Help With This Supposedly Easy Integration

[NeedPhysHelp8]

Homework Statement

Integrate: \exp(\sin(t)) / (1 + t^2)

The attempt at a solution
Ok so I tried substituting u=sin(t) du=cos(t)dt but I end up with (1 + arcsin^2(u)) on the bottom and I don't know how to integrate that.
I also tried letting t=cos(u) dt=-sin(u)du but then I end up with e^(sin(cos(t)) which I've never seen before!
If anyone knows how to do this please just give me a hint or the first step to take and I will try to do the rest! Thanks

 

[vela]

I don't think you can integrate that analytically. Is this integral part of a larger problem?

 

[NeedPhysHelp8]

Ok well it was a differential equation problem that I reduced to that but here is the initial problem:

dy/dt + y\cos(t) = 1/ (1+t^2)

so I got an integrating factor of e^(sint) which led to this integral! Hope this helps maybe I did something wrong in first part.

 

[vela]

Hmm, perhaps you're expected to leave the solution in terms of the integral.

 

[NeedPhysHelp8]

I don't think so since the prof asked to solve it in terms of t explicitly. Maybe she made a mistake in writing the problem if this cannot be solved analytically.