Optimal. Leaf size=55 \[ \frac{a^2 \sin ^4(c+d x)}{4 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.0482555, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2833, 12, 43} \[ \frac{a^2 \sin ^4(c+d x)}{4 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^2(c+d x)}{2 d} \]
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))^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (a+x)^2}{a} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac{\operatorname{Subst}\left (\int x (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^2 x+2 a x^2+x^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac{a^2 \sin ^2(c+d x)}{2 d}+\frac{2 a^2 \sin ^3(c+d x)}{3 d}+\frac{a^2 \sin ^4(c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.0506013, size = 38, normalized size = 0.69 \[ \frac{a^2 \sin ^2(c+d x) \left (3 \sin ^2(c+d x)+8 \sin (c+d x)+6\right )}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 45, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{2}}{4}}+{\frac{2\,{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14817, size = 61, normalized size = 1.11 \begin{align*} \frac{3 \, a^{2} \sin \left (d x + c\right )^{4} + 8 \, a^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64804, size = 134, normalized size = 2.44 \begin{align*} \frac{3 \, a^{2} \cos \left (d x + c\right )^{4} - 12 \, a^{2} \cos \left (d x + c\right )^{2} - 8 \,{\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2}\right )} \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.09734, size = 87, normalized size = 1.58 \begin{align*} \begin{cases} \frac{2 a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} - \frac{a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} - \frac{a^{2} \cos ^{4}{\left (c + d x \right )}}{4 d} - \frac{a^{2} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{2} \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 = 1.13873, size = 61, normalized size = 1.11 \begin{align*} \frac{3 \, a^{2} \sin \left (d x + c\right )^{4} + 8 \, a^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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