3.335 \(\int \cos ^4\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx\)

Optimal. Leaf size=20 \[ \frac{3 x}{8}+\frac{\cos (x)}{2}-\frac{1}{8} \sin (x) \cos (x) \]

[Out]

(3*x)/8 + Cos[x]/2 - (Cos[x]*Sin[x])/8

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Rubi [B]  time = 0.0289181, antiderivative size = 64, normalized size of antiderivative = 3.2, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{3 x}{8}+\frac{1}{2} \sin \left (\frac{x}{2}+\frac{\pi }{4}\right ) \cos ^3\left (\frac{x}{2}+\frac{\pi }{4}\right )+\frac{3}{4} \sin \left (\frac{x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{x}{2}+\frac{\pi }{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Cos[Pi/4 + x/2]^4,x]

[Out]

(3*x)/8 + (3*Cos[Pi/4 + x/2]*Sin[Pi/4 + x/2])/4 + (Cos[Pi/4 + x/2]^3*Sin[Pi/4 +
x/2])/2

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Rubi in Sympy [A]  time = 0.735055, size = 44, normalized size = 2.2 \[ \frac{3 x}{8} + \frac{\sin{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \cos ^{3}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{2} + \frac{3 \sin{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \cos{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(1/4*pi+1/2*x)**4,x)

[Out]

3*x/8 + sin(x/2 + pi/4)*cos(x/2 + pi/4)**3/2 + 3*sin(x/2 + pi/4)*cos(x/2 + pi/4)
/4

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Mathematica [A]  time = 0.0211051, size = 21, normalized size = 1.05 \[ \frac{1}{16} (6 x+8 \cos (x)-2 \sin (x) \cos (x)+3 \pi ) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[Pi/4 + x/2]^4,x]

[Out]

(3*Pi + 6*x + 8*Cos[x] - 2*Cos[x]*Sin[x])/16

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Maple [B]  time = 0.01, size = 39, normalized size = 2. \[{\frac{1}{2} \left ( \left ( \cos \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) \right ) ^{3}+{\frac{3}{2}\cos \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) } \right ) \sin \left ({\frac{\pi }{4}}+{\frac{x}{2}} \right ) }+{\frac{3\,\pi }{16}}+{\frac{3\,x}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(1/4*Pi+1/2*x)^4,x)

[Out]

1/2*(cos(1/4*Pi+1/2*x)^3+3/2*cos(1/4*Pi+1/2*x))*sin(1/4*Pi+1/2*x)+3/16*Pi+3/8*x

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Maxima [A]  time = 1.44371, size = 31, normalized size = 1.55 \[ \frac{3}{16} \, \pi + \frac{3}{8} \, x + \frac{1}{16} \, \sin \left (\pi + 2 \, x\right ) + \frac{1}{2} \, \sin \left (\frac{1}{2} \, \pi + x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(1/4*pi + 1/2*x)^4,x, algorithm="maxima")

[Out]

3/16*pi + 3/8*x + 1/16*sin(pi + 2*x) + 1/2*sin(1/2*pi + x)

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Fricas [A]  time = 0.23287, size = 50, normalized size = 2.5 \[ \frac{1}{4} \,{\left (2 \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )^{3} + 3 \, \cos \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right )\right )} \sin \left (\frac{1}{4} \, \pi + \frac{1}{2} \, x\right ) + \frac{3}{8} \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(1/4*pi + 1/2*x)^4,x, algorithm="fricas")

[Out]

1/4*(2*cos(1/4*pi + 1/2*x)^3 + 3*cos(1/4*pi + 1/2*x))*sin(1/4*pi + 1/2*x) + 3/8*
x

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Sympy [A]  time = 0.946761, size = 99, normalized size = 4.95 \[ \frac{3 x \sin ^{4}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{8} + \frac{3 x \sin ^{2}{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \cos ^{2}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{4} + \frac{3 x \cos ^{4}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{8} + \frac{3 \sin ^{3}{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \cos{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{4} + \frac{5 \sin{\left (\frac{x}{2} + \frac{\pi }{4} \right )} \cos ^{3}{\left (\frac{x}{2} + \frac{\pi }{4} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(1/4*pi+1/2*x)**4,x)

[Out]

3*x*sin(x/2 + pi/4)**4/8 + 3*x*sin(x/2 + pi/4)**2*cos(x/2 + pi/4)**2/4 + 3*x*cos
(x/2 + pi/4)**4/8 + 3*sin(x/2 + pi/4)**3*cos(x/2 + pi/4)/4 + 5*sin(x/2 + pi/4)*c
os(x/2 + pi/4)**3/4

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GIAC/XCAS [A]  time = 0.201193, size = 19, normalized size = 0.95 \[ \frac{3}{8} \, x + \frac{1}{2} \, \cos \left (x\right ) - \frac{1}{16} \, \sin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(1/4*pi + 1/2*x)^4,x, algorithm="giac")

[Out]

3/8*x + 1/2*cos(x) - 1/16*sin(2*x)