Optimal. Leaf size=25 \[ \frac{\sin (a+b x) \cos (a+b x)}{2 b}+\frac{x}{2} \]
[Out]
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Rubi [A] time = 0.0182586, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sin (a+b x) \cos (a+b x)}{2 b}+\frac{x}{2} \]
Antiderivative was successfully verified.
[In] Int[Cos[a + b*x]^2,x]
[Out]
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Rubi in Sympy [A] time = 0.712147, size = 19, normalized size = 0.76 \[ \frac{x}{2} + \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(cos(b*x+a)**2,x)
[Out]
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Mathematica [A] time = 0.0222123, size = 23, normalized size = 0.92 \[ \frac{2 (a+b x)+\sin (2 (a+b x))}{4 b} \]
Antiderivative was successfully verified.
[In] Integrate[Cos[a + b*x]^2,x]
[Out]
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Maple [A] time = 0.004, size = 27, normalized size = 1.1 \[{\frac{1}{b} \left ({\frac{\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{2}}+{\frac{bx}{2}}+{\frac{a}{2}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(cos(b*x+a)^2,x)
[Out]
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Maxima [A] time = 1.38177, size = 30, normalized size = 1.2 \[ \frac{2 \, b x + 2 \, a + \sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(b*x + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.244026, size = 30, normalized size = 1.2 \[ \frac{b x + \cos \left (b x + a\right ) \sin \left (b x + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(b*x + a)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.263376, size = 46, normalized size = 1.84 \[ \begin{cases} \frac{x \sin ^{2}{\left (a + b x \right )}}{2} + \frac{x \cos ^{2}{\left (a + b x \right )}}{2} + \frac{\sin{\left (a + b x \right )} \cos{\left (a + b x \right )}}{2 b} & \text{for}\: b \neq 0 \\x \cos ^{2}{\left (a \right )} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(b*x+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.20044, size = 24, normalized size = 0.96 \[ \frac{1}{2} \, x + \frac{\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(cos(b*x + a)^2,x, algorithm="giac")
[Out]