Integrand size = 33, antiderivative size = 345 \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a^2 (15 A b+11 a B) \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 a \left (9 a^2 A+26 A b^2+33 a b B\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 a A (a+b \sec (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \] Output:
2/15*(21*A*a^2*b+9*A*b^3+7*B*a^3+27*B*a*b^2)*cos(d*x+c)^(1/2)*EllipticE(si n(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/231*(45*A*a^3+165*A*a*b^2+1 65*B*a^2*b+77*B*b^3)*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2 ))*sec(d*x+c)^(1/2)/d+2/99*a^2*(15*A*b+11*B*a)*sin(d*x+c)/d/sec(d*x+c)^(7/ 2)+2/77*a*(9*A*a^2+26*A*b^2+33*B*a*b)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/45*( 21*A*a^2*b+9*A*b^3+7*B*a^3+27*B*a*b^2)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/231 *(45*A*a^3+165*A*a*b^2+165*B*a^2*b+77*B*b^3)*sin(d*x+c)/d/sec(d*x+c)^(1/2) +2/11*a*A*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(9/2)
Time = 9.77 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (3696 \left (21 a^2 A b+9 A b^3+7 a^3 B+27 a b^2 B\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (45 a^3 A+165 a A b^2+165 a^2 b B+77 b^3 B\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (154 \left (129 a^2 A b+36 A b^3+43 a^3 B+108 a b^2 B\right ) \cos (c+d x)+180 a \left (16 a^2 A+33 A b^2+33 a b B\right ) \cos (2 (c+d x))+770 a^2 (3 A b+a B) \cos (3 (c+d x))+15 \left (531 a^3 A+1716 a A b^2+1716 a^2 b B+616 b^3 B+21 a^3 A \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{27720 d} \] Input:
Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2 ),x]
Output:
(Sqrt[Sec[c + d*x]]*(3696*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sq rt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (1 54*(129*a^2*A*b + 36*A*b^3 + 43*a^3*B + 108*a*b^2*B)*Cos[c + d*x] + 180*a* (16*a^2*A + 33*A*b^2 + 33*a*b*B)*Cos[2*(c + d*x)] + 770*a^2*(3*A*b + a*B)* Cos[3*(c + d*x)] + 15*(531*a^3*A + 1716*a*A*b^2 + 1716*a^2*b*B + 616*b^3*B + 21*a^3*A*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(27720*d)
Time = 2.12 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.90, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 4513, 27, 3042, 4562, 27, 3042, 4535, 3042, 4256, 3042, 4258, 3042, 3119, 4533, 3042, 4256, 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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {2}{11} \int -\frac {(a+b \sec (c+d x)) \left (b (5 a A+11 b B) \sec ^2(c+d x)+\left (9 A a^2+22 b B a+11 A b^2\right ) \sec (c+d x)+a (15 A b+11 a B)\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \sec (c+d x)) \left (b (5 a A+11 b B) \sec ^2(c+d x)+\left (9 A a^2+22 b B a+11 A b^2\right ) \sec (c+d x)+a (15 A b+11 a B)\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (9 A a^2+22 b B a+11 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (15 A b+11 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}-\frac {2}{9} \int -\frac {9 b^2 (5 a A+11 b B) \sec ^2(c+d x)+11 \left (7 B a^3+21 A b a^2+27 b^2 B a+9 A b^3\right ) \sec (c+d x)+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {9 b^2 (5 a A+11 b B) \sec ^2(c+d x)+11 \left (7 B a^3+21 A b a^2+27 b^2 B a+9 A b^3\right ) \sec (c+d x)+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+11 \left (7 B a^3+21 A b a^2+27 b^2 B a+9 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \sec ^2(c+d x)+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \int \frac {1}{\sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {3}{5} \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {3}{5} \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {3}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\int \frac {9 b^2 (5 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+9 a \left (9 A a^2+33 b B a+26 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4256 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4258 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )+\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {2 a^2 (11 a B+15 A b) \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{9} \left (\frac {18 a \left (9 a^2 A+33 a b B+26 A b^2\right ) \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {9}{7} \left (45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )+11 \left (7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right ) \left (\frac {2 \sin (c+d x)}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {6 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}\right )\right )\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]
Output:
(2*a*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (( 2*a^2*(15*A*b + 11*a*B)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((18*a*(9 *a^2*A + 26*A*b^2 + 33*a*b*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + 11* (21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*((6*Sqrt[Cos[c + d*x]]*Ellip ticE[(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*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*( (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
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_))^(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[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & & LeQ[n, -1]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(824\) vs. \(2(316)=632\).
