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 = 0.0512354, 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 \[ \frac{x}{16}+\frac{1}{8} \sin (2 x)+\frac{3}{32} \sin (4 x)+\frac{1}{24} \sin (6 x)+\frac{1}{128} \sin (8 x) \]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\sin{\left (2 x \right )}}{8} + \frac{3 \sin{\left (4 x \right )}}{32} + \frac{\sin{\left (6 x \right )}}{24} + \frac{\sin{\left (8 x \right )}}{128} + \int \frac{1}{16}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**4*cos(4*x),x)

[Out]

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

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Mathematica [A]  time = 0.0187952, 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 = 0.07, size = 29, normalized size = 0.8 \[{\frac{x}{16}}+{\frac{\sin \left ( 2\,x \right ) }{8}}+{\frac{3\,\sin \left ( 4\,x \right ) }{32}}+{\frac{\sin \left ( 6\,x \right ) }{24}}+{\frac{\sin \left ( 8\,x \right ) }{128}} \]

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 = 1.38624, size = 41, normalized size = 1.08 \[ -\frac{1}{6} \, \sin \left (2 \, x\right )^{3} + \frac{1}{16} \, x + \frac{1}{128} \, \sin \left (8 \, x\right ) + \frac{3}{32} \, \sin \left (4 \, x\right ) + \frac{1}{4} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(x)^4,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 = 0.252802, size = 42, normalized size = 1.11 \[ \frac{1}{48} \,{\left (48 \, \cos \left (x\right )^{7} - 8 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac{1}{16} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(x)^4,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 [A]  time = 45.8458, size = 139, normalized size = 3.66 \[ \frac{x \sin ^{4}{\left (x \right )} \cos{\left (4 x \right )}}{16} - \frac{x \sin ^{3}{\left (x \right )} \sin{\left (4 x \right )} \cos{\left (x \right )}}{4} - \frac{3 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} \cos{\left (4 x \right )}}{8} + \frac{x \sin{\left (x \right )} \sin{\left (4 x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac{x \cos ^{4}{\left (x \right )} \cos{\left (4 x \right )}}{16} - \frac{\sin ^{4}{\left (x \right )} \sin{\left (4 x \right )}}{64} - \frac{5 \sin ^{2}{\left (x \right )} \sin{\left (4 x \right )} \cos ^{2}{\left (x \right )}}{32} - \frac{\sin{\left (x \right )} \cos ^{3}{\left (x \right )} \cos{\left (4 x \right )}}{3} + \frac{61 \sin{\left (4 x \right )} \cos ^{4}{\left (x \right )}}{192} \]

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)/64 - 5*sin(x)**2*sin(4*x)*cos(x)**2/32 - sin(x)*cos(x)**3*cos(4*x)/3
+ 61*sin(4*x)*cos(x)**4/192

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GIAC/XCAS [A]  time = 0.196763, size = 38, normalized size = 1. \[ \frac{1}{16} \, x + \frac{1}{128} \, \sin \left (8 \, x\right ) + \frac{1}{24} \, \sin \left (6 \, x\right ) + \frac{3}{32} \, \sin \left (4 \, x\right ) + \frac{1}{8} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(4*x)*cos(x)^4,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)