\(\int \cos (x) \cos (4 x) \, dx\) [368]

Optimal result

Integrand size = 7, antiderivative size = 17 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]

[Out]

1/6*sin(3*x)+1/10*sin(5*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4368} \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]

[In]

Int[Cos[x]*Cos[4*x],x]

[Out]

Sin[3*x]/6 + Sin[5*x]/10

Rule 4368

Int[cos[(a_.) + (b_.)*(x_)]*cos[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
+ Simp[Sin[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*} \text {integral}& = \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]

[In]

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

[Out]

Sin[3*x]/6 + Sin[5*x]/10

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\)	\(14\)
risch	\(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\)	\(14\)
parallelrisch	\(\frac {\sin \left (3 x \right )}{6}+\frac {\sin \left (5 x \right )}{10}\)	\(14\)
norman	\(\frac {-\frac {8 \tan \left (2 x \right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15}+\frac {2 \left (\tan ^{2}\left (2 x \right )\right ) \tan \left (\frac {x}{2}\right )}{15}+\frac {8 \tan \left (2 x \right )}{15}-\frac {2 \tan \left (\frac {x}{2}\right )}{15}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (2 x \right )\right )}\)	\(59\)

[In]

int(cos(x)*cos(4*x),x,method=_RETURNVERBOSE)

[Out]

1/6*sin(3*x)+1/10*sin(5*x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{15} \, {\left (24 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) \]

[In]

integrate(cos(x)*cos(4*x),x, algorithm="fricas")

[Out]

1/15*(24*cos(x)^4 - 8*cos(x)^2 - 1)*sin(x)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int \cos (x) \cos (4 x) \, dx=- \frac {\sin {\left (x \right )} \cos {\left (4 x \right )}}{15} + \frac {4 \sin {\left (4 x \right )} \cos {\left (x \right )}}{15} \]

[In]

integrate(cos(x)*cos(4*x),x)

[Out]

-sin(x)*cos(4*x)/15 + 4*sin(4*x)*cos(x)/15

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]

[In]

integrate(cos(x)*cos(4*x),x, algorithm="maxima")

[Out]

1/10*sin(5*x) + 1/6*sin(3*x)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \cos (4 x) \, dx=\frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]

[In]

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

[Out]

1/10*sin(5*x) + 1/6*sin(3*x)

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \cos (x) \cos (4 x) \, dx=\frac {\sin \left (3\,x\right )}{6}+\frac {\sin \left (5\,x\right )}{10} \]

[In]

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

[Out]

sin(3*x)/6 + sin(5*x)/10