3.131 \(\int \cos (3 x) \sin (2 x) \, dx\)

Optimal. Leaf size=15 \[ \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \]

[Out]

1/2*cos(x)-1/10*cos(5*x)

________________________________________________________________________________________

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 = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4284} \[ \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]*Sin[2*x],x]

[Out]

Cos[x]/2 - Cos[5*x]/10

Rule 4284

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*} \int \cos (3 x) \sin (2 x) \, dx &=\frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]*Sin[2*x],x]

[Out]

Cos[x]/2 - Cos[5*x]/10

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 13, normalized size = 0.87 \[ -\frac {8}{5} \, \cos \relax (x)^{5} + 2 \, \cos \relax (x)^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="fricas")

[Out]

-8/5*cos(x)^5 + 2*cos(x)^3

________________________________________________________________________________________

giac [A]  time = 1.02, size = 11, normalized size = 0.73 \[ -\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="giac")

[Out]

-1/10*cos(5*x) + 1/2*cos(x)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 12, normalized size = 0.80 \[ \frac {\cos \relax (x )}{2}-\frac {\cos \left (5 x \right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*sin(2*x),x)

[Out]

1/2*cos(x)-1/10*cos(5*x)

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 11, normalized size = 0.73 \[ -\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="maxima")

[Out]

-1/10*cos(5*x) + 1/2*cos(x)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 13, normalized size = 0.87 \[ 2\,{\cos \relax (x)}^3-\frac {8\,{\cos \relax (x)}^5}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*sin(2*x),x)

[Out]

2*cos(x)^3 - (8*cos(x)^5)/5

________________________________________________________________________________________

sympy [B]  time = 0.52, size = 26, normalized size = 1.73 \[ \frac {3 \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{5} + \frac {2 \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x)

[Out]

3*sin(2*x)*sin(3*x)/5 + 2*cos(2*x)*cos(3*x)/5

________________________________________________________________________________________