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 = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 1.93, size = 15, normalized size = 0.58 \[ -\frac {i \, a e^{\left (i \, d x + i \, c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 24, normalized size = 0.92 \[ -\frac {i \, a \cos \left (d x + c\right )}{d} + \frac {a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 26, normalized size = 1.00 \[ -\frac {i a \cos \left (d x +c \right )}{d}+\frac {a \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 24, normalized size = 0.92 \[ -\frac {i \, a \cos \left (d x + c\right )}{d} + \frac {a \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.39, size = 20, normalized size = 0.77 \[ \frac {2\,a}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 26, normalized size = 1.00 \[ \begin {cases} - \frac {i a e^{i c} e^{i d x}}{d} & \text {for}\: d \neq 0 \\a x e^{i c} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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