The indefinite integral 
$$F(x)=(-1)^{3/4} \operatorname{arctanh}\left[(-1)^{1/4} \operatorname{cd}(x\mid i)\right]$$ 
of the Jacobi elliptic function $\operatorname{sn}(x\mid i)$ has a discontinuity of
$$\delta=2\lim_{z\rightarrow\infty}(-1)^{3/4} \operatorname{arctanh}\left[(-1)^{1/4} z\right]=
-\frac{(1+i) \pi }{\sqrt{2}}$$ at $x=7.0748\cdots$ [arising from the pole in $\operatorname{cd}(x\mid i)$], hence the definite integral equals
$$\int_0^{10} \operatorname{sn}(x\mid i) \, dx=F(10)-F(0)+\delta.$$
This evaluates to $-2.42813 - 2.31227\, i$, in agreement with a numerical evaluation of the integral.