3.42 \(\int \cos (5 x) \sin (3 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{4} \cos (2 x)-\frac{1}{16} \cos (8 x) \]

[Out]

Cos[2*x]/4 - Cos[8*x]/16

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Rubi [A]  time = 0.0085067, antiderivative size = 17, normalized size of antiderivative = 1., 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} \[ \frac{1}{4} \cos (2 x)-\frac{1}{16} \cos (8 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[5*x]*Sin[3*x],x]

[Out]

Cos[2*x]/4 - Cos[8*x]/16

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 (5 x) \sin (3 x) \, dx &=\frac{1}{4} \cos (2 x)-\frac{1}{16} \cos (8 x)\\ \end{align*}

Mathematica [A]  time = 0.0074548, size = 17, normalized size = 1. \[ \frac{\cos ^2(x)}{2}-\frac{1}{16} \cos (8 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[5*x]*Sin[3*x],x]

[Out]

Cos[x]^2/2 - Cos[8*x]/16

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Maple [A]  time = 0.074, size = 14, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( 2\,x \right ) }{4}}-{\frac{\cos \left ( 8\,x \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(5*x)*sin(3*x),x)

[Out]

1/4*cos(2*x)-1/16*cos(8*x)

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Maxima [A]  time = 0.933849, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{16} \, \cos \left (8 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/16*cos(8*x) + 1/4*cos(2*x)

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Fricas [A]  time = 2.69187, size = 76, normalized size = 4.47 \begin{align*} -8 \, \cos \left (x\right )^{8} + 16 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + \frac{5}{2} \, \cos \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x, algorithm="fricas")

[Out]

-8*cos(x)^8 + 16*cos(x)^6 - 10*cos(x)^4 + 5/2*cos(x)^2

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Sympy [B]  time = 0.512751, size = 26, normalized size = 1.53 \begin{align*} \frac{5 \sin{\left (3 x \right )} \sin{\left (5 x \right )}}{16} + \frac{3 \cos{\left (3 x \right )} \cos{\left (5 x \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x)

[Out]

5*sin(3*x)*sin(5*x)/16 + 3*cos(3*x)*cos(5*x)/16

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Giac [A]  time = 1.05937, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{16} \, \cos \left (8 \, x\right ) + \frac{1}{4} \, \cos \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x, algorithm="giac")

[Out]

-1/16*cos(8*x) + 1/4*cos(2*x)