Integrand size = 41, antiderivative size = 266 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 (9 A b+9 a B+7 b C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 (7 a A+5 b B+5 a C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{21 d}+\frac {2 (9 A b+9 a B+7 b C) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 (b B+a C) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {2 b C \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d} \]
2/21*(7*A*a+5*B*b+5*C*a)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/45*(9*A*b+9*B*a+7 *C*b)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/7*(B*b+C*a)*sec(d*x+c)^(7/2)*sin(d*x +c)/d+2/9*b*C*sec(d*x+c)^(9/2)*sin(d*x+c)/d+2/15*(9*A*b+9*B*a+7*C*b)*sin(d *x+c)*sec(d*x+c)^(1/2)/d-2/15*(9*A*b+9*B*a+7*C*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)/d+2/21*(7*A*a+5*B*b+5*C*a)*(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
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.57 (sec) , antiderivative size = 1262, normalized size of antiderivative = 4.74 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \sqrt {2} A b e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^3(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {2 \sqrt {2} a B e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^3(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{5 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {14 \sqrt {2} b C e^{-i d x} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^3(c+d x) \csc (c) \left (-3 \sqrt {1+e^{2 i (c+d x)}}+e^{2 i d x} \left (-1+e^{2 i c}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )\right ) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{45 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {4 a A \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{3 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {20 b B \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {20 a C \cos ^{\frac {7}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{21 d (b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4 (9 A b+9 a B+7 b C) \cos (d x) \csc (c)}{15 d}+\frac {4 b C \sec (c) \sec ^4(c+d x) \sin (d x)}{9 d}+\frac {4 \sec (c) \sec ^3(c+d x) (7 b C \sin (c)+9 b B \sin (d x)+9 a C \sin (d x))}{63 d}+\frac {4 \sec (c) \sec (c+d x) (63 A b \sin (c)+63 a B \sin (c)+49 b C \sin (c)+105 a A \sin (d x)+75 b B \sin (d x)+75 a C \sin (d x))}{315 d}+\frac {4 \sec (c) \sec ^2(c+d x) (45 b B \sin (c)+45 a C \sin (c)+63 A b \sin (d x)+63 a B \sin (d x)+49 b C \sin (d x))}{315 d}+\frac {4 (7 a A+5 b B+5 a C) \tan (c)}{21 d}\right )}{(b+a \cos (c+d x)) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {5}{2}}(c+d x)} \]
(2*Sqrt[2]*A*b*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^ ((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^(( 2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x ]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A* Cos[2*c + 2*d*x])) + (2*Sqrt[2]*a*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1 [1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (14*Sqrt[2]*b*C*Sqrt[E^(I*(c + d *x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x] ^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I) *c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(b + a*Co s[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*a*A*Co s[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(b + a*Cos[c + d*x] )*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*b*B*Cos[c + d*x ]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x...
Time = 1.44 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.90, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4255, 3042, 4258, 3042, 3119, 4534, 3042, 4255, 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.
