3.2.31 \(\int \cos (3 x) \sin (2 x) \, dx\) [131]

Optimal. Leaf size=15 \[ \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \]

[Out]

1/2*cos(x)-1/10*cos(5*x)

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4369} \begin {gather*} \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[3*x]*Sin[2*x],x]

[Out]

Cos[x]/2 - Cos[5*x]/10

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 (3 x) \sin (2 x) \, dx &=\frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \frac {\cos (x)}{2}-\frac {1}{10} \cos (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[3*x]*Sin[2*x],x]

[Out]

Cos[x]/2 - Cos[5*x]/10

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Maple [A]
time = 0.06, size = 12, normalized size = 0.80

method	result	size
default	\(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}\)	\(12\)
risch	\(\frac {\cos \left (x \right )}{2}-\frac {\cos \left (5 x \right )}{10}\)	\(12\)
norman	\(\frac {-\frac {4 \left (\tan ^{2}\left (x \right )\right )}{5}-\frac {4 \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{5}+\frac {12 \tan \left (x \right ) \tan \left (\frac {3 x}{2}\right )}{5}}{\left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right ) \left (1+\tan ^{2}\left (x \right )\right )}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*sin(2*x),x,method=_RETURNVERBOSE)

[Out]

1/2*cos(x)-1/10*cos(5*x)

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Maxima [A]
time = 2.09, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="maxima")

[Out]

-1/10*cos(5*x) + 1/2*cos(x)

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Fricas [A]
time = 0.47, size = 13, normalized size = 0.87 \begin {gather*} -\frac {8}{5} \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="fricas")

[Out]

-8/5*cos(x)^5 + 2*cos(x)^3

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (10) = 20\).
time = 0.13, size = 26, normalized size = 1.73 \begin {gather*} \frac {3 \sin {\left (2 x \right )} \sin {\left (3 x \right )}}{5} + \frac {2 \cos {\left (2 x \right )} \cos {\left (3 x \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x)

[Out]

3*sin(2*x)*sin(3*x)/5 + 2*cos(2*x)*cos(3*x)/5

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Giac [A]
time = 0.58, size = 11, normalized size = 0.73 \begin {gather*} -\frac {1}{10} \, \cos \left (5 \, x\right ) + \frac {1}{2} \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(3*x)*sin(2*x),x, algorithm="giac")

[Out]

-1/10*cos(5*x) + 1/2*cos(x)

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Mupad [B]
time = 0.04, size = 13, normalized size = 0.87 \begin {gather*} 2\,{\cos \left (x\right )}^3-\frac {8\,{\cos \left (x\right )}^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(3*x)*sin(2*x),x)

[Out]

2*cos(x)^3 - (8*cos(x)^5)/5

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