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How to solve the integral $\int_0^\infty\frac{e^{-x} \sin x}{(e^{3 x} + 1) x^{3/10}} dx$

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$$\mbox{How to solve the following integral ?}:\quad\int_{0}^{\infty}\frac{{\rm e}^{-x}\sin\left(x\right)}{\left({\rm e}^{3 x} + 1\right)x^{3/10}}{\rm d}x$$

  • I think it cannot be solved using elementary functions.
  • Solution may obtained through series or special functions ?.
  • $\tt Wolfram\ Alpha$ gives approximate value as $0.0605704$.

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