Integrals of goniometric functions

Convert the following integrals to integrals of a rational function and think about which partial fractions you get (which denominators). You no longer have to calculate the decomposition into partial fractions and the resulting primitive function.

• Variant 1

$$\int\frac{1}{\sin x \cos x} \, dx$$

• Variant 2

$$\int \frac{1}{\sin x} \, dx$$

• Variant 3

$$\int\frac{1}{\cos x \sin^3 x} \, dx$$

• Variant 4

$$\int \operatorname{tan}^5 x \, dx$$

• Variant 5

$$\int \frac{\cos^4 x + \sin^4 x}{\cos^2 x - \sin^2 x} \, dx$$

• Variant 6

$$\int \frac{\sin x}{1 + \sin x} \, dx$$

• Variant 7

$$\int \frac{1}{2 \sin x -\cos x + 5} \, dx$$

• Variant 8

$$\int \frac{\sin x \cos x}{1 + \sin^4 x} \, dx$$