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