\[ \int \frac {\tan (e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx \]
Optimal antiderivative \[ \frac {b^{2}}{4 a^{3} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {b}{a^{3} f \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}-\frac {\ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 a^{3} f} \]
command
integrate(tan(f*x+e)/(a+b*sec(f*x+e)**2)**3,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} \frac {\tilde {\infty } x \tan {\left (e \right )}}{\sec ^{6}{\left (e \right )}} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 a^{3} f} & \text {for}\: b = 0 \\- \frac {1}{6 b^{3} f \sec ^{6}{\left (e + f x \right )}} & \text {for}\: a = 0 \\\frac {x \tan {\left (e \right )}}{\left (a + b \sec ^{2}{\left (e \right )}\right )^{3}} & \text {for}\: f = 0 \\- \frac {2 a^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 a^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {2 a^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {3 a^{2}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {4 a b \log {\left (- \sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {4 a b \log {\left (\sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {4 a b \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {2 a b \sec ^{2}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 b^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} - \frac {2 b^{2} \log {\left (\sqrt {- \frac {a}{b}} + \sec {\left (e + f x \right )} \right )} \sec ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} + \frac {2 b^{2} \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \sec ^{4}{\left (e + f x \right )}}{4 a^{5} f + 8 a^{4} b f \sec ^{2}{\left (e + f x \right )} + 4 a^{3} b^{2} f \sec ^{4}{\left (e + f x \right )}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________