Optimal. Leaf size=26 \[ \frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d} \]
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Rubi [A] time = 0.0140323, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2637, 2638} \[ \frac{a \sin (c+d x)}{d}-\frac{i a \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a \cos (c+d x)+i a \sin (c+d x)) \, dx &=(i a) \int \sin (c+d x) \, dx+a \int \cos (c+d x) \, dx\\ &=-\frac{i a \cos (c+d x)}{d}+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0118994, size = 51, normalized size = 1.96 \[ \frac{i a \sin (c) \sin (d x)}{d}-\frac{i a \cos (c) \cos (d x)}{d}+\frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 26, normalized size = 1. \begin{align*}{\frac{-ia\cos \left ( dx+c \right ) }{d}}+{\frac{a\sin \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98745, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a \cos \left (d x + c\right )}{d} + \frac{a \sin \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 = 2.16157, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a e^{\left (i \, d x + i \, c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.144124, size = 26, normalized size = 1. \begin{align*} \begin{cases} - \frac{i a e^{i c} e^{i d x}}{d} & \text{for}\: d \neq 0 \\a x e^{i c} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12907, size = 32, normalized size = 1.23 \begin{align*} -\frac{i \, a \cos \left (d x + c\right )}{d} + \frac{a \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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