Integrand size = 43, antiderivative size = 361 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{495 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 (6 A b+11 a B) \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^2 \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))^3 \sin (c+d x)}{11 d} \]
2/15*(7*B*a^3+27*B*a*b^2+3*b^3*(3*A+5*C)+3*a^2*b*(7*A+9*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+ 2/231*(165*B*a^2*b+77*B*b^3+33*a*b^2*(5*A+7*C)+5*a^3*(9*A+11*C))*(cos(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+2/495*(24*A*b^3+77*B*a^3+242*B*a*b^2+33*a^2*b*(7*A+9*C))*cos(d*x+c)^( 3/2)*sin(d*x+c)/d+2/693*a*(24*A*b^2+143*B*a*b+9*a^2*(9*A+11*C))*cos(d*x+c) ^(5/2)*sin(d*x+c)/d+2/99*(6*A*b+11*B*a)*cos(d*x+c)^(3/2)*(b+a*cos(d*x+c))^ 2*sin(d*x+c)/d+2/11*A*cos(d*x+c)^(3/2)*(b+a*cos(d*x+c))^3*sin(d*x+c)/d+2/2 31*(165*B*a^2*b+77*B*b^3+33*a*b^2*(5*A+7*C)+5*a^3*(9*A+11*C))*sin(d*x+c)*c os(d*x+c)^(1/2)/d
Time = 4.43 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.79 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {154 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (154 \left (36 A b^3+43 a^3 B+108 a b^2 B+3 a^2 b (43 A+36 C)\right ) \cos (c+d x)+5 \left (36 a \left (33 A b^2+33 a b B+a^2 (16 A+11 C)\right ) \cos (2 (c+d x))+154 a^2 (3 A b+a B) \cos (3 (c+d x))+3 \left (1716 a^2 b B+616 b^3 B+132 a b^2 (13 A+14 C)+a^3 (531 A+572 C)+21 a^3 A \cos (4 (c+d x))\right )\right )\right ) \sin (c+d x)}{1155 d} \]
Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(154*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*Elli pticE[(c + d*x)/2, 2] + 10*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154 *(36*A*b^3 + 43*a^3*B + 108*a*b^2*B + 3*a^2*b*(43*A + 36*C))*Cos[c + d*x] + 5*(36*a*(33*A*b^2 + 33*a*b*B + a^2*(16*A + 11*C))*Cos[2*(c + d*x)] + 154 *a^2*(3*A*b + a*B)*Cos[3*(c + d*x)] + 3*(1716*a^2*b*B + 616*b^3*B + 132*a* b^2*(13*A + 14*C) + a^3*(531*A + 572*C) + 21*a^3*A*Cos[4*(c + d*x)])))*Sin [c + d*x])/12)/(1155*d)
Time = 2.04 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.95, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4600, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{11/2} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^3 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+b\right )^3 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \left ((6 A b+11 a B) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+b (3 A+11 C)\right )dx+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2 \left ((6 A b+11 a B) \cos ^2(c+d x)+(9 a A+11 b B+11 a C) \cos (c+d x)+b (3 A+11 C)\right )dx+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((6 A b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+b (3 A+11 C)\right )dx+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \left (\left (9 (9 A+11 C) a^2+143 b B a+24 A b^2\right ) \cos ^2(c+d x)+\left (77 B a^2+150 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 b (15 A b+33 C b+11 a B)\right )dx+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sqrt {\cos (c+d x)} (b+a \cos (c+d x)) \left (\left (9 (9 A+11 C) a^2+143 b B a+24 A b^2\right ) \cos ^2(c+d x)+\left (77 B a^2+150 A b a+198 b C a+99 b^2 B\right ) \cos (c+d x)+3 b (15 A b+33 C b+11 a B)\right )dx+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (9 (9 A+11 C) a^2+143 b B a+24 A b^2\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+150 A b a+198 b C a+99 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 b (15 A b+33 C b+11 a B)\right )dx+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 (15 A b+33 C b+11 a B) b^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right ) \cos ^2(c+d x)+9 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \cos (c+d x)\right )dx+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 (15 A b+33 C b+11 a B) b^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right ) \cos ^2(c+d x)+9 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \cos (c+d x)\right )dx+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 (15 A b+33 C b+11 a B) b^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (77 \left (7 B a^3+3 b (7 A+9 C) a^2+27 b^2 B a+3 b^3 (3 A+5 C)\right )+15 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (77 \left (7 B a^3+3 b (7 A+9 C) a^2+27 b^2 B a+3 b^3 (3 A+5 C)\right )+15 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 \left (7 B a^3+3 b (7 A+9 C) a^2+27 b^2 B a+3 b^3 (3 A+5 C)\right )+15 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (15 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{d}\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}\right )+\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 a \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d}+\frac {1}{7} \left (\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d}+\frac {3}{5} \left (\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{d}+15 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}\right )+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}\) |
(2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + ((2* (6*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/( 9*d) + ((2*a*(24*A*b^2 + 143*a*b*B + 9*a^2*(9*A + 11*C))*Cos[c + d*x]^(5/2 )*Sin[c + d*x])/(7*d) + ((14*(24*A*b^3 + 77*a^3*B + 242*a*b^2*B + 33*a^2*b *(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((154*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*EllipticE[(c + d*x) /2, 2])/d + 15*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin [c + d*x])/(3*d))))/5)/7)/9)/11
3.14.5.3.1 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[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
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)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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]
Timed out.
