\(\int \sin (x) \sin (3 x) \, dx\) [67]

Optimal result

Integrand size = 7, antiderivative size = 17 \[ \int \sin (x) \sin (3 x) \, dx=\frac {1}{4} \sin (2 x)-\frac {1}{8} \sin (4 x) \]

[Out]

1/4*sin(2*x)-1/8*sin(4*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, 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} \[ \int \sin (x) \sin (3 x) \, dx=\frac {1}{4} \sin (2 x)-\frac {1}{8} \sin (4 x) \]

[In]

Int[Sin[x]*Sin[3*x],x]

[Out]

Sin[2*x]/4 - Sin[4*x]/8

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 {1}{4} \sin (2 x)-\frac {1}{8} \sin (4 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \sin (x) \sin (3 x) \, dx=\frac {1}{4} \sin (2 x)-\frac {1}{8} \sin (4 x) \]

[In]

Integrate[Sin[x]*Sin[3*x],x]

[Out]

Sin[2*x]/4 - Sin[4*x]/8

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\sin \left (2 x \right )}{4}-\frac {\sin \left (4 x \right )}{8}\)	\(14\)
risch	\(\frac {\sin \left (2 x \right )}{4}-\frac {\sin \left (4 x \right )}{8}\)	\(14\)
parallelrisch	\(\frac {\sin \left (2 x \right )}{4}-\frac {\sin \left (4 x \right )}{8}\)	\(14\)
norman	\(\frac {\frac {3 \tan \left (\frac {x}{2}\right ) \tan \left (\frac {3 x}{2}\right )^{2}}{4}-\frac {\tan \left (\frac {x}{2}\right )^{2} \tan \left (\frac {3 x}{2}\right )}{4}-\frac {3 \tan \left (\frac {x}{2}\right )}{4}+\frac {\tan \left (\frac {3 x}{2}\right )}{4}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {3 x}{2}\right )^{2}\right )}\)	\(59\)

[In]

int(sin(x)*sin(3*x),x,method=_RETURNVERBOSE)

[Out]

1/4*sin(2*x)-1/8*sin(4*x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \sin (x) \sin (3 x) \, dx=-{\left (\cos \left (x\right )^{3} - \cos \left (x\right )\right )} \sin \left (x\right ) \]

[In]

integrate(sin(x)*sin(3*x),x, algorithm="fricas")

[Out]

-(cos(x)^3 - cos(x))*sin(x)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \sin (x) \sin (3 x) \, dx=- \frac {3 \sin {\left (x \right )} \cos {\left (3 x \right )}}{8} + \frac {\sin {\left (3 x \right )} \cos {\left (x \right )}}{8} \]

[In]

integrate(sin(x)*sin(3*x),x)

[Out]

-3*sin(x)*cos(3*x)/8 + sin(3*x)*cos(x)/8

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \sin (x) \sin (3 x) \, dx=-\frac {1}{8} \, \sin \left (4 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

[In]

integrate(sin(x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/8*sin(4*x) + 1/4*sin(2*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \sin (x) \sin (3 x) \, dx=-\frac {1}{8} \, \sin \left (4 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \]

[In]

integrate(sin(x)*sin(3*x),x, algorithm="giac")

[Out]

-1/8*sin(4*x) + 1/4*sin(2*x)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \sin (x) \sin (3 x) \, dx=\frac {\sin \left (2\,x\right )}{4}-\frac {\sin \left (4\,x\right )}{8} \]

[In]

int(sin(3*x)*sin(x),x)

[Out]

sin(2*x)/4 - sin(4*x)/8