Optimal. Leaf size=33 \[ \frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0358782, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2833, 12, 43} \[ \frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2833
Rule 12
Rule 43
Rubi steps
\begin{align*} \int \cos (c+d x) \sin (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+x)}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}(\int x (a+x) \, dx,x,a \sin (c+d x))}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a x+x^2\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.090569, size = 30, normalized size = 0.91 \[ \frac{4 a \sin ^3(c+d x)-3 a \cos (2 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 28, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}a}{3}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.29919, size = 38, normalized size = 1.15 \begin{align*} \frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64546, size = 93, normalized size = 2.82 \begin{align*} -\frac{3 \, a \cos \left (d x + c\right )^{2} + 2 \,{\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.3498, size = 41, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{a \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \sin{\left (c \right )} \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29005, size = 38, normalized size = 1.15 \begin{align*} \frac{2 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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