18.10 Problem number 228

\[ \int \frac {\csc (a+b x)}{(d \cos (a+b x))^{3/2}} \, dx \]

Optimal antiderivative \[ \frac {\arctan \! \left (\frac {\sqrt {d \cos \left (b x +a \right )}}{\sqrt {d}}\right )}{b \,d^{\frac {3}{2}}}-\frac {\arctanh \! \left (\frac {\sqrt {d \cos \left (b x +a \right )}}{\sqrt {d}}\right )}{b \,d^{\frac {3}{2}}}+\frac {2}{b d \sqrt {d \cos \! \left (b x +a \right )}} \]

command

integrate(csc(b*x+a)/(d*cos(b*x+a))^(3/2),x, algorithm="giac")

Giac 1.9.0-11 via sagemath 9.6 output

\[ \text {could not integrate} \]

Giac 1.7.0 via sagemath 9.3 output

\[ \frac {\frac {2 \, \arctan \left (-\frac {\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} + \frac {\log \left ({\left | -\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} + \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} \right |}\right )}{\sqrt {-d}} - \frac {8}{\sqrt {-d} \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{2} - \sqrt {-d \tan \left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )^{4} + d} - \sqrt {-d}}}{2 \, b d} \]