Integrate $e^{ax}\sin(bx)?$ Is there a general formula for finding the primitive of $$e^{ax}\sin(bx)?$$

I've done this manually with $a=9$ and $b=4$ using Euler's formulas. But it takes a bit of time. Is there a pattern here?
There is a pattern. Differentiating a function of the form $e^{ax}\sin (bx)$ yields a linear combination of a function of the same form, and a function $e^{ax}\cos (bx)$. The analogous property holds for functions $e^{ax}\cos (bx)$. So the primitive of $e^{ax}\sin (bx)$ will be a linear combination of $e^{ax}\sin (bx)$ and $e^{ax}\cos (bx)$ (plus a constant).

It remains to find the coefficients.

$$\begin{align}
\frac{d}{dx}\left(e^{ax}(m\sin (bx) + n\cos (bx)\right) &amp;= e^{ax}\left(a\bigl(m\sin(bx) + n\cos(bx)\bigr) + \bigl(bm\cos(bx) - bn\sin(bx)\bigr)\right)\\
&amp;= e^{ax}\left((am - bn)\sin (bx) + (an+bm)\cos(bx)\right)
\end{align}$$

Now solve the linear system

$$\begin{align}
am - bn &amp;= 1\\
an + bm &amp;= 0.
\end{align}$$
$e^{ax}(a\sin(bx)-b\cos(bx))/(a^2+b^2)$