Calculate $\int \frac{1}{(\sin (x)+\cos (x))^{100}}dx$

Question

I would to calculate the following integral with different ways

$$\int \frac{1}{(\sin x+\cos x)^{100}}dx$$

My thoughts:

$\cos(x)+\sin(x)=\sqrt{2}\cos(x+\pi/4)$

Answer 1

Following up on Jack D'Aurizio's comment below the OP, after the substitution, you wind up faced trying to integrate

$$\int{d\theta\over\cos^{100}\theta}=\int\sec^{98}\theta\sec^2\theta\, d\theta=\int(\tan^2\theta+1)^{49}\,d(\tan\theta)=\int(u^2+1)^{49}du=\cdots$$

This is clearly straightforward, but looks unpleasant. It might be worth experimenting with some smaller exponents, say $6$ instead of $100$, to see if the end result simplifies.