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

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

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Rubi [B]  time = 0.02, antiderivative size = 64, normalized size of antiderivative = 3.20, 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 8

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

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]

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*}

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Mathematica [A]  time = 0.02, 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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

IntegrateAlgebraic[Cos[Pi/4 + x/2]^4,x]

[Out]

Could not integrate

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fricas [B]  time = 0.62, size = 37, normalized size = 1.85 \[ \frac {1}{4} \, {\left (2 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )^{3} + 3 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right ) + \frac {3}{8} \, x \]

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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giac [A]  time = 0.59, size = 14, normalized size = 0.70 \[ \frac {3}{8} \, x + \frac {1}{2} \, \cos \relax (x) - \frac {1}{16} \, \sin \left (2 \, x\right ) \]

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)

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maple [A]  time = 0.34, size = 15, normalized size = 0.75

method	result	size
risch	\(\frac {3 x}{8}+\frac {\cos \relax (x )}{2}-\frac {\sin \left (2 x \right )}{16}\)	\(15\)
derivativedivides	\(\frac {\left (\cos ^{3}\left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {3 \cos \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}+\frac {3 \pi }{16}+\frac {3 x}{8}\)	\(39\)
default	\(\frac {\left (\cos ^{3}\left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {3 \cos \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}+\frac {3 \pi }{16}+\frac {3 x}{8}\)	\(39\)
norman	\(\frac {\frac {3 x}{8}-\frac {3 \left (\tan ^{3}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 \left (\tan ^{5}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}-\frac {5 \left (\tan ^{7}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 x \left (\tan ^{2}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {9 x \left (\tan ^{4}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{4}+\frac {3 x \left (\tan ^{6}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 x \left (\tan ^{8}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{8}+\frac {5 \tan \left (\frac {\pi }{8}+\frac {x}{4}\right )}{2}}{\left (1+\tan ^{2}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )^{4}}\)	\(118\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(1/4*Pi+1/2*x)^4,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.49, size = 23, normalized size = 1.15 \[ \frac {3}{16} \, \pi + \frac {3}{8} \, x + \frac {1}{16} \, \sin \left (\pi + 2 \, x\right ) + \frac {1}{2} \, \sin \left (\frac {1}{2} \, \pi + x\right ) \]

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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mupad [B]  time = 0.27, size = 20, normalized size = 1.00 \[ \frac {3\,x}{8}+\frac {\sin \left (\Pi +2\,x\right )}{16}+\frac {\sin \left (\frac {\Pi }{2}+x\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(Pi/4 + x/2)^4,x)

[Out]

(3*x)/8 + sin(Pi + 2*x)/16 + sin(Pi/2 + x)/2

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sympy [B]  time = 0.65, 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)*cos(x/2 + pi/4)**3/4

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