Optimal. Leaf size=29 \[ \frac{2 a \cos (d+e x)}{e}+2 a x+\frac{2 b \sin (d+e x)}{e} \]
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Rubi [A] time = 0.0144019, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2637, 2638} \[ \frac{2 a \cos (d+e x)}{e}+2 a x+\frac{2 b \sin (d+e x)}{e} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rubi steps
\begin{align*} \int (2 a+2 b \cos (d+e x)-2 a \sin (d+e x)) \, dx &=2 a x-(2 a) \int \sin (d+e x) \, dx+(2 b) \int \cos (d+e x) \, dx\\ &=2 a x+\frac{2 a \cos (d+e x)}{e}+\frac{2 b \sin (d+e x)}{e}\\ \end{align*}
Mathematica [A] time = 0.0119253, size = 53, normalized size = 1.83 \[ -\frac{2 a \sin (d) \sin (e x)}{e}+\frac{2 a \cos (d) \cos (e x)}{e}+2 a x+\frac{2 b \sin (d) \cos (e x)}{e}+\frac{2 b \cos (d) \sin (e x)}{e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 30, normalized size = 1. \begin{align*} 2\,ax+2\,{\frac{a\cos \left ( ex+d \right ) }{e}}+2\,{\frac{b\sin \left ( ex+d \right ) }{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990877, size = 39, normalized size = 1.34 \begin{align*} 2 \, a x + \frac{2 \, a \cos \left (e x + d\right )}{e} + \frac{2 \, b \sin \left (e x + d\right )}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14906, size = 63, normalized size = 2.17 \begin{align*} \frac{2 \,{\left (a e x + a \cos \left (e x + d\right ) + b \sin \left (e x + d\right )\right )}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.153007, size = 39, normalized size = 1.34 \begin{align*} 2 a x - 2 a \left (\begin{cases} - \frac{\cos{\left (d + e x \right )}}{e} & \text{for}\: e \neq 0 \\x \sin{\left (d \right )} & \text{otherwise} \end{cases}\right ) + 2 b \left (\begin{cases} \frac{\sin{\left (d + e x \right )}}{e} & \text{for}\: e \neq 0 \\x \cos{\left (d \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 = 1.12482, size = 39, normalized size = 1.34 \begin{align*} 2 \, a \cos \left (x e + d\right ) e^{\left (-1\right )} + 2 \, b e^{\left (-1\right )} \sin \left (x e + d\right ) + 2 \, a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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