Optimal. Leaf size=26 \[ \frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0106401, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ \frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2633
Rubi steps
\begin{align*} \int \cos ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (a+b x)\right )}{b}\\ &=\frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0074745, size = 26, normalized size = 1. \[ \frac{\sin (a+b x)}{b}-\frac{\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 22, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2+ \left ( \cos \left ( bx+a \right ) \right ) ^{2} \right ) \sin \left ( bx+a \right ) }{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.943241, size = 30, normalized size = 1.15 \begin{align*} -\frac{\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61269, size = 55, normalized size = 2.12 \begin{align*} \frac{{\left (\cos \left (b x + a\right )^{2} + 2\right )} \sin \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.466847, size = 36, normalized size = 1.38 \begin{align*} \begin{cases} \frac{2 \sin ^{3}{\left (a + b x \right )}}{3 b} + \frac{\sin{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \cos ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0726, size = 30, normalized size = 1.15 \begin{align*} -\frac{\sin \left (b x + a\right )^{3} - 3 \, \sin \left (b x + a\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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