Non solvable integral?

• A
Tomder
TL;DR Summary
I have a non linear system to which I implement a PD controller, but when applying the kinetic-work theorem I can't solve an integral.
The integral is (dx/dt)^2 dt, where x=x(t) so it can't be just x + C.

The non linear system for whom wants to know how did I get to that point is:

d(dx/dt)/dt = sqrt(a^2+b^2)*sin(x+alfa+phi) - Kd*(dx/dt); where alfa = atan(a/b), phi = constant angle, Kd = constant coefficient.
After applying the kinetic work theorem by multiplying both sides by dx/dt I get:

d(dx/dt)/dt *dx/dt = sqrt(a^2+b^2)*sin(x+alfa+phi)*dx/dt - Kd*(dx/dt)*dx/dt ; So, by integration by dt I get to:

1/2*(dx/dt)^2 = - sqrt(a^2+b^2)*cos(x+alfaphi) - INTEGRAL[(Kd*(dx/dt)^2]dt + C ;

Rearranging terms:

1/2*(dx/dt)^2 + sqrt(a^2+b^2)*cos(x+alfaphi) = C - INTEGRAL[(Kd*(dx/dt)^2]dt ;

And by this without the Kd term i could get the total energy of the system and the velocity at every point but I don't know how to proceed with the Kd term.
I'm sure maybe some of the theory may be wrong explained so I say sorry in advance.
For further explanation, my system doesn't lose energy when Kd = 0 because the total energy would be constant but with Kd term I assume it is like a frictional component that takes the energy out and the system would slowly stop oscillating in the equilibrium point.

Simplifying the constants and shifting the origin of $x$, your system is $$\ddot x = A \sin x - k\dot x.$$ This is the equation of motion of a pendulum in a resistive medium. The resistive force $-k\dot x$ effectively removes energy from the pendulum as it does work in moving through the resistive medium.
You can write this as the 2D system $$\begin{split} \dot x &= y \\ \dot y &= A\sin x - ky \end{split}$$ and use Dulac's criterion to show that no non-constant periodic solutions exist.