### 3.283 $$\int \cos (3 x) \cos (5 x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x)$

[Out]

Sin[2*x]/4 + Sin[8*x]/16

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Rubi [A]  time = 0.0078319, 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 = {4283} $\frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sin[2*x]/4 + Sin[8*x]/16

Rule 4283

Int[cos[(a_.) + (b_.)*(x_)]*cos[(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*} \int \cos (3 x) \cos (5 x) \, dx &=\frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x)\\ \end{align*}

Mathematica [A]  time = 0.0070379, size = 17, normalized size = 1. $\frac{1}{4} \sin (2 x)+\frac{1}{16} \sin (8 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Sin[2*x]/4 + Sin[8*x]/16

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Maple [A]  time = 0.041, size = 14, normalized size = 0.8 \begin{align*}{\frac{\sin \left ( 2\,x \right ) }{4}}+{\frac{\sin \left ( 8\,x \right ) }{16}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sin(2*x)+1/16*sin(8*x)

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Maxima [A]  time = 0.948322, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{16} \, \sin \left (8 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sin(8*x) + 1/4*sin(2*x)

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Fricas [A]  time = 2.00115, size = 65, normalized size = 3.82 \begin{align*}{\left (8 \, \cos \left (x\right )^{7} - 12 \, \cos \left (x\right )^{5} + 5 \, \cos \left (x\right )^{3}\right )} \sin \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(8*cos(x)^7 - 12*cos(x)^5 + 5*cos(x)^3)*sin(x)

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Sympy [B]  time = 0.532001, size = 26, normalized size = 1.53 \begin{align*} - \frac{3 \sin{\left (3 x \right )} \cos{\left (5 x \right )}}{16} + \frac{5 \sin{\left (5 x \right )} \cos{\left (3 x \right )}}{16} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.05692, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{16} \, \sin \left (8 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sin(8*x) + 1/4*sin(2*x)