Integrand size = 33, antiderivative size = 292 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {7 (7 A-17 B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{10 a^3 d}-\frac {(13 A-33 B) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{6 a^3 d}+\frac {7 (7 A-17 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{10 a^3 d}-\frac {(13 A-33 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}+\frac {(A-B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac {(A-2 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \sec (c+d x))^2}+\frac {7 (7 A-17 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \sec (c+d x)\right )} \]
-1/6*(13*A-33*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/a^3/d+1/5*(A-B)*sec(d*x+c)^(9 /2)*sin(d*x+c)/d/(a+a*sec(d*x+c))^3+1/3*(A-2*B)*sec(d*x+c)^(7/2)*sin(d*x+c )/a/d/(a+a*sec(d*x+c))^2+7/30*(7*A-17*B)*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(a^ 3+a^3*sec(d*x+c))+7/10*(7*A-17*B)*sin(d*x+c)*sec(d*x+c)^(1/2)/a^3/d-7/10*( 7*A-17*B)*(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)/a^3/d-1/6*(13*A-33 *B)*(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)/a^3/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.32 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.71 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\frac {e^{-i d x} (A+B \sec (c+d x)) \left (\frac {7 i (7 A-17 B) e^{i d x} \left (1+e^{i (c+d x)}\right )^6 \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )}{2 \sqrt {2} \left (1+e^{2 i (c+d x)}\right )^{3/2}}+\frac {e^{-\frac {1}{2} i (5 c+3 d x)} \cos \left (\frac {1}{2} (c+d x)\right ) \left (-i A \left (65+374 e^{i (c+d x)}+986 e^{2 i (c+d x)}+1658 e^{3 i (c+d x)}+2164 e^{4 i (c+d x)}+1954 e^{5 i (c+d x)}+1390 e^{6 i (c+d x)}+670 e^{7 i (c+d x)}+147 e^{8 i (c+d x)}\right )+i B \left (165+944 e^{i (c+d x)}+2476 e^{2 i (c+d x)}+4148 e^{3 i (c+d x)}+5134 e^{4 i (c+d x)}+4664 e^{5 i (c+d x)}+3340 e^{6 i (c+d x)}+1620 e^{7 i (c+d x)}+357 e^{8 i (c+d x)}\right )-5 (13 A-33 B) \left (1+e^{i (c+d x)}\right )^5 \left (1+e^{2 i (c+d x)}\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )\right ) \sec ^{\frac {5}{2}}(c+d x)}{8 \left (1+e^{2 i (c+d x)}\right )}\right )}{15 a^3 d (B+A \cos (c+d x)) (1+\sec (c+d x))^3} \]
((A + B*Sec[c + d*x])*((((7*I)/2)*(7*A - 17*B)*E^(I*d*x)*(1 + E^(I*(c + d* x)))^6*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Hypergeometric2F1[1 /2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/(Sqrt[2]*(1 + E^((2*I)*(c + d*x)))^(3 /2)) + (Cos[(c + d*x)/2]*((-I)*A*(65 + 374*E^(I*(c + d*x)) + 986*E^((2*I)* (c + d*x)) + 1658*E^((3*I)*(c + d*x)) + 2164*E^((4*I)*(c + d*x)) + 1954*E^ ((5*I)*(c + d*x)) + 1390*E^((6*I)*(c + d*x)) + 670*E^((7*I)*(c + d*x)) + 1 47*E^((8*I)*(c + d*x))) + I*B*(165 + 944*E^(I*(c + d*x)) + 2476*E^((2*I)*( c + d*x)) + 4148*E^((3*I)*(c + d*x)) + 5134*E^((4*I)*(c + d*x)) + 4664*E^( (5*I)*(c + d*x)) + 3340*E^((6*I)*(c + d*x)) + 1620*E^((7*I)*(c + d*x)) + 3 57*E^((8*I)*(c + d*x))) - 5*(13*A - 33*B)*(1 + E^(I*(c + d*x)))^5*(1 + E^( (2*I)*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sec[c + d* x]^(5/2))/(8*E^((I/2)*(5*c + 3*d*x))*(1 + E^((2*I)*(c + d*x))))))/(15*a^3* d*E^(I*d*x)*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^3)
Time = 1.66 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.515, Rules used = {3042, 4507, 27, 3042, 4507, 3042, 4507, 27, 3042, 4274, 3042, 4255, 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.
