3.2.7 \(\int \cos (x) \sin (3 x) \, dx\) [107]

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 = {4369} \begin {gather*} -\frac {1}{4} \cos (2 x)-\frac {1}{8} \cos (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*Cos[2*x] - Cos[4*x]/8

Rule 4369

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 \begin {gather*} -\frac {1}{2} \cos ^2(x)-\frac {1}{8} \cos (4 x) \end {gather*}

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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Maple [A]
time = 0.06, size = 13, normalized size = 0.76

method	result	size
derivativedivides	\(-\frac {\left (4 \left (\cos ^{2}\left (x \right )\right )-1\right )^{2}}{16}\)	\(13\)
default	\(-\frac {\left (4 \left (\cos ^{2}\left (x \right )\right )-1\right )^{2}}{16}\)	\(13\)
risch	\(-\frac {\cos \left (2 x \right )}{4}-\frac {\cos \left (4 x \right )}{8}\)	\(14\)
norman	\(\frac {\frac {3 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{4}-\frac {\tan \left (\frac {x}{2}\right ) \tan \left (\frac {3 x}{2}\right )}{2}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*sin(3*x),x,method=_RETURNVERBOSE)

[Out]

-1/16*(4*cos(x)^2-1)^2

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Maxima [A]
time = 1.56, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{8} \, \cos \left (4 \, x\right ) - \frac {1}{4} \, \cos \left (2 \, x\right ) \end {gather*}

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 = 0.60, size = 13, normalized size = 0.76 \begin {gather*} -\cos \left (x\right )^{4} + \frac {1}{2} \, \cos \left (x\right )^{2} \end {gather*}

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 = 0.12, size = 22, normalized size = 1.29 \begin {gather*} - \frac {\sin {\left (x \right )} \sin {\left (3 x \right )}}{8} - \frac {3 \cos {\left (x \right )} \cos {\left (3 x \right )}}{8} \end {gather*}

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 = 0.98, size = 13, normalized size = 0.76 \begin {gather*} -\cos \left (x\right )^{4} + \frac {1}{2} \, \cos \left (x\right )^{2} \end {gather*}

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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Mupad [B]
time = 0.03, size = 13, normalized size = 0.76 \begin {gather*} \frac {{\cos \left (x\right )}^2}{2}-{\cos \left (x\right )}^4 \end {gather*}

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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