3.99 \(\int \cos (x) \sin (4 x) \, dx\)

Optimal. Leaf size=17 \[ -\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x) \]

[Out]

-Cos[3*x]/6 - Cos[5*x]/10

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Rubi [A]  time = 0.0082679, 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} \[ -\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sin[4*x],x]

[Out]

-Cos[3*x]/6 - 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 (x) \sin (4 x) \, dx &=-\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0057735, size = 17, normalized size = 1. \[ -\frac{1}{6} \cos (3 x)-\frac{1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sin[4*x],x]

[Out]

-Cos[3*x]/6 - Cos[5*x]/10

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Maple [A]  time = 0.02, size = 14, normalized size = 0.8 \begin{align*} -{\frac{8\, \left ( \cos \left ( x \right ) \right ) ^{5}}{5}}+{\frac{4\, \left ( \cos \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(4*x),x)

[Out]

-8/5*cos(x)^5+4/3*cos(x)^3

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Maxima [A]  time = 1.00275, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{10} \, \cos \left (5 \, x\right ) - \frac{1}{6} \, \cos \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(4*x),x, algorithm="maxima")

[Out]

-1/10*cos(5*x) - 1/6*cos(3*x)

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Fricas [A]  time = 2.50008, size = 41, normalized size = 2.41 \begin{align*} -\frac{8}{5} \, \cos \left (x\right )^{5} + \frac{4}{3} \, \cos \left (x\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(4*x),x, algorithm="fricas")

[Out]

-8/5*cos(x)^5 + 4/3*cos(x)^3

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Sympy [A]  time = 0.689863, size = 22, normalized size = 1.29 \begin{align*} - \frac{\sin{\left (x \right )} \sin{\left (4 x \right )}}{15} - \frac{4 \cos{\left (x \right )} \cos{\left (4 x \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(4*x),x)

[Out]

-sin(x)*sin(4*x)/15 - 4*cos(x)*cos(4*x)/15

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Giac [A]  time = 1.11087, size = 18, normalized size = 1.06 \begin{align*} -\frac{1}{10} \, \cos \left (5 \, x\right ) - \frac{1}{6} \, \cos \left (3 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(4*x),x, algorithm="giac")

[Out]

-1/10*cos(5*x) - 1/6*cos(3*x)