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

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

[Out]

1/2*sin(x)-1/6*sin(3*x)

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 - Sin[3*x]/6

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fricas [A]  time = 0.44, size = 10, normalized size = 0.67 \[ -\frac {2}{3} \, {\left (\cos \relax (x)^{2} - 1\right )} \sin \relax (x) \]

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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giac [A]  time = 1.14, size = 6, normalized size = 0.40 \[ \frac {2}{3} \, \sin \relax (x)^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(x)^3

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maple [A]  time = 0.03, size = 7, normalized size = 0.47 \[ \frac {2 \left (\sin ^{3}\relax (x )\right )}{3} \]

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 = 0.42, size = 11, normalized size = 0.73 \[ -\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \relax (x) \]

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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mupad [B]  time = 0.03, size = 6, normalized size = 0.40 \[ \frac {2\,{\sin \relax (x)}^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.57, size = 20, normalized size = 1.33 \[ - \frac {2 \sin {\relax (x )} \cos {\left (2 x \right )}}{3} + \frac {\sin {\left (2 x \right )} \cos {\relax (x )}}{3} \]

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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