Integral of exp(a*x)*sin(b*x) dx

                                                          //                         /cos(b*x)   x*sin(b*x)                                                \
                                                          ||                         |-------- + ----------  for b != 0                                    |
                                                          ||                         |    2          b                                                     |
                                                          ||                         |   b                                                                 |
                                                          ||                         <                                                            for a = 0|
                                                          ||                         |          2                                                          |
                                                          ||                         |         x                                                           |
                                                          ||                         |         --            otherwise                                     |
                                                          ||                         \         2                                                           |
                                                          ||                                                                                               |
                                                          ||/                             x                                for And(a = 0, b = 0)           |
  /                       // x    for a = 0\              |||                                                                                              |
 |                        ||               |              |||            -I*b*x    -I*b*x                 -I*b*x                                           |
 |  a*x                   || a*x           |              |||x*cos(b*x)*e         e      *sin(b*x)   I*x*e      *sin(b*x)                                  |
 | e   *sin(b*x) dx = C + |
            

$$\int e^{a x} \sin{\left(b x \right)}\, dx = C - b \left(\begin{cases} \begin{cases} \frac{x \sin{\left(b x \right)}}{b} + \frac{\cos{\left(b x \right)}}{b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2} & \text{otherwise} \end{cases} & \text{for}\: a = 0 \\\frac{\begin{cases} x & \text{for}\: a = 0 \wedge b = 0 \\\frac{i x e^{- i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{- i b x} \cos{\left(b x \right)}}{2} + \frac{e^{- i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = - i b \\- \frac{i x e^{i b x} \sin{\left(b x \right)}}{2} + \frac{x e^{i b x} \cos{\left(b x \right)}}{2} + \frac{e^{i b x} \sin{\left(b x \right)}}{2 b} & \text{for}\: a = i b \\\frac{a e^{a x} \cos{\left(b x \right)}}{a^{2} + b^{2}} + \frac{b e^{a x} \sin{\left(b x \right)}}{a^{2} + b^{2}} & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right) + \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{e^{a x}}{a} & \text{otherwise} \end{cases}\right) \sin{\left(b x \right)}$$

Simplify