\[\large \int_{0}^{\pi} \frac{\sin x(\sin x + 1)e^{\sin x + \cos x}}{e^{\cos x}+1} dx = a + b\int_{0}^{\pi}e^{\sin x} dx\]

Given the equation above, where \(a\) and \(b\) are positive rational numbers, find the value of \(100(a^2+b^2).\)

**Note:** \(\displaystyle \int_{0}^{\pi}e^{\sin x} dx\) is irrational.

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