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]

-1/6*cos(3*x)-1/10*cos(5*x)

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {1}{6} \cos (3 x)-\frac {1}{10} \cos (5 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*Cos[3*x] - Cos[5*x]/10

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fricas [A]  time = 0.55, size = 13, normalized size = 0.76 \[ -\frac {8}{5} \, \cos \relax (x)^{5} + \frac {4}{3} \, \cos \relax (x)^{3} \]

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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giac [A]  time = 0.12, size = 13, normalized size = 0.76 \[ -\frac {8}{5} \, \cos \relax (x)^{5} + \frac {4}{3} \, \cos \relax (x)^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/5*cos(x)^5 + 4/3*cos(x)^3

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maple [A]  time = 0.06, size = 14, normalized size = 0.82 \[ -\frac {8 \left (\cos ^{5}\relax (x )\right )}{5}+\frac {4 \left (\cos ^{3}\relax (x )\right )}{3} \]

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 = 0.31, size = 13, normalized size = 0.76 \[ -\frac {1}{10} \, \cos \left (5 \, x\right ) - \frac {1}{6} \, \cos \left (3 \, x\right ) \]

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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mupad [B]  time = 0.02, size = 14, normalized size = 0.82 \[ -\frac {4\,{\cos \relax (x)}^3\,\left (6\,{\cos \relax (x)}^2-5\right )}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*cos(x)^3*(6*cos(x)^2 - 5))/15

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sympy [A]  time = 0.41, size = 22, normalized size = 1.29 \[ - \frac {\sin {\relax (x )} \sin {\left (4 x \right )}}{15} - \frac {4 \cos {\relax (x )} \cos {\left (4 x \right )}}{15} \]

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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