Integrand size = 43, antiderivative size = 401 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a \left (24 A b^2+143 a b B+9 a^2 (9 A+11 C)\right ) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (24 A b^3+77 a^3 B+242 a b^2 B+33 a^2 b (7 A+9 C)\right ) \sin (c+d x)}{495 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (6 A b+11 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
2/693*a*(24*A*b^2+143*B*a*b+9*a^2*(9*A+11*C))*sin(d*x+c)/d/sec(d*x+c)^(5/2 )+2/495*(24*A*b^3+77*B*a^3+242*B*a*b^2+33*a^2*b*(7*A+9*C))*sin(d*x+c)/d/se c(d*x+c)^(3/2)+2/99*(6*A*b+11*B*a)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x +c)^(7/2)+2/11*A*(a+b*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/231*(1 65*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)/d/sec( d*x+c)^(1/2)+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))*cos(d*x+c)^(1/2)*sec(d*x+c)^(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))*cos(d*x+c)^(1/2)*sec(d*x +c)^(1/2)/d
Time = 15.05 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (7392 \left (7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)+3 a^2 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+480 \left (165 a^2 b B+77 b^3 B+33 a b^2 (5 A+7 C)+5 a^3 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (154 \left (36 A b^3+43 a^3 B+108 a b^2 B+3 a^2 b (43 A+36 C)\right ) \cos (c+d x)+5 \left (36 a \left (33 A b^2+33 a b B+a^2 (16 A+11 C)\right ) \cos (2 (c+d x))+154 a^2 (3 A b+a B) \cos (3 (c+d x))+3 \left (1716 a^2 b B+616 b^3 B+132 a b^2 (13 A+14 C)+a^3 (531 A+572 C)+21 a^3 A \cos (4 (c+d x))\right )\right )\right ) \sin (2 (c+d x))\right )}{27720 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {9}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sec[c + d*x]^(11/2),x]
((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(7392*(7*a ^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*( 5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*(36*A*b^3 + 43*a^3*B + 108*a*b^2*B + 3*a^2*b*(43*A + 36*C))*C os[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[2*(c + d*x)]))/(27720*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B *Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))
Time = 2.61 (sec) , antiderivative size = 380, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4562, 27, 3042, 4535, 3042, 4256, 3042, 4258, 3042, 3120, 4533, 3042, 4258, 3042, 3119}
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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {2}{11} \int \frac {(a+b \sec (c+d x))^2 \left (b (3 A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+6 A b+11 a B\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \sec (c+d x))^2 \left (b (3 A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+6 A b+11 a B\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (3 A+11 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+6 A b+11 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \sec (c+d x)) \left (9 (9 A+11 C) a^2+143 b B a+24 A b^2+3 b (15 A b+33 C b+11 a B) \sec ^2(c+d x)+\left (77 B a^2+150 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \sec (c+d x)) \left (9 (9 A+11 C) a^2+143 b B a+24 A b^2+3 b (15 A b+33 C b+11 a B) \sec ^2(c+d x)+\left (77 B a^2+150 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (9 (9 A+11 C) a^2+143 b B a+24 A b^2+3 b (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+150 A b a+198 b C a+99 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int -\frac {21 b^2 (15 A b+33 C b+11 a B) \sec ^2(c+d x)+9 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \sec (c+d x)+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 (15 A b+33 C b+11 a B) \sec ^2(c+d x)+9 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \sec (c+d x)+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 \left (5 (9 A+11 C) a^3+165 b B a^2+33 b^2 (5 A+7 C) a+77 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+\int \frac {21 b^2 (15 A b+33 C b+11 a B) \sec ^2(c+d x)+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 (15 A b+33 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 B a^3+33 b (7 A+9 C) a^2+242 b^2 B a+24 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {14 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \sqrt {\cos (c+d x)}dx+\frac {14 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {14 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2 a \sin (c+d x) \left (9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (\frac {14 \sin (c+d x) \left (77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right )}{5 d}+9 \left (5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )\right )+\frac {2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
(2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + ((2* (6*A*b + 11*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7 /2)) + ((2*a*(24*A*b^2 + 143*a*b*B + 9*a^2*(9*A + 11*C))*Sin[c + d*x])/(7* d*Sec[c + d*x]^(5/2)) + ((462*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec [c + d*x]])/(5*d) + (14*(24*A*b^3 + 77*a^3*B + 242*a*b^2*B + 33*a^2*b*(7*A + 9*C))*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + 9*(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*Sqrt[Cos[c + d*x]]*Elli pticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sin[c + d*x])/(3*d*Sq rt[Sec[c + d*x]])))/7)/9)/11
3.11.3.3.1 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[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[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n) Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1081\) vs. \(2(421)=842\).
Time = 17.37 (sec) , antiderivative size = 1082, normalized size of antiderivative = 2.70
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1082\) |
| parts | \(\text {Expression too large to display}\) | \(1220\) |
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x ,method=_RETURNVERBOSE)
-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*A*a ^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-50400*A*a^3-36960*A*a^2*b-12 320*B*a^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*A*a^3+73920*A*a ^2*b+23760*A*a*b^2+24640*B*a^3+23760*B*a^2*b+7920*C*a^3)*sin(1/2*d*x+1/2*c )^8*cos(1/2*d*x+1/2*c)+(-34920*A*a^3-68376*A*a^2*b-35640*A*a*b^2-5544*A*b^ 3-22792*B*a^3-35640*B*a^2*b-16632*B*a*b^2-11880*C*a^3-16632*C*a^2*b)*sin(1 /2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a^3+31416*A*a^2*b+27720*A*a*b^ 2+5544*A*b^3+10472*B*a^3+27720*B*a^2*b+16632*B*a*b^2+4620*B*b^3+9240*C*a^3 +16632*C*a^2*b+13860*C*a*b^2)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-27 90*A*a^3-5544*A*a^2*b-7920*A*a*b^2-1386*A*b^3-1848*B*a^3-7920*B*a^2*b-4158 *B*a*b^2-2310*B*b^3-2640*C*a^3-4158*C*a^2*b-6930*C*a*b^2)*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)+675*A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2* d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+2475*a*A*b^2*( sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos (1/2*d*x+1/2*c),2^(1/2))-4851*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d* x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b-2079*A*(si n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1 /2*d*x+1/2*c),2^(1/2))*b^3+2475*B*a^2*b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*si n(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+1155*B*b ^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellipt...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.10 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {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 ) - \frac {2 \, {\left (315 \, A a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 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 )^{3} + 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 )^{2} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{3} + 165 \, B a^{2} b + 33 \, {\left (5 \, A + 7 \, C\right )} a b^{2} + 77 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 1/2),x, algorithm="fricas")
-1/3465*(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, weierstrassPInverse(-4, 0, co s(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, weierstr assPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(315*A*a^3*cos(d*x + c)^5 + 385*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^4 + 45*((9*A + 11*C)*a^3 + 3 3*B*a^2*b + 33*A*a*b^2)*cos(d*x + c)^3 + 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)^2 + 15*(5*(9*A + 11*C)*a^3 + 165*B*a ^2*b + 33*(5*A + 7*C)*a*b^2 + 77*B*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(co s(d*x + c)))/d
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 1/2),x, algorithm="maxima")
\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 1/2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3/s ec(d*x + c)^(11/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
int(((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(11/2),x)