Integrand size = 43, antiderivative size = 310 \[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=-\frac {8 a^4 (24 A+19 B+16 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (187 A+132 B+113 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 a (11 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (33 A+55 B+43 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{231 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {4 (891 A+946 B+769 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{3465 d \cos ^{\frac {5}{2}}(c+d x)} \]
-8/15*a^4*(24*A+19*B+16*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c) *EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/231*a^4*(187*A+132*B+113*C)*(co s(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c), 2^(1/2))/d+4/1155*a^4*(913*A+803*B+667*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/ 99*a*(11*B+8*C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/11*C*(a +a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(11/2)+2/231*(33*A+55*B+43*C)*(a^ 2+a^2*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(7/2)+4/3465*(891*A+946*B+769* C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(5/2)+8/15*a^4*(24*A+19*B+ 16*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 14.22 (sec) , antiderivative size = 1795, normalized size of antiderivative = 5.79 \[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx =\text {Too large to display} \]
Integrate[((a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sqrt[Cos[c + d*x]],x]
(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Se c[c + d*x] + C*Sec[c + d*x]^2)*(((24*A + 19*B + 16*C)*Csc[c]*Sec[c])/(15*d ) + (C*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^5*(9* C*Sin[c] + 11*B*Sin[d*x] + 44*C*Sin[d*x]))/(396*d) + (Sec[c]*Sec[c + d*x]^ 4*(77*B*Sin[c] + 308*C*Sin[c] + 99*A*Sin[d*x] + 396*B*Sin[d*x] + 675*C*Sin [d*x]))/(2772*d) + (Sec[c]*Sec[c + d*x]^3*(495*A*Sin[c] + 1980*B*Sin[c] + 3375*C*Sin[c] + 2772*A*Sin[d*x] + 4697*B*Sin[d*x] + 4928*C*Sin[d*x]))/(138 60*d) + (Sec[c]*Sec[c + d*x]*(2585*A*Sin[c] + 2640*B*Sin[c] + 2260*C*Sin[c ] + 7392*A*Sin[d*x] + 5852*B*Sin[d*x] + 4928*C*Sin[d*x]))/(4620*d) + (Sec[ c]*Sec[c + d*x]^2*(2772*A*Sin[c] + 4697*B*Sin[c] + 4928*C*Sin[c] + 7755*A* Sin[d*x] + 7920*B*Sin[d*x] + 6780*C*Sin[d*x]))/(13860*d)))/(A + 2*C + 2*B* Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (17*A*Cos[c + d*x]^6*Csc[c]*Hypergeom etricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2 ]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot [c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c] ]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[ c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, S in[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 ...
Time = 2.24 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.03, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 4600, 3042, 3522, 27, 3042, 3454, 27, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 27, 3042, 3227, 3042, 3116, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {13}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (11 B+8 C)+a (11 A+C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (11 B+8 C)+a (11 A+C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (11 B+8 C)+a (11 A+C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {(\cos (c+d x) a+a)^3 \left (3 (33 A+55 B+43 C) a^2+(99 A+11 B+17 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {(\cos (c+d x) a+a)^3 \left (3 (33 A+55 B+43 C) a^2+(99 A+11 B+17 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 (33 A+55 B+43 C) a^2+(99 A+11 B+17 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((891 A+946 B+769 C) a^3+(396 A+121 B+124 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((891 A+946 B+769 C) a^3+(396 A+121 B+124 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left (3 (913 A+803 B+667 C) a^4+(957 A+517 B+463 C) \cos (c+d x) a^4\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {(\cos (c+d x) a+a) \left (3 (913 A+803 B+667 C) a^4+(957 A+517 B+463 C) \cos (c+d x) a^4\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (3 (913 A+803 B+667 C) a^4+(957 A+517 B+463 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {(957 A+517 B+463 C) \cos ^2(c+d x) a^5+3 (913 A+803 B+667 C) a^5+\left ((957 A+517 B+463 C) a^5+3 (913 A+803 B+667 C) a^5\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \int \frac {(957 A+517 B+463 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+3 (913 A+803 B+667 C) a^5+\left ((957 A+517 B+463 C) a^5+3 (913 A+803 B+667 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {3 \left (77 (24 A+19 B+16 C) a^5+5 (187 A+132 B+113 C) \cos (c+d x) a^5\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \int \frac {77 (24 A+19 B+16 C) a^5+5 (187 A+132 B+113 C) \cos (c+d x) a^5}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \int \frac {77 (24 A+19 B+16 C) a^5+5 (187 A+132 B+113 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (77 a^5 (24 A+19 B+16 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 a^5 (187 A+132 B+113 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (77 a^5 (24 A+19 B+16 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 a^5 (187 A+132 B+113 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (187 A+132 B+113 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 a^5 (24 A+19 B+16 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (187 A+132 B+113 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 a^5 (24 A+19 B+16 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (\frac {3}{5} \left (2 \left (5 a^5 (187 A+132 B+113 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 a^5 (24 A+19 B+16 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 a^2 (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {1}{9} \left (\frac {2}{7} \left (\frac {2 (891 A+946 B+769 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {3}{5} \left (\frac {2 a^5 (913 A+803 B+667 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+2 \left (\frac {10 a^5 (187 A+132 B+113 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+77 a^5 (24 A+19 B+16 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {6 a^3 (33 A+55 B+43 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )}{11 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
(2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2 *a^2*(11*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^( 9/2)) + ((6*a^3*(33*A + 55*B + 43*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/ (7*d*Cos[c + d*x]^(7/2)) + (2*((2*(891*A + 946*B + 769*C)*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (3*((2*a^5*(913*A + 803* B + 667*C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + 2*((10*a^5*(187*A + 132* B + 113*C)*EllipticF[(c + d*x)/2, 2])/d + 77*a^5*(24*A + 19*B + 16*C)*((-2 *EllipticE[(c + d*x)/2, 2])/d + (2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])))) )/5))/7)/9)/(11*a)
3.13.16.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(1477\) vs. \(2(334)=668\).
