Optimal. Leaf size=22 \[ \frac{(a \sin (c+d x)+a)^2}{2 a d} \]
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Rubi [A] time = 0.0163306, antiderivative size = 28, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2667} \[ \frac{a \sin ^2(c+d x)}{2 d}+\frac{a \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2667
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sin (c+d x)) \, dx &=\frac{\operatorname{Subst}(\int (a+x) \, dx,x,a \sin (c+d x))}{a d}\\ &=\frac{a \sin (c+d x)}{d}+\frac{a \sin ^2(c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0133389, size = 39, normalized size = 1.77 \[ -\frac{a \cos ^2(c+d x)}{2 d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 25, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}a}{2}}+a\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94099, size = 27, normalized size = 1.23 \begin{align*} \frac{{\left (a \sin \left (d x + c\right ) + a\right )}^{2}}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61103, size = 62, normalized size = 2.82 \begin{align*} -\frac{a \cos \left (d x + c\right )^{2} - 2 \, a \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.220222, size = 34, normalized size = 1.55 \begin{align*} \begin{cases} \frac{a \sin ^{2}{\left (c + d x \right )}}{2 d} + \frac{a \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\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 = 1.11971, size = 34, normalized size = 1.55 \begin{align*} \frac{a \sin \left (d x + c\right )^{2} + 2 \, a \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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