Optimal. Leaf size=24 \[ \frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d} \]
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Rubi [A] time = 0.0142294, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2637, 2638} \[ \frac{a \sin (c+d x)}{d}-\frac{b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (a \cos (c+d x)+b \sin (c+d x)) \, dx &=a \int \cos (c+d x) \, dx+b \int \sin (c+d x) \, dx\\ &=-\frac{b \cos (c+d x)}{d}+\frac{a \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0121349, size = 46, normalized size = 1.92 \[ \frac{a \sin (c) \cos (d x)}{d}+\frac{a \cos (c) \sin (d x)}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 25, normalized size = 1. \begin{align*} -{\frac{b\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 = 0.98092, size = 32, normalized size = 1.33 \begin{align*} -\frac{b \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 = 1.92775, size = 51, normalized size = 2.12 \begin{align*} -\frac{b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.179612, size = 31, normalized size = 1.29 \begin{align*} a \left (\begin{cases} \frac{\sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \cos{\left (c \right )} & \text{otherwise} \end{cases}\right ) + b \left (\begin{cases} - \frac{\cos{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \sin{\left (c \right )} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13581, size = 32, normalized size = 1.33 \begin{align*} -\frac{b \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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