3.368 \(\int \cos (x) \cos (4 x) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]

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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 = {4283} \[ \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[3*x]/6 + Sin[5*x]/10

Rule 4283

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*} \int \cos (x) \cos (4 x) \, dx &=\frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 17, normalized size = 1.00 \[ \frac {1}{6} \sin (3 x)+\frac {1}{10} \sin (5 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[3*x]/6 + Sin[5*x]/10

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos (x) \cos (4 x) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[x]*Cos[4*x],x]

[Out]

Could not integrate

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fricas [A]  time = 0.59, size = 18, normalized size = 1.06 \[ \frac {1}{15} \, {\left (24 \, \cos \relax (x)^{4} - 8 \, \cos \relax (x)^{2} - 1\right )} \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.61, size = 13, normalized size = 0.76 \[ \frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sin(5*x) + 1/6*sin(3*x)

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maple [A]  time = 0.12, size = 14, normalized size = 0.82

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 13, normalized size = 0.76 \[ \frac {1}{10} \, \sin \left (5 \, x\right ) + \frac {1}{6} \, \sin \left (3 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*sin(5*x) + 1/6*sin(3*x)

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mupad [B]  time = 0.19, size = 13, normalized size = 0.76 \[ \frac {\sin \left (3\,x\right )}{6}+\frac {\sin \left (5\,x\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(3*x)/6 + sin(5*x)/10

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sympy [A]  time = 0.55, size = 20, normalized size = 1.18 \[ - \frac {\sin {\relax (x )} \cos {\left (4 x \right )}}{15} + \frac {4 \sin {\left (4 x \right )} \cos {\relax (x )}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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