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$$\displaystyle \int\frac{1}{(a+b\sin x)^4}dx,~~~~\text{and}~~~~\displaystyle \int\frac{1}{(a+b\cos x)^4}dx,$$

although i have tried using Trg. substution. but nothing get

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This question has an open bounty worth +50 reputation from Brahmagupta ending in 7 days.

Looking for an answer drawing from credible and/or official sources.

Use $u=\tan(x/2)$ substitution. – user72870 Jun 4 '13 at 10:23

Use $\sin(x)= \frac{2\tan\frac{x}{2}}{1 +\tan^2 \frac {x}{2}}$ and $\cos(x)= \frac{1-\tan ^{2}\frac{x}{2}}{1 +\tan^{2}\frac {x}{2}}$

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