40.90 Problem number 293

\[ \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{5/2}} \, dx \]

Optimal antiderivative \[ \frac {4 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d e \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {2 a^{2} \arcsinh \left (\frac {\sqrt {e \cos \left (d x +c \right )}}{\sqrt {e}}\right ) \sqrt {1+\cos \left (d x +c \right )}\, \sqrt {a +a \sin \left (d x +c \right )}}{d \,e^{\frac {5}{2}} \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )}-\frac {2 a^{2} \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {e}}{\sqrt {e \cos \left (d x +c \right )}\, \sqrt {1+\cos \left (d x +c \right )}}\right ) \sqrt {1+\cos \left (d x +c \right )}\, \sqrt {a +a \sin \left (d x +c \right )}}{d \,e^{\frac {5}{2}} \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right )} \]

command

integrate((a+a*sin(d*x+c))^(5/2)/(e*cos(d*x+c))^(5/2),x, algorithm="fricas")

Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output

\[ \text {output too large to display} \]

Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]