\[ \int \frac {1}{(b \cos (x)+a \sin (x))^2} \, dx \]
Optimal antiderivative \[ \frac {\sin \left (x \right )}{b \left (b \cos \left (x \right )+a \sin \left (x \right )\right )} \]
command
integrate(1/(b*cos(x)+a*sin(x))**2,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty } \tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} & \text {for}\: a = 0 \wedge b = 0 \\\frac {x \tan ^{4}{\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 x \tan ^{2}{\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} + \frac {x}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} + \frac {2 \tan ^{3}{\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} - \frac {2 \tan {\left (\frac {x}{2} \right )}}{2 b^{2} \sin ^{2}{\left (x \right )} \tan ^{4}{\left (\frac {x}{2} \right )} - 4 b^{2} \sin ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )} + 2 b^{2} \sin ^{2}{\left (x \right )} + 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan ^{3}{\left (\frac {x}{2} \right )} - 8 b^{2} \sin {\left (x \right )} \cos {\left (x \right )} \tan {\left (\frac {x}{2} \right )} + 8 b^{2} \cos ^{2}{\left (x \right )} \tan ^{2}{\left (\frac {x}{2} \right )}} & \text {for}\: a = \frac {b \left (\tan {\left (\frac {x}{2} \right )} - \frac {1}{\tan {\left (\frac {x}{2} \right )}}\right )}{2} \\\frac {\frac {\tan {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tan {\left (\frac {x}{2} \right )}}}{a^{2}} & \text {for}\: b = 0 \\\frac {2 \tan {\left (\frac {x}{2} \right )}}{2 a b \tan {\left (\frac {x}{2} \right )} - b^{2} \tan ^{2}{\left (\frac {x}{2} \right )} + b^{2}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \int \frac {1}{\left (a \sin {\left (x \right )} + b \cos {\left (x \right )}\right )^{2}}\, dx \]________________________________________________________________________________________