Evaluate $\int_{-\pi/4}^{\pi/4} \frac{x\sin(x)}{1+\exp(x^2\sin(x))} \mathrm{d}x$

Question

Can any one obtain an analytical expression for the area under the curve $$f(x)=\frac{x\sin(x)}{1+\exp(x^2\sin(x))}$$ from $x = -\frac{\pi}{4}$ to $x = \frac{\pi}{4}$, as I can't ! The numerical value is 0.15175. Any suggestions would be most welcome.

Answer 1

I don't know how the antiderivative could be obtained the CAS I used did not give anything even using special functions).

However, over the given range, the integrand can be quite nicely represented using Padé approximants. For example $$f(x)\approx\frac{\frac{x^2}{2}-\frac{65 x^3}{8407}+\frac{1399 x^4}{16814}-\frac{25091 x^5}{100884}-\frac{119389 x^6}{6053040} }{1-\frac{130 x}{8407}+\frac{16801 x^2}{50442} }$$ would lead to an explicit expression which evaluates as $0.151743$.

Using $[n,2]$ Padé approximants, the following numerical values are obtained $$\left( \begin{array}{cc} n & \text{integral} \\ 4 & 0.158864 \\ 5 & 0.151865 \\ 6 & 0.151743 \\ 7 & 0.151746 \\ 8 & 0.151746 \\ 9 & 0.151747 \\ 10 & 0.151746 \end{array} \right)$$

Simpler would be a Taylor expansion of the integrand $$f(x)=\frac{x^2}{2}-\frac{x^4}{12}-\frac{x^5}{4}+\frac{x^6}{240}+\frac{x^7}{12}-\frac{x ^8}{10080}-\frac{x^9}{90}+\frac{x^{10}}{725760}+\frac{109 x^{11}}{5040}+O\left(x^{12}\right)$$ which leads to $$I=\frac{\pi ^3}{192}-\frac{\pi ^5}{30720}+\frac{\pi ^7}{13762560}-\frac{\pi ^9}{11890851840}+\frac{\pi ^{11}}{16742319390720}\approx 0.151746414000$$ while the given solution is $\approx 0.151746413917$.

Answer 2

Just observe the symmetry: $$\int_{ - \pi /4}^{\pi /4} {\frac{{x\sin x}}{{1 + {e^{{x^2}\sin x}}}}dx} = \int_0^{\pi /4} {\left( {\frac{{x\sin x}}{{1 + {e^{{x^2}\sin x}}}} + \frac{{x\sin x}}{{1 + {e^{ - {x^2}\sin x}}}}} \right)dx} = \int_0^{\pi /4} {x\sin xdx} = \frac{{4 - \pi }}{{4\sqrt 2 }}$$