Optimal. Leaf size=45 \[ -\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2} \]
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Rubi [A] time = 0.0136893, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2644} \[ -\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac{3 a^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 2644
Rubi steps
\begin{align*} \int (a+a \sin (c+d x))^2 \, dx &=\frac{3 a^2 x}{2}-\frac{2 a^2 \cos (c+d x)}{d}-\frac{a^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.19256, size = 34, normalized size = 0.76 \[ -\frac{a^2 (-6 (c+d x)+\sin (2 (c+d x))+8 \cos (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 52, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) -2\,\cos \left ( dx+c \right ){a}^{2}+{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08459, size = 63, normalized size = 1.4 \begin{align*} a^{2} x + \frac{{\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{4 \, d} - \frac{2 \, a^{2} \cos \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55815, size = 97, normalized size = 2.16 \begin{align*} \frac{3 \, a^{2} d x - a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 4 \, a^{2} \cos \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 = 0.697112, size = 78, normalized size = 1.73 \begin{align*} \begin{cases} \frac{a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + a^{2} x - \frac{a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} - \frac{2 a^{2} \cos{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.85333, size = 51, normalized size = 1.13 \begin{align*} \frac{3}{2} \, a^{2} x - \frac{2 \, a^{2} \cos \left (d x + c\right )}{d} - \frac{a^{2} \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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