Optimal. Leaf size=36 \[ \frac{b e^x \sin (a+b x)}{b^2+1}+\frac{e^x \cos (a+b x)}{b^2+1} \]
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Rubi [A] time = 0.0123241, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {4433} \[ \frac{b e^x \sin (a+b x)}{b^2+1}+\frac{e^x \cos (a+b x)}{b^2+1} \]
Antiderivative was successfully verified.
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Rule 4433
Rubi steps
\begin{align*} \int e^x \cos (a+b x) \, dx &=\frac{e^x \cos (a+b x)}{1+b^2}+\frac{b e^x \sin (a+b x)}{1+b^2}\\ \end{align*}
Mathematica [A] time = 0.0504032, size = 26, normalized size = 0.72 \[ \frac{e^x (b \sin (a+b x)+\cos (a+b x))}{b^2+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 35, normalized size = 1. \begin{align*}{\frac{{{\rm e}^{x}}\cos \left ( bx+a \right ) }{{b}^{2}+1}}+{\frac{{{\rm e}^{x}}b\sin \left ( bx+a \right ) }{{b}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09355, size = 34, normalized size = 0.94 \begin{align*} \frac{{\left (b \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} e^{x}}{b^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.461251, size = 69, normalized size = 1.92 \begin{align*} \frac{b e^{x} \sin \left (b x + a\right ) + \cos \left (b x + a\right ) e^{x}}{b^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.47465, size = 114, normalized size = 3.17 \begin{align*} \begin{cases} - \frac{i x e^{x} \sin{\left (a - i x \right )}}{2} + \frac{x e^{x} \cos{\left (a - i x \right )}}{2} + \frac{e^{x} \cos{\left (a - i x \right )}}{2} & \text{for}\: b = - i \\\frac{i x e^{x} \sin{\left (a + i x \right )}}{2} + \frac{x e^{x} \cos{\left (a + i x \right )}}{2} - \frac{i e^{x} \sin{\left (a + i x \right )}}{2} & \text{for}\: b = i \\\frac{b e^{x} \sin{\left (a + b x \right )}}{b^{2} + 1} + \frac{e^{x} \cos{\left (a + b x \right )}}{b^{2} + 1} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13738, size = 45, normalized size = 1.25 \begin{align*}{\left (\frac{b \sin \left (b x + a\right )}{b^{2} + 1} + \frac{\cos \left (b x + a\right )}{b^{2} + 1}\right )} e^{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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