Optimal. Leaf size=25 \[ \frac{x^2}{4}+\frac{\cos ^2(x)}{4}+\frac{1}{2} x \sin (x) \cos (x) \]
[Out]
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Rubi [A] time = 0.0245117, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x^2}{4}+\frac{\cos ^2(x)}{4}+\frac{1}{2} x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
[In] Int[x*Cos[x]^2,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{x \sin{\left (x \right )} \cos{\left (x \right )}}{2} + \frac{\cos ^{2}{\left (x \right )}}{4} + \frac{\int x\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*cos(x)**2,x)
[Out]
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Mathematica [A] time = 0.00512229, size = 25, normalized size = 1. \[ \frac{x^2}{4}+\frac{1}{4} x \sin (2 x)+\frac{1}{8} \cos (2 x) \]
Antiderivative was successfully verified.
[In] Integrate[x*Cos[x]^2,x]
[Out]
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Maple [A] time = 0.005, size = 25, normalized size = 1. \[ x \left ({\frac{x}{2}}+{\frac{\cos \left ( x \right ) \sin \left ( x \right ) }{2}} \right ) -{\frac{{x}^{2}}{4}}-{\frac{ \left ( \sin \left ( x \right ) \right ) ^{2}}{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*cos(x)^2,x)
[Out]
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Maxima [A] time = 1.41086, size = 26, normalized size = 1.04 \[ \frac{1}{4} \, x^{2} + \frac{1}{4} \, x \sin \left (2 \, x\right ) + \frac{1}{8} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*cos(x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218863, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, x \cos \left (x\right ) \sin \left (x\right ) + \frac{1}{4} \, x^{2} + \frac{1}{4} \, \cos \left (x\right )^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*cos(x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.405291, size = 36, normalized size = 1.44 \[ \frac{x^{2} \sin ^{2}{\left (x \right )}}{4} + \frac{x^{2} \cos ^{2}{\left (x \right )}}{4} + \frac{x \sin{\left (x \right )} \cos{\left (x \right )}}{2} - \frac{\sin ^{2}{\left (x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*cos(x)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.200048, size = 26, normalized size = 1.04 \[ \frac{1}{4} \, x^{2} + \frac{1}{4} \, x \sin \left (2 \, x\right ) + \frac{1}{8} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*cos(x)^2,x, algorithm="giac")
[Out]