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

Optimal result

Integrand size = 7, antiderivative size = 15 \[ \int \sin (x) \sin (2 x) \, dx=\frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4367} \[ \int \sin (x) \sin (2 x) \, dx=\frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \]

[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*} \text {integral}& = \frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \sin (x) \sin (2 x) \, dx=\frac {\sin (x)}{2}-\frac {1}{6} \sin (3 x) \]

[In]

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

[Out]

Sin[x]/2 - Sin[3*x]/6

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.67 \[ \int \sin (x) \sin (2 x) \, dx=-\frac {2}{3} \, {\left (\cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]

[In]

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

[Out]

-2/3*(cos(x)^2 - 1)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.33 \[ \int \sin (x) \sin (2 x) \, dx=- \frac {2 \sin {\left (x \right )} \cos {\left (2 x \right )}}{3} + \frac {\sin {\left (2 x \right )} \cos {\left (x \right )}}{3} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.73 \[ \int \sin (x) \sin (2 x) \, dx=-\frac {1}{6} \, \sin \left (3 \, x\right ) + \frac {1}{2} \, \sin \left (x\right ) \]

[In]

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

[Out]

-1/6*sin(3*x) + 1/2*sin(x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.40 \[ \int \sin (x) \sin (2 x) \, dx=\frac {2}{3} \, \sin \left (x\right )^{3} \]

[In]

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

[Out]

2/3*sin(x)^3

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.40 \[ \int \sin (x) \sin (2 x) \, dx=\frac {2\,{\sin \left (x\right )}^3}{3} \]

[In]

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

[Out]

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