Optimal. Leaf size=43 \[ -\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]
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Rubi [A] time = 0.0330301, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2669, 2635, 8} \[ -\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx &=-\frac{a \cos ^3(c+d x)}{3 d}+a \int \cos ^2(c+d x) \, dx\\ &=-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}+\frac{1}{2} a \int 1 \, dx\\ &=\frac{a x}{2}-\frac{a \cos ^3(c+d x)}{3 d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0484589, size = 46, normalized size = 1.07 \[ \frac{a (c+d x)}{2 d}+\frac{a \sin (2 (c+d x))}{4 d}-\frac{a \cos ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 41, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( -{\frac{a \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{3}}+a \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959197, size = 50, normalized size = 1.16 \begin{align*} -\frac{4 \, a \cos \left (d x + c\right )^{3} - 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65802, size = 96, normalized size = 2.23 \begin{align*} -\frac{2 \, a \cos \left (d x + c\right )^{3} - 3 \, a d x - 3 \, a \cos \left (d x + c\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 = 0.685955, size = 71, normalized size = 1.65 \begin{align*} \begin{cases} \frac{a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} - \frac{a \cos ^{3}{\left (c + d x \right )}}{3 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right ) \cos ^{2}{\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 = 1.1155, size = 63, normalized size = 1.47 \begin{align*} \frac{1}{2} \, a x - \frac{a \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac{a \cos \left (d x + c\right )}{4 \, d} + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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