hanged
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.19 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \, {\left (315 \, A a^{3} \cos \left (d x + c\right )^{4} + 75 \, {\left (9 \, A + 11 \, C\right )} a^{3} + 2475 \, B a^{2} b + 495 \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 1155 \, B b^{3} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{3} + 33 \, B a^{2} b + 33 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (7 \, B a^{3} + 3 \, {\left (7 \, A + 9 \, C\right )} a^{2} b + 27 \, B a b^{2} + 9 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (5 i \, {\left (9 \, A + 11 \, C\right )} a^{3} + 165 i \, B a^{2} b + 33 i \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 77 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 \, \sqrt {2} {\left (-5 i \, {\left (9 \, A + 11 \, C\right )} a^{3} - 165 i \, B a^{2} b - 33 i \, {\left (5 \, A + 7 \, C\right )} a b^{2} - 77 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-7 i \, B a^{3} - 3 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b - 27 i \, B a b^{2} - 3 i \, {\left (3 \, A + 5 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 \, \sqrt {2} {\left (7 i \, B a^{3} + 3 i \, {\left (7 \, A + 9 \, C\right )} a^{2} b + 27 i \, B a b^{2} + 3 i \, {\left (3 \, A + 5 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{3465 \, d} \]
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="fricas")
1/3465*(2*(315*A*a^3*cos(d*x + c)^4 + 75*(9*A + 11*C)*a^3 + 2475*B*a^2*b + 495*(5*A + 7*C)*a*b^2 + 1155*B*b^3 + 385*(B*a^3 + 3*A*a^2*b)*cos(d*x + c) ^3 + 45*((9*A + 11*C)*a^3 + 33*B*a^2*b + 33*A*a*b^2)*cos(d*x + c)^2 + 77*( 7*B*a^3 + 3*(7*A + 9*C)*a^2*b + 27*B*a*b^2 + 9*A*b^3)*cos(d*x + c))*sqrt(c os(d*x + c))*sin(d*x + c) - 15*sqrt(2)*(5*I*(9*A + 11*C)*a^3 + 165*I*B*a^2 *b + 33*I*(5*A + 7*C)*a*b^2 + 77*I*B*b^3)*weierstrassPInverse(-4, 0, cos(d *x + c) + I*sin(d*x + c)) - 15*sqrt(2)*(-5*I*(9*A + 11*C)*a^3 - 165*I*B*a^ 2*b - 33*I*(5*A + 7*C)*a*b^2 - 77*I*B*b^3)*weierstrassPInverse(-4, 0, cos( d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-7*I*B*a^3 - 3*I*(7*A + 9*C)*a^2 *b - 27*I*B*a*b^2 - 3*I*(3*A + 5*C)*b^3)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2)*(7*I*B*a^3 + 3*I*(7*A + 9*C)*a^2*b + 27*I*B*a*b^2 + 3*I*(3*A + 5*C)*b^3)*weierstrassZ eta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))))/d
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="maxima")
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \]
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c )^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3*c os(d*x + c)^(11/2), x)
Time = 20.68 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.42 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (C\,b^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a\,b^2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a\,b^2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {B\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a^2\,b\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,B\,a\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^2\,b\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*(C*b^3*ellipticE(c/2 + (d*x)/2, 2) + C*a*b^2*ellipticF(c/2 + (d*x)/2, 2 ) + C*a*b^2*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (B*b^3*((2*cos(c + d*x)^ (1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*a^3*c os(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^ 2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (2*A*b^3*cos(c + d*x)^(7/2)*sin(c + d* x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2 )) - (2*B*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4 , cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^( 9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a*b^2*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([ 1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (6*A*a^2* b*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d* x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (6*B*a*b^2*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2) ^(1/2)) - (2*B*a^2*b*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (6*C*a^2*b*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d* (sin(c + d*x)^2)^(1/2))