69.230 Problem number 1215

\[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx \]

Optimal antiderivative \[ -\frac {8 a^{4} \left (21 A +24 B +19 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {8 a^{4} \left (28 A +17 B +12 C \right ) \sqrt {\frac {\cos \left (d x +c \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) d}+\frac {2 a \left (9 B +8 C \right ) \left (a +a \cos \left (d x +c \right )\right )^{3} \sin \left (d x +c \right )}{63 d \cos \left (d x +c \right )^{\frac {7}{2}}}+\frac {2 C \left (a +a \cos \left (d x +c \right )\right )^{4} \sin \left (d x +c \right )}{9 d \cos \left (d x +c \right )^{\frac {9}{2}}}+\frac {2 \left (63 A +117 B +97 C \right ) \left (a^{2}+a^{2} \cos \left (d x +c \right )\right )^{2} \sin \left (d x +c \right )}{315 d \cos \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 \left (21 A +24 B +19 C \right ) \left (a^{4}+a^{4} \cos \left (d x +c \right )\right ) \sin \left (d x +c \right )}{45 d \cos \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 a^{4} \left (287 A +253 B +193 C \right ) \sin \left (d x +c \right )}{105 d \sqrt {\cos \left (d x +c \right )}} \]

command

integrate(cos(d*x+c)^(1/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

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

\[ -\frac {2 \, {\left (30 i \, \sqrt {2} {\left (28 \, A + 17 \, B + 12 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (28 \, A + 17 \, B + 12 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 42 i \, \sqrt {2} {\left (21 \, A + 24 \, B + 19 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 42 i \, \sqrt {2} {\left (21 \, A + 24 \, B + 19 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (21 \, {\left (99 \, A + 96 \, B + 76 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \, {\left (28 \, A + 47 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, A + 36 \, B + 61 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 45 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 35 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )^{5}} \]

Fricas 1.3.7 via sagemath 9.3 output

\[ {\rm integral}\left ({\left (C a^{4} \sec \left (d x + c\right )^{6} + {\left (B + 4 \, C\right )} a^{4} \sec \left (d x + c\right )^{5} + {\left (A + 4 \, B + 6 \, C\right )} a^{4} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, A + 3 \, B + 2 \, C\right )} a^{4} \sec \left (d x + c\right )^{3} + {\left (6 \, A + 4 \, B + C\right )} a^{4} \sec \left (d x + c\right )^{2} + {\left (4 \, A + B\right )} a^{4} \sec \left (d x + c\right ) + A a^{4}\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]