3.98 \(\int \cos (x) \sin (3 x) \, dx\)

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

[Out]

-1/4*cos(2*x)-1/8*cos(4*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} \[ -\frac {1}{4} \cos (2 x)-\frac {1}{8} \cos (4 x) \]

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*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ -\frac {1}{2} \cos ^2(x)-\frac {1}{8} \cos (4 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Cos[x]^2 - Cos[4*x]/8

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fricas [A]  time = 0.90, size = 13, normalized size = 0.76 \[ -\cos \relax (x)^{4} + \frac {1}{2} \, \cos \relax (x)^{2} \]

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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giac [A]  time = 0.14, size = 13, normalized size = 0.76 \[ -\cos \relax (x)^{4} + \frac {1}{2} \, \cos \relax (x)^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cos(x)^4 + 1/2*cos(x)^2

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maple [A]  time = 0.07, size = 14, normalized size = 0.82 \[ -\left (\cos ^{4}\relax (x )\right )+\frac {\left (\cos ^{2}\relax (x )\right )}{2} \]

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 = 0.31, size = 13, normalized size = 0.76 \[ -\frac {1}{8} \, \cos \left (4 \, x\right ) - \frac {1}{4} \, \cos \left (2 \, x\right ) \]

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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mupad [B]  time = 0.03, size = 13, normalized size = 0.76 \[ \frac {{\cos \relax (x)}^2}{2}-{\cos \relax (x)}^4 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos(x)^2/2 - cos(x)^4

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sympy [A]  time = 0.42, size = 22, normalized size = 1.29 \[ - \frac {\sin {\relax (x )} \sin {\left (3 x \right )}}{8} - \frac {3 \cos {\relax (x )} \cos {\left (3 x \right )}}{8} \]

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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