Recurrence relation for the integral, $ I_n=\int\frac{dx}{(1+x^2)^n} $

Express recurrence relation of the integral

$$
I_n=\int\frac{dx}{(1+x^2)^n}
$$

[My Answer]

$$
I_n = \int\frac{1+x^2}{(1+x^2)^n}dx-\int\frac{x^2}{(1+x^2)^n}dx
$$

$$
I_n=I_{n-1}-\int x\cdot\frac{x}{(1+x^2)^n}dx
$$

$$
I_n=I_{n-1}-\frac{x}{2(1-n)(x^2+1)^{n-1}}+\frac{1}{2(1-n)}I_{n-1}
$$

$$
I_n=\frac{2n-3}{2(n-1)}I_{n-1}+\frac{x}{2(n-1)(x^2+1)^{n-1}} \ \ \ \ (n>1)
$$

$$
I_1=\arctan(x)
$$

Is my answer correct?

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