28.6 Problem number 734

\[ \int \frac {\sec (3 x) \tan (3 x)}{\sqrt {1+5 \cos ^2(3 x)}} \, dx \]

Optimal antiderivative \[ \frac {\sec \! \left (3 x \right ) \sqrt {1+5 \left (\cos ^{2}\left (3 x \right )\right )}}{3} \]

command

integrate(sec(3*x)*tan(3*x)/(1+5*cos(3*x)^2)^(1/2),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

\[ -\frac {2 \, \sqrt {5}}{3 \, {\left ({\left (\sqrt {5} \cos \left (3 \, x\right ) - \sqrt {5 \, \cos \left (3 \, x\right )^{2} + 1}\right )}^{2} - 1\right )}} \]