Integrand size = 43, antiderivative size = 444 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
2/3465*a*(192*A*b^3+539*B*a^3+1353*B*a*b^2+2*a^2*b*(673*A+891*C))*sin(d*x+ c)/d/sec(d*x+c)^(3/2)+2/231*(16*A*b^2+55*B*a*b+3*a^2*(9*A+11*C))*(a+b*sec( d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/99*(8*A*b+11*B*a)*(a+b*sec(d*x+c ))^3*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/11*A*(a+b*sec(d*x+c))^4*sin(d*x+c)/d/ sec(d*x+c)^(9/2)+2/693*(64*A*b^4+660*B*a^3*b+682*B*a*b^3+15*a^4*(9*A+11*C) +9*a^2*b^2*(101*A+143*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(7*B*a^4+54*B *a^2*b^2+15*B*b^4+12*a*b^3*(3*A+5*C)+4*a^3*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*(220*B*a^3*b+308*B*a*b^3+77*b^4*(A+3*C )+66*a^2*b^2*(5*A+7*C)+5*a^4*(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 = 11.67 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (7392 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+480 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (154 a \left (144 A b^3+43 a^3 B+216 a b^2 B+4 a^2 b (43 A+36 C)\right ) \cos (c+d x)+5 \left (36 a^2 \left (66 A b^2+44 a b B+a^2 (16 A+11 C)\right ) \cos (2 (c+d x))+154 a^3 (4 A b+a B) \cos (3 (c+d x))+3 \left (616 A b^4+2288 a^3 b B+2464 a b^3 B+264 a^2 b^2 (13 A+14 C)+a^4 (531 A+572 C)+21 a^4 A \cos (4 (c+d x))\right )\right )\right ) \sin (2 (c+d x))\right )}{27720 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {11}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^4*(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])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(7392*(7*a ^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C ))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(220*a^3*b*B + 308*a *b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*S qrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(154*a*(144*A*b^3 + 43*a^3 *B + 216*a*b^2*B + 4*a^2*b*(43*A + 36*C))*Cos[c + d*x] + 5*(36*a^2*(66*A*b ^2 + 44*a*b*B + a^2*(16*A + 11*C))*Cos[2*(c + d*x)] + 154*a^3*(4*A*b + a*B )*Cos[3*(c + d*x)] + 3*(616*A*b^4 + 2288*a^3*b*B + 2464*a*b^3*B + 264*a^2* b^2*(13*A + 14*C) + a^4*(531*A + 572*C) + 21*a^4*A*Cos[4*(c + d*x)])))*Sin [2*(c + d*x)]))/(27720*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x ] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(11/2))
Time = 3.27 (sec) , antiderivative size = 458, normalized size of antiderivative = 1.03, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4562, 27, 3042, 4535, 3042, 4258, 3042, 3119, 4533, 3042, 4258, 3042, 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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {2}{11} \int \frac {(a+b \sec (c+d x))^3 \left (b (A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+8 A b+11 a B\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \sec (c+d x))^3 \left (b (A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+8 A b+11 a B\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (A+11 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+8 A b+11 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \sec (c+d x))^2 \left (b (17 A b+99 C b+11 a B) \sec ^2(c+d x)+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \sec (c+d x))^2 \left (b (17 A b+99 C b+11 a B) \sec ^2(c+d x)+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (17 A b+99 C b+11 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+146 A b a+198 b C a+99 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+55 b B a+16 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \sec (c+d x)) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \sec (c+d x)) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (539 B a^3+2 b (673 A+891 C) a^2+1353 b^2 B a+192 A b^3+b \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 (9 A+11 C) a^3+1441 b B a^2+b^2 (1381 A+2079 C) a+693 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \sec (c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \sec (c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+231 \left (7 B a^4+4 b (7 A+9 C) a^3+54 b^2 B a^2+12 b^3 (3 A+5 C) a+15 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+231 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (231 \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+231 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+231 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 \left (9 (9 A+11 C) a^2+242 b B a+b^2 (167 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 (9 A+11 C) a^4+660 b B a^3+9 b^2 (101 A+143 C) a^2+682 b^3 B a+64 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \sqrt {\sec (c+d x)}}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \sqrt {\sec (c+d x)}}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \sqrt {\sec (c+d x)}}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \sqrt {\sec (c+d x)}}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{d}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{d}\right )\right )\right )+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
(2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + ((2* (8*A*b + 11*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7 /2)) + ((6*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Sec[c + d*x]) ^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sin[c + d*x])/(5*d*Sec[c + d*x]^( 3/2)) + ((462*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec [c + d*x]])/d + (30*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2 *b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d *x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3* B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(d*Sqrt [Sec[c + d*x]]))/5)/7)/9)/11
3.11.10.3.1 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[(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. \(1272\) vs. \(2(464)=928\).
