3.370 \(\int \cos ^4(x) \cos (4 x) \, dx\)

Optimal. Leaf size=38 \[ \frac{x}{16}+\frac{1}{8} \sin (2 x)+\frac{3}{32} \sin (4 x)+\frac{1}{24} \sin (6 x)+\frac{1}{128} \sin (8 x) \]

[Out]

x/16 + Sin[2*x]/8 + (3*Sin[4*x])/32 + Sin[6*x]/24 + Sin[8*x]/128

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Rubi [A]  time = 0.0302545, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4354, 2637} \[ \frac{x}{16}+\frac{1}{8} \sin (2 x)+\frac{3}{32} \sin (4 x)+\frac{1}{24} \sin (6 x)+\frac{1}{128} \sin (8 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/16 + Sin[2*x]/8 + (3*Sin[4*x])/32 + Sin[6*x]/24 + Sin[8*x]/128

Rule 4354

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[ActivateT
rig[F[a + b*x]^p*G[c + d*x]^q], x], x] /; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, si
n] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cos ^4(x) \cos (4 x) \, dx &=\int \left (\frac{1}{16}+\frac{1}{4} \cos (2 x)+\frac{3}{8} \cos (4 x)+\frac{1}{4} \cos (6 x)+\frac{1}{16} \cos (8 x)\right ) \, dx\\ &=\frac{x}{16}+\frac{1}{16} \int \cos (8 x) \, dx+\frac{1}{4} \int \cos (2 x) \, dx+\frac{1}{4} \int \cos (6 x) \, dx+\frac{3}{8} \int \cos (4 x) \, dx\\ &=\frac{x}{16}+\frac{1}{8} \sin (2 x)+\frac{3}{32} \sin (4 x)+\frac{1}{24} \sin (6 x)+\frac{1}{128} \sin (8 x)\\ \end{align*}

Mathematica [A]  time = 0.0103867, size = 38, normalized size = 1. \[ \frac{x}{16}+\frac{1}{8} \sin (2 x)+\frac{3}{32} \sin (4 x)+\frac{1}{24} \sin (6 x)+\frac{1}{128} \sin (8 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/16 + Sin[2*x]/8 + (3*Sin[4*x])/32 + Sin[6*x]/24 + Sin[8*x]/128

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Maple [A]  time = 0.062, size = 29, normalized size = 0.8 \begin{align*}{\frac{x}{16}}+{\frac{\sin \left ( 2\,x \right ) }{8}}+{\frac{3\,\sin \left ( 4\,x \right ) }{32}}+{\frac{\sin \left ( 6\,x \right ) }{24}}+{\frac{\sin \left ( 8\,x \right ) }{128}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x+1/8*sin(2*x)+3/32*sin(4*x)+1/24*sin(6*x)+1/128*sin(8*x)

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Maxima [A]  time = 0.958005, size = 41, normalized size = 1.08 \begin{align*} -\frac{1}{6} \, \sin \left (2 \, x\right )^{3} + \frac{1}{16} \, x + \frac{1}{128} \, \sin \left (8 \, x\right ) + \frac{3}{32} \, \sin \left (4 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sin(2*x)^3 + 1/16*x + 1/128*sin(8*x) + 3/32*sin(4*x) + 1/4*sin(2*x)

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Fricas [A]  time = 2.47478, size = 99, normalized size = 2.61 \begin{align*} \frac{1}{48} \,{\left (48 \, \cos \left (x\right )^{7} - 8 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{16} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(48*cos(x)^7 - 8*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 1/16*x

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Sympy [B]  time = 27.5657, size = 144, normalized size = 3.79 \begin{align*} \frac{x \sin ^{4}{\left (x \right )} \cos{\left (4 x \right )}}{16} - \frac{x \sin ^{3}{\left (x \right )} \sin{\left (4 x \right )} \cos{\left (x \right )}}{4} - \frac{3 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} \cos{\left (4 x \right )}}{8} + \frac{x \sin{\left (x \right )} \sin{\left (4 x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac{x \cos ^{4}{\left (x \right )} \cos{\left (4 x \right )}}{16} - \frac{\sin ^{4}{\left (x \right )} \sin{\left (4 x \right )}}{3} - \frac{61 \sin ^{3}{\left (x \right )} \cos{\left (x \right )} \cos{\left (4 x \right )}}{48} + \frac{7 \sin ^{2}{\left (x \right )} \sin{\left (4 x \right )} \cos ^{2}{\left (x \right )}}{4} + \frac{15 \sin{\left (x \right )} \cos ^{3}{\left (x \right )} \cos{\left (4 x \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sin(x)**4*cos(4*x)/16 - x*sin(x)**3*sin(4*x)*cos(x)/4 - 3*x*sin(x)**2*cos(x)**2*cos(4*x)/8 + x*sin(x)*sin(4*
x)*cos(x)**3/4 + x*cos(x)**4*cos(4*x)/16 - sin(x)**4*sin(4*x)/3 - 61*sin(x)**3*cos(x)*cos(4*x)/48 + 7*sin(x)**
2*sin(4*x)*cos(x)**2/4 + 15*sin(x)*cos(x)**3*cos(4*x)/16

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Giac [A]  time = 1.08375, size = 38, normalized size = 1. \begin{align*} \frac{1}{16} \, x + \frac{1}{128} \, \sin \left (8 \, x\right ) + \frac{1}{24} \, \sin \left (6 \, x\right ) + \frac{3}{32} \, \sin \left (4 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*x + 1/128*sin(8*x) + 1/24*sin(6*x) + 3/32*sin(4*x) + 1/8*sin(2*x)