Integrand size = 43, antiderivative size = 336 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (24 A b^2+99 a b B+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (8 A b^3+15 a^3 B+54 a b^2 B+9 a^2 b (5 A+7 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (2 A b+3 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \]
2/315*a*(24*A*b^2+99*B*a*b+7*a^2*(7*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+ 2/21*(2*A*b+3*B*a)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/9*A* (a+b*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/63*(8*A*b^3+15*B*a^3+54 *B*a*b^2+9*a^2*b*(5*A+7*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(27*B*a^2*b +15*B*b^3+9*a*b^2*(3*A+5*C)+a^3*(7*A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(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)*se c(d*x+c)^(1/2)/d+2/21*(5*B*a^3+21*B*a*b^2+7*b^3*(A+3*C)+3*a^2*b*(5*A+7*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 = 7.77 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (336 \left (27 a^2 b B+15 b^3 B+9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (5 a^3 B+21 a b^2 B+7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (7 a \left (108 A b^2+108 a b B+a^2 (43 A+36 C)\right ) \cos (c+d x)+5 \left (84 A b^3+78 a^3 B+252 a b^2 B+6 a^2 (39 A b+42 b C)+18 a^2 (3 A b+a B) \cos (2 (c+d x))+7 a^3 A \cos (3 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1260 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {9}{2}}(c+d x)} \]
Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sec[c + d*x]^(9/2),x]
((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(336*(27*a ^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d* x]]*EllipticE[(c + d*x)/2, 2] + 240*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C ) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2* (7*a*(108*A*b^2 + 108*a*b*B + a^2*(43*A + 36*C))*Cos[c + d*x] + 5*(84*A*b^ 3 + 78*a^3*B + 252*a*b^2*B + 6*a^2*(39*A*b + 42*b*C) + 18*a^2*(3*A*b + a*B )*Cos[2*(c + d*x)] + 7*a^3*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*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 = 2.26 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \sec (c+d x))^2 \left (b (A+9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+3 (2 A b+3 a B)\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \sec (c+d x))^2 \left (b (A+9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+3 (2 A b+3 a B)\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (A+9 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+3 (2 A b+3 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \sec (c+d x)) \left (7 (7 A+9 C) a^2+99 b B a+24 A b^2+b (13 A b+63 C b+9 a B) \sec ^2(c+d x)+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \sec (c+d x)) \left (7 (7 A+9 C) a^2+99 b B a+24 A b^2+b (13 A b+63 C b+9 a B) \sec ^2(c+d x)+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (7 (7 A+9 C) a^2+99 b B a+24 A b^2+b (13 A b+63 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+86 A b a+126 b C a+63 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 b^2 (13 A b+63 C b+9 a B) \sec ^2(c+d x)+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \sec (c+d x)+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 (13 A b+63 C b+9 a B) \sec ^2(c+d x)+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \sec (c+d x)+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 b^2 (13 A b+63 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+21 \left ((7 A+9 C) a^3+27 b B a^2+9 b^2 (3 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 (13 A b+63 C b+9 a B) \sec ^2(c+d x)+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+21 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {5 b^2 (13 A b+63 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 (13 A b+63 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 (13 A b+63 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {5 b^2 (13 A b+63 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (15 B a^3+9 b (5 A+7 C) a^2+54 b^2 B a+8 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \sin (c+d x) \left (7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \left (15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right )}{d}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right )}{d}\right )\right )+\frac {6 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac {7}{2}}(c+d x)}\) |
(2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((6*( 2*A*b + 3*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2 )) + ((2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(5*d*Se c[c + d*x]^(3/2)) + ((42*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^ 3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d *x]])/d + (30*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C ))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (1 0*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*x])/(d *Sqrt[Sec[c + d*x]]))/5)/7)/9
3.11.2.3.1 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. \(974\) vs. \(2(360)=720\).
Time = 11.85 (sec) , antiderivative size = 975, normalized size of antiderivative = 2.90
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(975\) |
| parts | \(\text {Expression too large to display}\) | \(1128\) |
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9/2),x, method=_RETURNVERBOSE)
-2/315*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120*A*a^ 3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(2240*A*a^3+2160*A*a^2*b+720*B* a^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-2072*A*a^3-3240*A*a^2*b-151 2*A*a*b^2-1080*B*a^3-1512*B*a^2*b-504*C*a^3)*sin(1/2*d*x+1/2*c)^6*cos(1/2* d*x+1/2*c)+(952*A*a^3+2520*A*a^2*b+1512*A*a*b^2+420*A*b^3+840*B*a^3+1512*B *a^2*b+1260*B*a*b^2+504*C*a^3+1260*C*a^2*b)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d *x+1/2*c)+(-168*A*a^3-720*A*a^2*b-378*A*a*b^2-210*A*b^3-240*B*a^3-378*B*a^ 2*b-630*B*a*b^2-126*C*a^3-630*C*a^2*b)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/ 2*c)+225*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*E llipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b+105*A*(2*sin(1/2*d*x+1/2*c)^2-1 )^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)) *b^3-147*A*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*E llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3-567*A*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a *b^2+75*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*El lipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^3+315*B*(2*sin(1/2*d*x+1/2*c)^2-1)^( 1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a* b^2-567*B*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*El lipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^2*b-315*B*(2*sin(1/2*d*x+1/2*c)^2-1) ^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.15 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{3} + 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 i \, B a b^{2} + 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{3} - 3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b - 21 i \, B a b^{2} - 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{3} - 27 i \, B a^{2} b - 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} - 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{3} + 27 i \, B a^{2} b + 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2} + 15 i \, B b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, A a^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 27 \, B a^{2} b + 27 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{3} + 3 \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 21 \, B a b^{2} + 7 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9 /2),x, algorithm="fricas")
-1/315*(15*sqrt(2)*(5*I*B*a^3 + 3*I*(5*A + 7*C)*a^2*b + 21*I*B*a*b^2 + 7*I *(A + 3*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*B*a^3 - 3*I*(5*A + 7*C)*a^2*b - 21*I*B*a*b^2 - 7*I*(A + 3*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21* sqrt(2)*(-I*(7*A + 9*C)*a^3 - 27*I*B*a^2*b - 9*I*(3*A + 5*C)*a*b^2 - 15*I* B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I* sin(d*x + c))) + 21*sqrt(2)*(I*(7*A + 9*C)*a^3 + 27*I*B*a^2*b + 9*I*(3*A + 5*C)*a*b^2 + 15*I*B*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , cos(d*x + c) - I*sin(d*x + c))) - 2*(35*A*a^3*cos(d*x + c)^4 + 45*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^3 + 7*((7*A + 9*C)*a^3 + 27*B*a^2*b + 27*A*a*b^ 2)*cos(d*x + c)^2 + 15*(5*B*a^3 + 3*(5*A + 7*C)*a^2*b + 21*B*a*b^2 + 7*A*b ^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9 /2),x, algorithm="maxima")
\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(9 /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)^(9/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \]
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))^(9/2),x)