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

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 = 0.0160949, 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, Rules used = {2635, 8} \[ \frac{3 x}{8}+\frac{1}{2} \sin \left (\frac{x}{2}+\frac{\pi }{4}\right ) \cos ^3\left (\frac{x}{2}+\frac{\pi }{4}\right )+\frac{3}{4} \sin \left (\frac{x}{2}+\frac{\pi }{4}\right ) \cos \left (\frac{x}{2}+\frac{\pi }{4}\right ) \]

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^4\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx &=\frac{1}{2} \cos ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \sin \left (\frac{\pi }{4}+\frac{x}{2}\right )+\frac{3}{4} \int \sin ^2\left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx\\ &=\frac{3}{4} \cos \left (\frac{\pi }{4}+\frac{x}{2}\right ) \sin \left (\frac{\pi }{4}+\frac{x}{2}\right )+\frac{1}{2} \cos ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \sin \left (\frac{\pi }{4}+\frac{x}{2}\right )+\frac{3 \int 1 \, dx}{8}\\ &=\frac{3 x}{8}+\frac{3}{4} \cos \left (\frac{\pi }{4}+\frac{x}{2}\right ) \sin \left (\frac{\pi }{4}+\frac{x}{2}\right )+\frac{1}{2} \cos ^3\left (\frac{\pi }{4}+\frac{x}{2}\right ) \sin \left (\frac{\pi }{4}+\frac{x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.017465, 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. \begin{align*}{\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}} \end{align*}

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 = 0.933846, size = 31, normalized size = 1.55 \begin{align*} \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 ) \end{align*}

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 [B]  time = 1.67131, size = 112, normalized size = 5.6 \begin{align*} \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 \end{align*}

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 [B]  time = 0.615847, size = 99, normalized size = 4.95 \begin{align*} \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} \end{align*}

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)*cos(x/2 + pi/4)**3/4

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Giac [A]  time = 1.05228, size = 19, normalized size = 0.95 \begin{align*} \frac{3}{8} \, x + \frac{1}{2} \, \cos \left (x\right ) - \frac{1}{16} \, \sin \left (2 \, x\right ) \end{align*}

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)