Time = 7.51 (sec) , antiderivative size = 1478, normalized size of antiderivative = 4.77
int((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1/2),x, method=_RETURNVERBOSE)
-32*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(1/16*A* (sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2 *d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^( 1/2))+1/16*C*(-1/352*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d *x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^6-9/616*cos(1/2*d*x+1/2*c)*( -2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2- 1/2)^4-15/154*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2* c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(sin(1/2*d*x+1/2*c)^2)^(1/2 )*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1 /2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+(1/16*B+1/4*C)*(-1/1 44*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) /(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2 *c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1 /2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d *x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c )^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*Elliptic F(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/ 2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^( 1/2))))+(1/4*A+1/16*B)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.05 \[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \, {\left (30 i \, \sqrt {2} {\left (187 \, A + 132 \, B + 113 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (187 \, A + 132 \, B + 113 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 462 i \, \sqrt {2} {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 462 i \, \sqrt {2} {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (924 \, {\left (24 \, A + 19 \, B + 16 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 15 \, {\left (517 \, A + 528 \, B + 452 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 77 \, {\left (36 \, A + 61 \, B + 64 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 45 \, {\left (11 \, A + 44 \, B + 75 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 385 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 315 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d \cos \left (d x + c\right )^{6}} \]
integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1 /2),x, algorithm="fricas")
-2/3465*(30*I*sqrt(2)*(187*A + 132*B + 113*C)*a^4*cos(d*x + c)^6*weierstra ssPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 30*I*sqrt(2)*(187*A + 1 32*B + 113*C)*a^4*cos(d*x + c)^6*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 462*I*sqrt(2)*(24*A + 19*B + 16*C)*a^4*cos(d*x + c)^6*w eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 462*I*sqrt(2)*(24*A + 19*B + 16*C)*a^4*cos(d*x + c)^6*weierstrass Zeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - ( 924*(24*A + 19*B + 16*C)*a^4*cos(d*x + c)^5 + 15*(517*A + 528*B + 452*C)*a ^4*cos(d*x + c)^4 + 77*(36*A + 61*B + 64*C)*a^4*cos(d*x + c)^3 + 45*(11*A + 44*B + 75*C)*a^4*cos(d*x + c)^2 + 385*(B + 4*C)*a^4*cos(d*x + c) + 315*C *a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \]
integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1 /2),x, algorithm="maxima")
\[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \]
integrate((a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(1 /2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^4/s qrt(cos(d*x + c)), x)
Time = 23.95 (sec) , antiderivative size = 830, normalized size of antiderivative = 2.68 \[ \int \frac {(a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Too large to display} \]
(8*((11*B*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*B*a^4*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)))*hyperg eom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(21*d) - (8*((61*B*a^4*sin(c + d*x) )/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*B*a^4*sin(c + d*x))/(co s(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-1/4, 1/2], 7/4, cos( c + d*x)^2))/(135*d) + (8*((13*C*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(si n(c + d*x)^2)^(1/2)) + (5*C*a^4*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(45*d) + (8 *((75*C*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (7 *C*a^4*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)))*hypergeo m([-3/4, 1/2], 5/4, cos(c + d*x)^2))/(231*d) + (2*hypergeom([-1/4, 1/2], 3 /4, cos(c + d*x)^2)*((289*B*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (66*B*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x )^2)^(1/2)) + (5*B*a^4*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^ (1/2))))/(45*d) + (2*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((377*C*a ^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (218*C*a^4* sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)) + (21*C*a^4*sin( c + d*x))/(cos(c + d*x)^(11/2)*(sin(c + d*x)^2)^(1/2))))/(231*d) + (2*A*a^ 4*ellipticF(c/2 + (d*x)/2, 2))/d + (8*A*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2...