Optimal. Leaf size=27 \[ a x+\frac {b \sin (d+e x)}{e}-\frac {c \cos (d+e x)}{e} \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2637, 2638} \[ a x+\frac {b \sin (d+e x)}{e}-\frac {c \cos (d+e x)}{e} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin {align*} \int (a+b \cos (d+e x)+c \sin (d+e x)) \, dx &=a x+b \int \cos (d+e x) \, dx+c \int \sin (d+e x) \, dx\\ &=a x-\frac {c \cos (d+e x)}{e}+\frac {b \sin (d+e x)}{e}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 49, normalized size = 1.81 \[ a x+\frac {b \sin (d) \cos (e x)}{e}+\frac {b \cos (d) \sin (e x)}{e}+\frac {c \sin (d) \sin (e x)}{e}-\frac {c \cos (d) \cos (e x)}{e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 26, normalized size = 0.96 \[ \frac {a e x - c \cos \left (e x + d\right ) + b \sin \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 27, normalized size = 1.00 \[ -c \cos \left (x e + d\right ) e^{\left (-1\right )} + b e^{\left (-1\right )} \sin \left (x e + d\right ) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 1.04 \[ a x -\frac {c \cos \left (e x +d \right )}{e}+\frac {b \sin \left (e x +d \right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 27, normalized size = 1.00 \[ a x - \frac {c \cos \left (e x + d\right )}{e} + \frac {b \sin \left (e x + d\right )}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.51, size = 40, normalized size = 1.48 \[ a\,x-\frac {2\,c-2\,b\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e\,\left ({\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 34, normalized size = 1.26 \[ a x + b \left (\begin {cases} \frac {\sin {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \cos {\relax (d )} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} - \frac {\cos {\left (d + e x \right )}}{e} & \text {for}\: e \neq 0 \\x \sin {\relax (d )} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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