3.4.70 \(\int \cos ^4(x) \cos (4 x) \, dx\) [370]

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]

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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Rubi [A]
time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 = {4439, 2717} \begin {gather*} \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 {gather*}

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 2717

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

Rule 4439

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]

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*}

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Mathematica [A]
time = 0.02, size = 38, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.09, size = 29, normalized size = 0.76

method	result	size
default	\(\frac {x}{16}+\frac {\sin \left (2 x \right )}{8}+\frac {3 \sin \left (4 x \right )}{32}+\frac {\sin \left (6 x \right )}{24}+\frac {\sin \left (8 x \right )}{128}\)	\(29\)
risch	\(\frac {x}{16}+\frac {\sin \left (2 x \right )}{8}+\frac {3 \sin \left (4 x \right )}{32}+\frac {\sin \left (6 x \right )}{24}+\frac {\sin \left (8 x \right )}{128}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.54, size = 30, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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 = 1.11, size = 31, normalized size = 0.82 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (31) = 62\).
time = 2.24, size = 139, normalized size = 3.66 \begin {gather*} \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 )}}{24} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )} \cos {\left (4 x \right )}}{48} - \frac {11 \sin {\left (x \right )} \cos ^{3}{\left (x \right )} \cos {\left (4 x \right )}}{48} + \frac {7 \sin {\left (4 x \right )} \cos ^{4}{\left (x \right )}}{24} \end {gather*}

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)/24 - 5*sin(x)**3*cos(x)*cos(4*x)/48 - 11*sin(x)*
cos(x)**3*cos(4*x)/48 + 7*sin(4*x)*cos(x)**4/24

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Giac [A]
time = 0.86, size = 28, normalized size = 0.74 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 0.30, size = 36, normalized size = 0.95 \begin {gather*} \frac {x}{16}+\frac {\frac {{\mathrm {tan}\left (x\right )}^7}{16}+\frac {11\,{\mathrm {tan}\left (x\right )}^5}{48}+\frac {5\,{\mathrm {tan}\left (x\right )}^3}{48}+\frac {15\,\mathrm {tan}\left (x\right )}{16}}{{\left ({\mathrm {tan}\left (x\right )}^2+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/16 + ((15*tan(x))/16 + (5*tan(x)^3)/48 + (11*tan(x)^5)/48 + tan(x)^7/16)/(tan(x)^2 + 1)^4

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