\[ \int \frac {\cos ^5(a+b x)-\sin ^5(a+b x)}{\cos ^5(a+b x)+\sin ^5(a+b x)} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\cos \left (b x +a \right )\right )}{b}+\frac {\ln \left (1+\tan \left (b x +a \right )\right )}{5 b}-\frac {4 \ln \left (2-\left (-\sqrt {5}+1\right ) \tan \left (b x +a \right )+2 \left (\tan ^{2}\left (b x +a \right )\right )\right )}{5 b \left (-\sqrt {5}+1\right )}-\frac {4 \ln \left (2-\left (\sqrt {5}+1\right ) \tan \left (b x +a \right )+2 \left (\tan ^{2}\left (b x +a \right )\right )\right )}{5 b \left (\sqrt {5}+1\right )} \]
command
integrate((cos(b*x+a)**5-sin(b*x+a)**5)/(cos(b*x+a)**5+sin(b*x+a)**5),x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {47 \log {\left (\sin {\left (a + b x \right )} + \cos {\left (a + b x \right )} \right )}}{- 235 b + 105 \sqrt {5} b} + \frac {21 \sqrt {5} \log {\left (\sin {\left (a + b x \right )} + \cos {\left (a + b x \right )} \right )}}{- 235 b + 105 \sqrt {5} b} - \frac {26 \sqrt {5} \log {\left (16 \sin ^{2}{\left (a + b x \right )} - 8 \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} + 8 \sqrt {5} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} + 16 \cos ^{2}{\left (a + b x \right )} \right )}}{- 235 b + 105 \sqrt {5} b} + \frac {58 \log {\left (16 \sin ^{2}{\left (a + b x \right )} - 8 \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} + 8 \sqrt {5} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} + 16 \cos ^{2}{\left (a + b x \right )} \right )}}{- 235 b + 105 \sqrt {5} b} - \frac {152 \log {\left (16 \sin ^{2}{\left (a + b x \right )} - 8 \sqrt {5} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} - 8 \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} + 16 \cos ^{2}{\left (a + b x \right )} \right )}}{- 235 b + 105 \sqrt {5} b} + \frac {68 \sqrt {5} \log {\left (16 \sin ^{2}{\left (a + b x \right )} - 8 \sqrt {5} \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} - 8 \sin {\left (a + b x \right )} \cos {\left (a + b x \right )} + 16 \cos ^{2}{\left (a + b x \right )} \right )}}{- 235 b + 105 \sqrt {5} b} & \text {for}\: b \neq 0 \\\frac {x \left (- \sin ^{5}{\left (a \right )} + \cos ^{5}{\left (a \right )}\right )}{\sin ^{5}{\left (a \right )} + \cos ^{5}{\left (a \right )}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________