\[ \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^4} \, dx \]
Optimal antiderivative \[ -\frac {b c}{15 x^{\frac {5}{2}}}-\frac {b \,c^{3}}{9 x^{\frac {3}{2}}}+\frac {b \,c^{6} \arctanh \left (c \sqrt {x}\right )}{3}+\frac {-a -b \arctanh \left (c \sqrt {x}\right )}{3 x^{3}}-\frac {b \,c^{5}}{3 \sqrt {x}} \]
command
integrate((a+b*atanh(c*x**(1/2)))/x**4,x)
Sympy 1.10.1 under Python 3.10.4 output
\[ \begin {cases} - \frac {a}{3 x^{3}} + \frac {b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{3 x^{3}} & \text {for}\: c = - \sqrt {\frac {1}{x}} \\- \frac {a}{3 x^{3}} - \frac {b \operatorname {atanh}{\left (\sqrt {x} \sqrt {\frac {1}{x}} \right )}}{3 x^{3}} & \text {for}\: c = \sqrt {\frac {1}{x}} \\- \frac {15 a c^{2} x^{\frac {3}{2}}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {15 a \sqrt {x}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {15 b c^{8} x^{\frac {9}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} - \frac {15 b c^{7} x^{4}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} - \frac {15 b c^{6} x^{\frac {7}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {10 b c^{5} x^{3}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {2 b c^{3} x^{2}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} - \frac {15 b c^{2} x^{\frac {3}{2}} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {3 b c x}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} + \frac {15 b \sqrt {x} \operatorname {atanh}{\left (c \sqrt {x} \right )}}{45 c^{2} x^{\frac {9}{2}} - 45 x^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Sympy 1.8 under Python 3.8.8 output
\[ \text {Timed out} \]________________________________________________________________________________________