If $f(x)$ always satisfies this equation, what is $\int_{-1}^1 f(x)\: dx$?

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For all real values of \(x\), the function \(f(x)\) satisfies

\[6+f(x)=2f(-x)+3x^2\left(\int_{-1}^1 f(t)\: dt\right).\]

It follows that \(\displaystyle\int_{-1}^1 f(x)\: dx\) equals

\[\text{(a)}\quad 4, \quad\text{(b)}\quad 6, \quad\text{(c)}\quad 11, \quad\text{(d)}\quad \dfrac{17}{2}, \quad\text{(e)}\quad 23.\]