3.1.42 \(\int \cos (5 x) \sin (3 x) \, dx\) [42]

Optimal. Leaf size=17 \[ \frac {1}{4} \cos (2 x)-\frac {1}{16} \cos (8 x) \]

[Out]

1/4*cos(2*x)-1/16*cos(8*x)

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 17, 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 {1}{4} \cos (2 x)-\frac {1}{16} \cos (8 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[5*x]*Sin[3*x],x]

[Out]

Cos[2*x]/4 - Cos[8*x]/16

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 (5 x) \sin (3 x) \, dx &=\frac {1}{4} \cos (2 x)-\frac {1}{16} \cos (8 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\cos ^2(x)}{2}-\frac {1}{16} \cos (8 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[5*x]*Sin[3*x],x]

[Out]

Cos[x]^2/2 - Cos[8*x]/16

________________________________________________________________________________________

Mathics [A]
time = 1.96, size = 13, normalized size = 0.76 \begin {gather*} -\frac {\text {Cos}\left [8 x\right ]}{16}+\frac {\text {Cos}\left [2 x\right ]}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Sin[3*x]*Cos[5*x],x]')

[Out]

-Cos[8 x] / 16 + Cos[2 x] / 4

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {\cos \left (2 x \right )}{4}-\frac {\cos \left (8 x \right )}{16}\)	\(14\)
risch	\(\frac {\cos \left (2 x \right )}{4}-\frac {\cos \left (8 x \right )}{16}\)	\(14\)
norman	\(\frac {-\frac {3 \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{8}-\frac {3 \left (\tan ^{2}\left (\frac {5 x}{2}\right )\right )}{8}+\frac {5 \tan \left (\frac {3 x}{2}\right ) \tan \left (\frac {5 x}{2}\right )}{4}}{\left (1+\tan ^{2}\left (\frac {5 x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {3 x}{2}\right )\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(5*x)*sin(3*x),x,method=_RETURNVERBOSE)

[Out]

1/4*cos(2*x)-1/16*cos(8*x)

________________________________________________________________________________________

Maxima [A]
time = 0.25, size = 13, normalized size = 0.76 \begin {gather*} -\frac {1}{16} \, \cos \left (8 \, 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(5*x)*sin(3*x),x, algorithm="maxima")

[Out]

-1/16*cos(8*x) + 1/4*cos(2*x)

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 25, normalized size = 1.47 \begin {gather*} -8 \, \cos \left (x\right )^{8} + 16 \, \cos \left (x\right )^{6} - 10 \, \cos \left (x\right )^{4} + \frac {5}{2} \, \cos \left (x\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x, algorithm="fricas")

[Out]

-8*cos(x)^8 + 16*cos(x)^6 - 10*cos(x)^4 + 5/2*cos(x)^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\)
time = 0.14, size = 26, normalized size = 1.53 \begin {gather*} \frac {5 \sin {\left (3 x \right )} \sin {\left (5 x \right )}}{16} + \frac {3 \cos {\left (3 x \right )} \cos {\left (5 x \right )}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x)

[Out]

5*sin(3*x)*sin(5*x)/16 + 3*cos(3*x)*cos(5*x)/16

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 16, normalized size = 0.94 \begin {gather*} -\frac {\cos \left (8 x\right )}{16}+\frac {\cos \left (2 x\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(5*x)*sin(3*x),x)

[Out]

-1/16*cos(8*x) + 1/4*cos(2*x)

________________________________________________________________________________________

Mupad [B]
time = 0.04, size = 13, normalized size = 0.76 \begin {gather*} \frac {\cos \left (2\,x\right )}{4}-\frac {\cos \left (8\,x\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(5*x)*sin(3*x),x)

[Out]

cos(2*x)/4 - cos(8*x)/16

________________________________________________________________________________________