Optimal. Leaf size=15 \[ a x+\frac{b \sin (c+d x)}{d} \]
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Rubi [A] time = 0.0085139, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2637} \[ a x+\frac{b \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \, dx &=a x+b \int \cos (c+d x) \, dx\\ &=a x+\frac{b \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0063226, size = 26, normalized size = 1.73 \[ a x+\frac{b \sin (c) \cos (d x)}{d}+\frac{b \cos (c) \sin (d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 16, normalized size = 1.1 \begin{align*} ax+{\frac{b\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 = 0.947299, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \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 = 1.86152, size = 38, normalized size = 2.53 \begin{align*} \frac{a d x + b \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 = 0.125019, size = 17, normalized size = 1.13 \begin{align*} a x + b \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36589, size = 20, normalized size = 1.33 \begin{align*} a x + \frac{b \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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