Optimal. Leaf size=45 \[ \frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d} \]
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Rubi [A] time = 0.0365742, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2667, 43} \[ \frac{2 (a \sin (c+d x)+a)^3}{3 a^2 d}-\frac{(a \sin (c+d x)+a)^4}{4 a^3 d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 43
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int (a-x) (a+x)^2 \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (2 a (a+x)^2-(a+x)^3\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac{2 (a+a \sin (c+d x))^3}{3 a^2 d}-\frac{(a+a \sin (c+d x))^4}{4 a^3 d}\\ \end{align*}
Mathematica [A] time = 0.0173511, size = 44, normalized size = 0.98 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{a \cos ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 36, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{4}}+{\frac{a \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.942905, size = 65, normalized size = 1.44 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 6 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64002, size = 97, normalized size = 2.16 \begin{align*} -\frac{3 \, a \cos \left (d x + c\right )^{4} - 4 \,{\left (a \cos \left (d x + c\right )^{2} + 2 \, a\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 = 1.20515, size = 82, normalized size = 1.82 \begin{align*} \begin{cases} \frac{a \sin ^{4}{\left (c + d x \right )}}{4 d} + \frac{2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} + \frac{a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \cos ^{3}{\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 = 1.11346, size = 65, normalized size = 1.44 \begin{align*} -\frac{3 \, a \sin \left (d x + c\right )^{4} + 4 \, a \sin \left (d x + c\right )^{3} - 6 \, a \sin \left (d x + c\right )^{2} - 12 \, a \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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