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$$\int \sin (x^3)dx$$

I have tried some substitutions, but I haven’t reached the goal… Can you help me?

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By combining Euler’s formula with the integral expression for the $\Gamma$ function, we have, for $n>1$ $$\int_0^\infty\sin(x^n)~dx~=~\bigg(\frac1n\bigg){\large!}~\sin\frac\pi{2n}$$ and $$\int_0^\infty\cos(x^n)~dx~=~\bigg(\frac1n\bigg){\large!}~\cos\frac\pi{2n}$$ Given the fact that your integral is indefinite, its expression is very similar, but it involves the incomplete $\Gamma$ function, rather than the classical one.

Possibly this is not useful to you, but…

$$\sin(x^3)=\sum_{n=0}^\infty(-1)^n\frac{x^{6n+3}}{(2n+1)!}$$

so

$$\int\sin(x^3)dx=C+\sum_{n=0}^\infty(-1)^n\frac{x^{6n+4}}{(2n+1)!(6n+4)}$$

Your integral is the imaginary part of the Fresnel integral:

$$ \int \exp{(\mathrm{i} x^3)} \, \mathrm{d}x = x \, {}_1F_1(1/3,4/3,\mathrm{i}x^3), $$ where I have substituted $n=3$ and $m = 0$ from ** here**·

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