| \(\displaystyle \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 4564 |
| \(\displaystyle \frac {2}{9} \int \frac {1}{2} \sec ^{\frac {5}{2}}(c+d x) \left (9 (b B+a C) \sec ^2(c+d x)+(9 A b+7 C b+9 a B) \sec (c+d x)+9 a A\right )dx+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \sec ^{\frac {5}{2}}(c+d x) \left (9 (b B+a C) \sec ^2(c+d x)+(9 A b+7 C b+9 a B) \sec (c+d x)+9 a A\right )dx+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 A b+7 C b+9 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+9 a A\right )dx+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{9} \left ((9 a B+9 A b+7 b C) \int \sec ^{\frac {7}{2}}(c+d x)dx+\int \sec ^{\frac {5}{2}}(c+d x) \left (9 (b B+a C) \sec ^2(c+d x)+9 a A\right )dx\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left ((9 a B+9 A b+7 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}dx+\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {3}{5} \int \sec ^{\frac {3}{2}}(c+d x)dx+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {3}{5} \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (9 (b B+a C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a A\right )dx+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4534 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \int \sec ^{\frac {5}{2}}(c+d x)dx+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}\right )+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {9}{7} (7 a A+5 a C+5 b B) \left (\frac {2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+(9 a B+9 A b+7 b C) \left (\frac {2 \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {18 (a C+b B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{7 d}\right )+\frac {2 b C \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{9 d}\) |
(2*b*C*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d) + ((18*(b*B + a*C)*Sec[c + d *x]^(7/2)*Sin[c + d*x])/(7*d) + (9*(7*a*A + 5*b*B + 5*a*C)*((2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sec[c + d *x]^(3/2)*Sin[c + d*x])/(3*d)))/7 + (9*A*b + 9*a*B + 7*b*C)*((2*Sec[c + d* x]^(5/2)*Sin[c + d*x])/(5*d) + (3*((-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))/9
3.10.81.3.1 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_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) )), x] + Simp[(C*m + A*(m + 1))/(m + 1) Int[(b*Csc[e + f*x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, 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[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ n/(f*(n + 2))), x] + Simp[1/(n + 2) Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && !LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1019\) vs. \(2(286)=572\).
Time = 52.80 (sec) , antiderivative size = 1020, normalized size of antiderivative = 3.83
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1020\) |
| parts | \(\text {Expression too large to display}\) | \(1213\) |
int(sec(d*x+c)^(5/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,me thod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*C*b*(-1/144* cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(c os(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c) ^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2* d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+ 1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(c os(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d *x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)* (EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2 ))))+2*a*A*(-1/6*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1 /2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+ 1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2/5*(A*b+B*a)/(8*si n(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4+6*sin(1/2*d*x+1/2*c)^2-1)/sin(1 /2*d*x+1/2*c)^2*(24*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)-12*EllipticE(c os(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)*sin(1/2*d*x+1/2*c)^4-24*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2* c)^4+12*EllipticE(cos(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)*sin(1/2*d*x+1/2*c)^2+8*sin(1/2*d*x+1/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.17 \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {15 \, \sqrt {2} {\left (i \, {\left (7 \, A + 5 \, C\right )} a + 5 i \, B b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 5 \, C\right )} a - 5 i \, B b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (9 i \, B a + i \, {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-9 i \, B a - i \, {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (21 \, {\left (9 \, B a + {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left ({\left (7 \, A + 5 \, C\right )} a + 5 \, B b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, B a + {\left (9 \, A + 7 \, C\right )} b\right )} \cos \left (d x + c\right )^{2} + 35 \, C b + 45 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}} \]
integrate(sec(d*x+c)^(5/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="fricas")
-1/315*(15*sqrt(2)*(I*(7*A + 5*C)*a + 5*I*B*b)*cos(d*x + c)^4*weierstrassP Inverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-I*(7*A + 5*C) *a - 5*I*B*b)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*s in(d*x + c)) + 21*sqrt(2)*(9*I*B*a + I*(9*A + 7*C)*b)*cos(d*x + c)^4*weier strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c) )) + 21*sqrt(2)*(-9*I*B*a - I*(9*A + 7*C)*b)*cos(d*x + c)^4*weierstrassZet a(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(2 1*(9*B*a + (9*A + 7*C)*b)*cos(d*x + c)^4 + 15*((7*A + 5*C)*a + 5*B*b)*cos( d*x + c)^3 + 7*(9*B*a + (9*A + 7*C)*b)*cos(d*x + c)^2 + 35*C*b + 45*(C*a + B*b)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^4)
Timed out. \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
integrate(sec(d*x+c)^(5/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)*sec (d*x + c)^(5/2), x)
\[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {5}{2}} \,d x } \]
integrate(sec(d*x+c)^(5/2)*(a+b*sec(d*x+c))*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 ),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)*sec (d*x + c)^(5/2), x)
Timed out. \[ \int \sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x)) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]