\(\int \cos (x) \sin (3 x) \, dx\) [107]

Optimal result

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

[Out]

-1/4*cos(2*x)-1/8*cos(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 = {4369} \[ \int \cos (x) \sin (3 x) \, dx=-\frac {1}{4} \cos (2 x)-\frac {1}{8} \cos (4 x) \]

[In]

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

[Out]

-1/4*Cos[2*x] - Cos[4*x]/8

Rule 4369

Int[cos[(c_.) + (d_.)*(x_)]*sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[-Cos[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Cos[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} \cos (2 x)-\frac {1}{8} \cos (4 x) \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

-1/2*Cos[x]^2 - Cos[4*x]/8

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
default	\(-\frac {\cos \left (2 x \right )}{4}-\frac {\cos \left (4 x \right )}{8}\)	\(14\)
risch	\(-\frac {\cos \left (2 x \right )}{4}-\frac {\cos \left (4 x \right )}{8}\)	\(14\)
parallelrisch	\(-\frac {\cos \left (4 x \right )}{8}+\frac {3}{8}-\frac {\cos \left (2 x \right )}{4}\)	\(15\)
norman	\(\frac {\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{4}-\frac {\tan \left (\frac {x}{2}\right ) \tan \left (\frac {3 x}{2}\right )}{2}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right )}\)	\(49\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin (3 x) \, dx=-\cos \left (x\right )^{4} + \frac {1}{2} \, \cos \left (x\right )^{2} \]

[In]

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

[Out]

-cos(x)^4 + 1/2*cos(x)^2

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.29 \[ \int \cos (x) \sin (3 x) \, dx=- \frac {\sin {\left (x \right )} \sin {\left (3 x \right )}}{8} - \frac {3 \cos {\left (x \right )} \cos {\left (3 x \right )}}{8} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin (3 x) \, dx=-\frac {1}{8} \, \cos \left (4 \, x\right ) - \frac {1}{4} \, \cos \left (2 \, x\right ) \]

[In]

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

[Out]

-1/8*cos(4*x) - 1/4*cos(2*x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin (3 x) \, dx=-\cos \left (x\right )^{4} + \frac {1}{2} \, \cos \left (x\right )^{2} \]

[In]

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

[Out]

-cos(x)^4 + 1/2*cos(x)^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \sin (3 x) \, dx=\frac {{\cos \left (x\right )}^2}{2}-{\cos \left (x\right )}^4 \]

[In]

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

[Out]

cos(x)^2/2 - cos(x)^4