| \(\displaystyle \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle \frac {\int \frac {\sec ^{\frac {7}{2}}(c+d x) (7 a (A-B)-a (3 A-13 B) \sec (c+d x))}{2 (\sec (c+d x) a+a)^2}dx}{5 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sec ^{\frac {7}{2}}(c+d x) (7 a (A-B)-a (3 A-13 B) \sec (c+d x))}{(\sec (c+d x) a+a)^2}dx}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (7 a (A-B)-a (3 A-13 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle \frac {\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (25 a^2 (A-2 B)-3 a^2 (8 A-23 B) \sec (c+d x)\right )}{\sec (c+d x) a+a}dx}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (25 a^2 (A-2 B)-3 a^2 (8 A-23 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4507 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {3}{2} \sec ^{\frac {3}{2}}(c+d x) \left (7 a^3 (7 A-17 B)-5 a^3 (13 A-33 B) \sec (c+d x)\right )dx}{a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \sec ^{\frac {3}{2}}(c+d x) \left (7 a^3 (7 A-17 B)-5 a^3 (13 A-33 B) \sec (c+d x)\right )dx}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 a^3 (7 A-17 B)-5 a^3 (13 A-33 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \int \sec ^{\frac {3}{2}}(c+d x)dx-5 a^3 (13 A-33 B) \int \sec ^{\frac {5}{2}}(c+d x)dx\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx-5 a^3 (13 A-33 B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )-5 a^3 (13 A-33 B) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )-5 a^3 (13 A-33 B) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )-5 a^3 (13 A-33 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-5 a^3 (13 A-33 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {\frac {3 \left (7 a^3 (7 A-17 B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-5 a^3 (13 A-33 B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )\right )}{2 a^2}+\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {7 a^2 (7 A-17 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac {3 \left (7 a^3 (7 A-17 B) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-5 a^3 (13 A-33 B) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )}{2 a^2}}{3 a^2}+\frac {10 a (A-2 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}\) |
((A - B)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ( (10*a*(A - 2*B)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x]) ^2) + ((7*a^2*(7*A - 17*B)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[ c + d*x])) + (3*(7*a^3*(7*A - 17*B)*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d) - 5*a^3*(13*A - 33*B)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sq rt[Sec[c + d*x]])/(3*d) + (2*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d))))/(2* a^2))/(3*a^2))/(10*a^2)
3.3.16.3.1 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)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[d*(A*b - a*B)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^(n - 1)/(a*f*( 2*m + 1))), x] - Simp[1/(a*b*(2*m + 1)) Int[(a + b*Csc[e + f*x])^(m + 1)* (d*Csc[e + f*x])^(n - 1)*Simp[A*(a*d*(n - 1)) - B*(b*d*(n - 1)) - d*(a*B*(m - n + 1) + 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] && LtQ[m, -2^(-1)] && G tQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(312)=624\).
Time = 46.96 (sec) , antiderivative size = 876, normalized size of antiderivative = 3.00
-1/60*(4*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d *x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2* d*x+1/2*c),2^(1/2))-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-165*B*Elli pticF(cos(1/2*d*x+1/2*c),2^(1/2))+357*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 2)))*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-10*(-2*sin(1/2*d*x+1/2*c)^4+s in(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A*EllipticE (cos(1/2*d*x+1/2*c),2^(1/2))-165*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+3 57*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*sin(1/2*d*x+1/2*c)^4*cos(1/2*d *x+1/2*c)+8*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/ 2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(65*A*EllipticF(cos(1 /2*d*x+1/2*c),2^(1/2))-147*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-165*B*E llipticF(cos(1/2*d*x+1/2*c),2^(1/2))+357*B*EllipticE(cos(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*(-2*sin(1/2*d*x+1/2*c)^4 +sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(65*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*A*Ellipti cE(cos(1/2*d*x+1/2*c),2^(1/2))-165*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)) +357*B*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))*cos(1/2*d*x+1/2*c)-168*(-2*s in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(7*A-17*B)*sin(1/2*d*x+1/2 *c)^10+8*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(482*A-11...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.84 \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=-\frac {5 \, {\left (\sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-13 i \, A + 33 i \, B\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (13 i \, A - 33 i \, B\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (7 i \, A - 17 i \, B\right )} \cos \left (d x + c\right )\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 \, {\left (\sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{4} + 3 \, \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-7 i \, A + 17 i \, B\right )} \cos \left (d x + c\right )\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 (21 \, {\left (7 \, A - 17 \, B\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (188 \, A - 453 \, B\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left (59 \, A - 139 \, B\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (A - 2 \, B\right )} \cos \left (d x + c\right ) + 20 \, B\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{60 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d \cos \left (d x + c\right )\right )}} \]
-1/60*(5*(sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c)^4 + 3*sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-13*I*A + 33*I*B)*cos(d*x + c)^2 + sq rt(2)*(-13*I*A + 33*I*B)*cos(d*x + c))*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*(sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c)^4 + 3*s qrt(2)*(13*I*A - 33*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(13*I*A - 33*I*B)*cos( d*x + c)^2 + sqrt(2)*(13*I*A - 33*I*B)*cos(d*x + c))*weierstrassPInverse(- 4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*(sqrt(2)*(7*I*A - 17*I*B)*cos(d* x + c)^4 + 3*sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c)^2 + sqrt(2)*(7*I*A - 17*I*B)*cos(d*x + c))*weierstras sZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*(sqrt(2)*(-7*I*A + 17*I*B)*cos(d*x + c)^4 + 3*sqrt(2)*(-7*I*A + 17*I*B) *cos(d*x + c)^3 + 3*sqrt(2)*(-7*I*A + 17*I*B)*cos(d*x + c)^2 + sqrt(2)*(-7 *I*A + 17*I*B)*cos(d*x + c))*weierstrassZeta(-4, 0, weierstrassPInverse(-4 , 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(21*(7*A - 17*B)*cos(d*x + c)^4 + 2*(188*A - 453*B)*cos(d*x + c)^3 + 5*(59*A - 139*B)*cos(d*x + c)^2 + 60*( A - 2*B)*cos(d*x + c) + 20*B)*sin(d*x + c)/sqrt(cos(d*x + c)))/(a^3*d*cos( d*x + c)^4 + 3*a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + a^3*d*cos(d *x + c))
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\sec ^{\frac {9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \]