# Finding $\int \frac{\sqrt{\cot(x)} - \sqrt{\tan(x)}}{4+3 \sin^2 (x)} \ \mathrm d x$

How can we find the indefinite integral for:

$$\int \frac{\sqrt{\cot(x)} - \sqrt{\tan(x)}}{4+3 \sin^2 (x)} \ \mathrm d x$$

I tried expressing the denominator in $$\sin x+\cos x$$ form since we have its derivative in numerator but I am not able to proceed. Please help me out.

• How to use MathJax. Where is this problem from? Sep 7 '20 at 9:03
• It is from my Calculus Course book . I am in 12th Grade so I just know some basic integration Sep 7 '20 at 9:15
• @ZAhmed I believe the combination of the hypergeometric functions in Mathematica simplify to an elementary expression. Sep 7 '20 at 9:39

By some basic trigonometric identities, one has $$\int \frac{\sqrt{\cot(x)} - \sqrt{\tan(x)}}{4+3 \sin^2 (x)}~dx=\int \frac{\sec^2(x)(1-\tan(x))}{\sqrt{\tan(x)}(7\tan^2(x)+4)}~dx,$$ which motivates the substitution $$u=\tan(x)$$. This leads to $$\int \frac{1-u}{\sqrt{u}(7u^2+4)}~du,$$ whose integrand can be converted to a rational expression via the substitution $$u=v^2$$. The resulting expression can be solved using partial fractions.