Time = 17.39 (sec) , antiderivative size = 1273, normalized size of antiderivative = 2.87
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1273\) |
| parts | \(\text {Expression too large to display}\) | \(1380\) |
int((a+b*sec(d*x+c))^4*(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 ^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-50400*A*a^4-49280*A*a^3*b-12 320*B*a^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*A*a^4+98560*A*a ^3*b+47520*A*a^2*b^2+24640*B*a^4+31680*B*a^3*b+7920*C*a^4)*sin(1/2*d*x+1/2 *c)^8*cos(1/2*d*x+1/2*c)+(-34920*A*a^4-91168*A*a^3*b-71280*A*a^2*b^2-22176 *A*a*b^3-22792*B*a^4-47520*B*a^3*b-33264*B*a^2*b^2-11880*C*a^4-22176*C*a^3 *b)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(13860*A*a^4+41888*A*a^3*b+554 40*A*a^2*b^2+22176*A*a*b^3+4620*A*b^4+10472*B*a^4+36960*B*a^3*b+33264*B*a^ 2*b^2+18480*B*a*b^3+9240*C*a^4+22176*C*a^3*b+27720*C*a^2*b^2)*sin(1/2*d*x+ 1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2790*A*a^4-7392*A*a^3*b-15840*A*a^2*b^2-5544 *A*a*b^3-2310*A*b^4-1848*B*a^4-10560*B*a^3*b-8316*B*a^2*b^2-9240*B*a*b^3-2 640*C*a^4-5544*C*a^3*b-13860*C*a^2*b^2)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1 /2*c)+675*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)* EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^4+4950*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)) *a^2*b^2+1155*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^4-6468*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^3*b-8316*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*b^3+3300*B*(sin(1/2*d*x+1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \sec (c+d x))^4 \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^{4} + 220 i \, B a^{3} b + 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 i \, B a b^{3} + 77 i \, {\left (A + 3 \, C\right )} b^{4}\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^{4} - 220 i \, B a^{3} b - 66 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} - 308 i \, B a b^{3} - 77 i \, {\left (A + 3 \, C\right )} b^{4}\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^{4} - 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} - 15 i \, B b^{4}\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^{4} + 4 i \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 12 i \, {\left (3 \, A + 5 \, C\right )} a b^{3} + 15 i \, B b^{4}\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^{4} \cos \left (d x + c\right )^{5} + 385 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left ({\left (9 \, A + 11 \, C\right )} a^{4} + 44 \, B a^{3} b + 66 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, B a^{4} + 4 \, {\left (7 \, A + 9 \, C\right )} a^{3} b + 54 \, B a^{2} b^{2} + 36 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, {\left (9 \, A + 11 \, C\right )} a^{4} + 220 \, B a^{3} b + 66 \, {\left (5 \, A + 7 \, C\right )} a^{2} b^{2} + 308 \, B a b^{3} + 77 \, A b^{4}\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))^4*(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^4 + 220*I*B*a^3*b + 66*I*(5*A + 7* C)*a^2*b^2 + 308*I*B*a*b^3 + 77*I*(A + 3*C)*b^4)*weierstrassPInverse(-4, 0 , cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*(9*A + 11*C)*a^4 - 220 *I*B*a^3*b - 66*I*(5*A + 7*C)*a^2*b^2 - 308*I*B*a*b^3 - 77*I*(A + 3*C)*b^4 )*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)* (-7*I*B*a^4 - 4*I*(7*A + 9*C)*a^3*b - 54*I*B*a^2*b^2 - 12*I*(3*A + 5*C)*a* b^3 - 15*I*B*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d* x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(7*I*B*a^4 + 4*I*(7*A + 9*C)*a^3*b + 54*I*B*a^2*b^2 + 12*I*(3*A + 5*C)*a*b^3 + 15*I*B*b^4)*weierstrassZeta(- 4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(315* A*a^4*cos(d*x + c)^5 + 385*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^4 + 45*((9*A + 11*C)*a^4 + 44*B*a^3*b + 66*A*a^2*b^2)*cos(d*x + c)^3 + 77*(7*B*a^4 + 4*( 7*A + 9*C)*a^3*b + 54*B*a^2*b^2 + 36*A*a*b^3)*cos(d*x + c)^2 + 15*(5*(9*A + 11*C)*a^4 + 220*B*a^3*b + 66*(5*A + 7*C)*a^2*b^2 + 308*B*a*b^3 + 77*A*b^ 4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
integrate((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 1/2),x, algorithm="maxima")
\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^4*(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)^4/s ec(d*x + c)^(11/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
int(((a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(11/2),x)