.549111, size = 20, normalized size = 1.33 \begin{align*} - \frac{2 \sin{\left (x \right )} \cos{\left (2 x \right )}}{3} + \frac{\sin{\left (2 x \right )} \cos{\left (x \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12975, size = 15, normalized size = 1. \begin{align*} -\frac{1}{6} \, \sin \left (3 \, x\right ) + \frac{1}{2} \, \sin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sin(3*x) + 1/2*sin(x)