Optimal. Leaf size=15 \[ \frac{\sin ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.0183016, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2564, 30} \[ \frac{\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 2564
Rule 30
Rubi steps
\begin{align*} \int \cos (a+b x) \sin ^2(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int x^2 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{\sin ^3(a+b x)}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0035197, size = 15, normalized size = 1. \[ \frac{\sin ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 14, normalized size = 0.9 \begin{align*}{\frac{ \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{3\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.976664, size = 18, normalized size = 1.2 \begin{align*} \frac{\sin \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85827, size = 57, normalized size = 3.8 \begin{align*} -\frac{{\left (\cos \left (b x + a\right )^{2} - 1\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 = 0.514821, size = 20, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sin ^{3}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos{\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 = 1.22347, size = 18, normalized size = 1.2 \begin{align*} \frac{\sin \left (b x + a\right )^{3}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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