### 3.107 $$\int \cos (x) \sin (3 x) \, dx$$

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

[Out]

-Cos[2*x]/4 - Cos[4*x]/8

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Rubi [A]  time = 0.0075752, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {4284} $-\frac{1}{4} \cos (2 x)-\frac{1}{8} \cos (4 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cos[2*x]/4 - Cos[4*x]/8

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

Mathematica [A]  time = 0.0051436, size = 17, normalized size = 1. $-\frac{1}{2} \cos ^2(x)-\frac{1}{8} \cos (4 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cos[x]^2/2 - Cos[4*x]/8

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Maple [A]  time = 0.029, size = 14, normalized size = 0.8 \begin{align*} - \left ( \cos \left ( x \right ) \right ) ^{4}+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)^4+1/2*cos(x)^2

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Maxima [A]  time = 0.930472, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{8} \, \cos \left (4 \, 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(x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/8*cos(4*x) - 1/4*cos(2*x)

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Fricas [A]  time = 1.93429, size = 35, normalized size = 2.06 \begin{align*} -\cos \left (x\right )^{4} + \frac{1}{2} \, \cos \left (x\right )^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)^4 + 1/2*cos(x)^2

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Sympy [A]  time = 0.512396, size = 22, normalized size = 1.29 \begin{align*} - \frac{\sin{\left (x \right )} \sin{\left (3 x \right )}}{8} - \frac{3 \cos{\left (x \right )} \cos{\left (3 x \right )}}{8} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07167, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{8} \, \cos \left (4 \, 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(x)*sin(3*x),x, algorithm="giac")

[Out]

-1/8*cos(4*x) - 1/4*cos(2*x)