3.1.91 \(\int \sin (x) \sin (2 x) \, dx\) [91]

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 = 0.00, 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 = {4367} \begin {gather*} \frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 - Sin[3*x]/6

Rule 4367

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 \begin {gather*} \frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[x]/2 - Sin[3*x]/6

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Mathics [A]
time = 1.95, size = 6, normalized size = 0.40 \begin {gather*} \frac {2 \text {Sin}\left [x\right ]^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[x]*Sin[2*x],x]')

[Out]

2 Sin[x] ^ 3 / 3

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Maple [A]
time = 0.03, size = 7, normalized size = 0.47

method	result	size
derivativedivides	\(\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(7\)
default	\(\frac {2 \left (\sin ^{3}\left (x \right )\right )}{3}\)	\(7\)
risch	\(\frac {\sin \left (x \right )}{2}-\frac {\sin \left (3 x \right )}{6}\)	\(12\)
norman	\(\frac {-\frac {2 \tan \left (x \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}+\frac {4 \left (\tan ^{2}\left (x \right )\right ) \tan \left (\frac {x}{2}\right )}{3}+\frac {2 \tan \left (x \right )}{3}-\frac {4 \tan \left (\frac {x}{2}\right )}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (x \right )\right )}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)*sin(2*x),x,method=_RETURNVERBOSE)

[Out]

2/3*sin(x)^3

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Maxima [A]
time = 0.26, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \end {gather*}

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 = 0.35, size = 10, normalized size = 0.67 \begin {gather*} -\frac {2}{3} \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \end {gather*}

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 = 0.14, size = 20, normalized size = 1.33 \begin {gather*} - \frac {2 \sin {\left (x \right )} \cos {\left (2 x \right )}}{3} + \frac {\sin {\left (2 x \right )} \cos {\left (x \right )}}{3} \end {gather*}

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 = 0.00, size = 8, normalized size = 0.53 \begin {gather*} \frac {2}{3} \sin ^{3}x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sin(x)^3

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Mupad [B]
time = 0.03, size = 6, normalized size = 0.40 \begin {gather*} \frac {2\,{\sin \left (x\right )}^3}{3} \end {gather*}

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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