Optimal. Leaf size=42 \[ \frac{a e^{a x} \sin (b x)}{a^2+b^2}-\frac{b e^{a x} \cos (b x)}{a^2+b^2} \]
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Rubi [A] time = 0.0113484, antiderivative size = 42, 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 = {4432} \[ \frac{a e^{a x} \sin (b x)}{a^2+b^2}-\frac{b e^{a x} \cos (b x)}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 4432
Rubi steps
\begin{align*} \int e^{a x} \sin (b x) \, dx &=-\frac{b e^{a x} \cos (b x)}{a^2+b^2}+\frac{a e^{a x} \sin (b x)}{a^2+b^2}\\ \end{align*}
Mathematica [A] time = 0.0241032, size = 29, normalized size = 0.69 \[ \frac{e^{a x} (a \sin (b x)-b \cos (b x))}{a^2+b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 41, normalized size = 1. \begin{align*} -{\frac{{{\rm e}^{ax}}b\cos \left ( bx \right ) }{{a}^{2}+{b}^{2}}}+{\frac{a{{\rm e}^{ax}}\sin \left ( bx \right ) }{{a}^{2}+{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9364, size = 39, normalized size = 0.93 \begin{align*} -\frac{{\left (b \cos \left (b x\right ) - a \sin \left (b x\right )\right )} e^{\left (a x\right )}}{a^{2} + b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.506773, size = 76, normalized size = 1.81 \begin{align*} -\frac{b \cos \left (b x\right ) e^{\left (a x\right )} - a e^{\left (a x\right )} \sin \left (b x\right )}{a^{2} + b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.36466, size = 136, normalized size = 3.24 \begin{align*} \begin{cases} 0 & \text{for}\: a = 0 \wedge b = 0 \\\frac{x e^{- i b x} \sin{\left (b x \right )}}{2} - \frac{i x e^{- i b x} \cos{\left (b x \right )}}{2} - \frac{e^{- i b x} \cos{\left (b x \right )}}{2 b} & \text{for}\: a = - i b \\\frac{x e^{i b x} \sin{\left (b x \right )}}{2} + \frac{i x e^{i b x} \cos{\left (b x \right )}}{2} - \frac{e^{i b x} \cos{\left (b x \right )}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \sin{\left (b x \right )}}{a^{2} + b^{2}} - \frac{b e^{a x} \cos{\left (b x \right )}}{a^{2} + b^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06469, size = 51, normalized size = 1.21 \begin{align*} -{\left (\frac{b \cos \left (b x\right )}{a^{2} + b^{2}} - \frac{a \sin \left (b x\right )}{a^{2} + b^{2}}\right )} e^{\left (a x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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