\[ \int \frac {\text {PolyLog}\left (2,a x^2\right )}{x^6} \, dx \]
Optimal antiderivative \[ -\frac {4 a}{75 x^{3}}-\frac {4 a^{2}}{25 x}+\frac {4 a^{\frac {5}{2}} \arctanh \left (x \sqrt {a}\right )}{25}+\frac {2 \ln \left (-a \,x^{2}+1\right )}{25 x^{5}}-\frac {\polylog \left (2, a \,x^{2}\right )}{5 x^{5}} \]
command
integrate(polylog(2,a*x**2)/x**6,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {\pi ^{2}}{30 x^{5}} & \text {for}\: a = \frac {1}{x^{2}} \\0 & \text {for}\: a = 0 \\- \frac {12 a^{3} x^{7} \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{75 x^{7} - \frac {75 x^{5}}{a}} - \frac {6 a^{3} x^{7} \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{75 x^{7} - \frac {75 x^{5}}{a}} - \frac {12 a^{2} x^{6}}{75 x^{7} - \frac {75 x^{5}}{a}} + \frac {12 a^{2} x^{5} \sqrt {\frac {1}{a}} \log {\left (x - \sqrt {\frac {1}{a}} \right )}}{75 x^{7} - \frac {75 x^{5}}{a}} + \frac {6 a^{2} x^{5} \sqrt {\frac {1}{a}} \operatorname {Li}_{1}\left (a x^{2}\right )}{75 x^{7} - \frac {75 x^{5}}{a}} + \frac {8 a x^{4}}{75 x^{7} - \frac {75 x^{5}}{a}} - \frac {6 x^{2} \operatorname {Li}_{1}\left (a x^{2}\right )}{75 x^{7} - \frac {75 x^{5}}{a}} - \frac {15 x^{2} \operatorname {Li}_{2}\left (a x^{2}\right )}{75 x^{7} - \frac {75 x^{5}}{a}} + \frac {4 x^{2}}{75 x^{7} - \frac {75 x^{5}}{a}} + \frac {6 \operatorname {Li}_{1}\left (a x^{2}\right )}{75 a x^{7} - 75 x^{5}} + \frac {15 \operatorname {Li}_{2}\left (a x^{2}\right )}{75 a x^{7} - 75 x^{5}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________