Integrand size = 43, antiderivative size = 291 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {4 a^2 (7 A+8 B+9 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {4 a^2 (50 A+55 B+66 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a^2 (89 A+121 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2 (7 A+8 B+9 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2 (50 A+55 B+66 C) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+a \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 (4 A+11 B) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)} \]
2/693*a^2*(89*A+121*B+99*C)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+4/45*a^2*(7*A+8* B+9*C)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/11*A*(a+a*sec(d*x+c))^2*sin(d*x+c)/ d/sec(d*x+c)^(9/2)+2/99*(4*A+11*B)*(a^2+a^2*sec(d*x+c))*sin(d*x+c)/d/sec(d *x+c)^(7/2)+4/231*a^2*(50*A+55*B+66*C)*sin(d*x+c)/d/sec(d*x+c)^(1/2)+4/15* a^2*(7*A+8*B+9*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elliptic E(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+4/231*a^ 2*(50*A+55*B+66*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellipti cF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 5.92 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.93 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {a^2 e^{-i d x} \sqrt {\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left (480 (50 A+55 B+66 C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-2464 i (7 A+8 B+9 C) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+\cos (c+d x) (51744 i A+59136 i B+66528 i C+30 (941 A+22 (46 B+51 C)) \sin (c+d x)+308 (38 A+37 B+36 C) \sin (2 (c+d x))+4545 A \sin (3 (c+d x))+3960 B \sin (3 (c+d x))+1980 C \sin (3 (c+d x))+1540 A \sin (4 (c+d x))+770 B \sin (4 (c+d x))+315 A \sin (5 (c+d x)))\right )}{27720 d} \]
Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) /Sec[c + d*x]^(11/2),x]
(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(480*(50*A + 55*B + 66*C)* Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(7*A + 8*B + 9*C)* E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((51744*I)*A + (59136*I)*B + (66 528*I)*C + 30*(941*A + 22*(46*B + 51*C))*Sin[c + d*x] + 308*(38*A + 37*B + 36*C)*Sin[2*(c + d*x)] + 4545*A*Sin[3*(c + d*x)] + 3960*B*Sin[3*(c + d*x) ] + 1980*C*Sin[3*(c + d*x)] + 1540*A*Sin[4*(c + d*x)] + 770*B*Sin[4*(c + d *x)] + 315*A*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))
Time = 1.53 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.99, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4574, 27, 3042, 4505, 27, 3042, 4484, 25, 3042, 4274, 3042, 4256, 3042, 4258, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(a \sec (c+d x)+a)^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {2 \int \frac {(\sec (c+d x) a+a)^2 (a (4 A+11 B)+a (5 A+11 C) \sec (c+d x))}{2 \sec ^{\frac {9}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\sec (c+d x) a+a)^2 (a (4 A+11 B)+a (5 A+11 C) \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (4 A+11 B)+a (5 A+11 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {(\sec (c+d x) a+a) \left ((89 A+121 B+99 C) a^2+(65 A+55 B+99 C) \sec (c+d x) a^2\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {(\sec (c+d x) a+a) \left ((89 A+121 B+99 C) a^2+(65 A+55 B+99 C) \sec (c+d x) a^2\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((89 A+121 B+99 C) a^2+(65 A+55 B+99 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int -\frac {77 (7 A+8 B+9 C) a^3+9 (50 A+55 B+66 C) \sec (c+d x) a^3}{\sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {77 (7 A+8 B+9 C) a^3+9 (50 A+55 B+66 C) \sec (c+d x) a^3}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {77 (7 A+8 B+9 C) a^3+9 (50 A+55 B+66 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+8 B+9 C) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)}dx+9 a^3 (50 A+55 B+66 C) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+8 B+9 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+9 a^3 (50 A+55 B+66 C) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+8 B+9 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (50 A+55 B+66 C) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+8 B+9 C) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (50 A+55 B+66 C) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+8 B+9 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (50 A+55 B+66 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+8 B+9 C) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+9 a^3 (50 A+55 B+66 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (9 a^3 (50 A+55 B+66 C) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+77 a^3 (7 A+8 B+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2 a^3 (89 A+121 B+99 C) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2}{7} \left (9 a^3 (50 A+55 B+66 C) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+77 a^3 (7 A+8 B+9 C) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )\right )+\frac {2 (4 A+11 B) \sin (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{9 d \sec ^{\frac {7}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
(2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + ((2* (4*A + 11*B)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2 )) + ((2*a^3*(89*A + 121*B + 99*C)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(77*a^3*(7*A + 8*B + 9*C)*((6*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/ 2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2) )) + 9*a^3*(50*A + 55*B + 66*C)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x) /2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x] ]))))/7)/9)/(11*a)
3.6.48.3.1 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_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && 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/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 11.75 (sec) , antiderivative size = 545, normalized size of antiderivative = 1.87
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{2} \left (10080 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-37520 A -6160 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (57040 A +20240 B +3960 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-46192 A -26048 B -11484 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (22022 A +17248 B +12474 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-4563 A -4257 B -3861 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+750 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1617 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+825 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1848 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+990 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-2079 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(545\) |
| parts | \(\text {Expression too large to display}\) | \(1063\) |
int((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/2),x ,method=_RETURNVERBOSE)
-4/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^2*(10080 *A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-37520*A-6160*B)*sin(1/2*d*x+ 1/2*c)^10*cos(1/2*d*x+1/2*c)+(57040*A+20240*B+3960*C)*sin(1/2*d*x+1/2*c)^8 *cos(1/2*d*x+1/2*c)+(-46192*A-26048*B-11484*C)*sin(1/2*d*x+1/2*c)^6*cos(1/ 2*d*x+1/2*c)+(22022*A+17248*B+12474*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)+(-4563*A-4257*B-3861*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+750*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(c os(1/2*d*x+1/2*c),2^(1/2))-1617*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))+825*B*(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))-1848*B*(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))+990*C*(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))-2079*C*(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)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/ 2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/ d
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.93 \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (50 \, A + 55 \, B + 66 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (50 \, A + 55 \, B + 66 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (7 \, A + 8 \, B + 9 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 8 \, B + 9 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {{\left (315 \, A a^{2} \cos \left (d x + c\right )^{5} + 385 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{4} + 45 \, {\left (20 \, A + 22 \, B + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 154 \, {\left (7 \, A + 8 \, B + 9 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 30 \, {\left (50 \, A + 55 \, B + 66 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}\right )}}{3465 \, d} \]
integrate((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 1/2),x, algorithm="fricas")
-2/3465*(15*I*sqrt(2)*(50*A + 55*B + 66*C)*a^2*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(50*A + 55*B + 66*C)*a^2*wei erstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt(2)*(7* A + 8*B + 9*C)*a^2*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d *x + c) + I*sin(d*x + c))) + 231*I*sqrt(2)*(7*A + 8*B + 9*C)*a^2*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (315*A*a^2*cos(d*x + c)^5 + 385*(2*A + B)*a^2*cos(d*x + c)^4 + 45*(20*A + 22*B + 11*C)*a^2*cos(d*x + c)^3 + 154*(7*A + 8*B + 9*C)*a^2*cos(d*x + c)^ 2 + 30*(50*A + 55*B + 66*C)*a^2*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(1 1/2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^2/s ec(d*x + c)^(11/2), x)
\[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+a*sec(d*x+c))^2*(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)*(a*sec(d*x + c) + a)^2/s ec(d*x + c)^(11/2), x)
Timed out. \[ \int \frac {(a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
int(((a + a/cos(c + d*x))^2*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co s(c + d*x))^(11/2),x)