Optimal. Leaf size=15 \[ a x+\frac {b \sin (c+d x)}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.76, size = 17, normalized size = 1.13 \[ \frac {a d x + b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 15, normalized size = 1.00 \[ a x + \frac {b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 16, normalized size = 1.07 \[ a x +\frac {b \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 15, normalized size = 1.00 \[ a x + \frac {b \sin \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 17, normalized size = 1.13 \[ \frac {b\,\sin \left (c+d\,x\right )+a\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 1.13 \[ a x + b \left (\begin {cases} \frac {\sin {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cos {\relax (c )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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