Time = 41.40 (sec) , antiderivative size = 825, normalized size of antiderivative = 2.39
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(825\) |
| parts | \(\text {Expression too large to display}\) | \(1063\) |
Input:
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x,method=_RETURN VERBOSE)
Output:
-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*a^3 *A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-50400*A*a^3-36960*A*a^2*b-12 320*B*a^3)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(56880*A*a^3+73920*A*a ^2*b+23760*A*a*b^2+24640*B*a^3+23760*B*a^2*b)*sin(1/2*d*x+1/2*c)^8*cos(1/2 *d*x+1/2*c)+(-34920*A*a^3-68376*A*a^2*b-35640*A*a*b^2-5544*A*b^3-22792*B*a ^3-35640*B*a^2*b-16632*B*a*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(1 3860*A*a^3+31416*A*a^2*b+27720*A*a*b^2+5544*A*b^3+10472*B*a^3+27720*B*a^2* b+16632*B*a*b^2+4620*B*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-2790 *A*a^3-5544*A*a^2*b-7920*A*a*b^2-1386*A*b^3-1848*B*a^3-7920*B*a^2*b-4158*B *a*b^2-2310*B*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+675*A*EllipticF (cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1 /2*c)^2-1)^(1/2)*a^3+2475*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*a*b^2-4851*A*Ellipti cE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x +1/2*c)^2-1)^(1/2)*a^2*b-2079*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin (1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*b^3+2475*B*Ellip ticF(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d *x+1/2*c)^2-1)^(1/2)*a^2*b+1155*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*(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)*b^3-1617*B*Ell ipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.07 \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (45 i \, A a^{3} + 165 i \, B a^{2} b + 165 i \, A a b^{2} + 77 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-45 i \, A a^{3} - 165 i \, B a^{2} b - 165 i \, A a b^{2} - 77 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-7 i \, B a^{3} - 21 i \, A a^{2} b - 27 i \, B a b^{2} - 9 i \, A b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (7 i \, B a^{3} + 21 i \, A a^{2} b + 27 i \, B a b^{2} + 9 i \, A b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, A a^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{4} + 135 \, {\left (3 \, A a^{3} + 11 \, B a^{2} b + 11 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (7 \, B a^{3} + 21 \, A a^{2} b + 27 \, B a b^{2} + 9 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (45 \, A a^{3} + 165 \, B a^{2} b + 165 \, A a b^{2} + 77 \, B b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \] Input:
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorit hm="fricas")
Output:
-1/3465*(15*sqrt(2)*(45*I*A*a^3 + 165*I*B*a^2*b + 165*I*A*a*b^2 + 77*I*B*b ^3)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2) *(-45*I*A*a^3 - 165*I*B*a^2*b - 165*I*A*a*b^2 - 77*I*B*b^3)*weierstrassPIn verse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-7*I*B*a^3 - 21 *I*A*a^2*b - 27*I*B*a*b^2 - 9*I*A*b^3)*weierstrassZeta(-4, 0, weierstrassP Inverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(7*I*B*a^3 + 21*I*A*a^2*b + 27*I*B*a*b^2 + 9*I*A*b^3)*weierstrassZeta(-4, 0, weierstras sPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(315*A*a^3*cos(d*x + c)^5 + 385*(B*a^3 + 3*A*a^2*b)*cos(d*x + c)^4 + 135*(3*A*a^3 + 11*B*a^2*b + 11*A*a*b^2)*cos(d*x + c)^3 + 77*(7*B*a^3 + 21*A*a^2*b + 27*B*a*b^2 + 9*A *b^3)*cos(d*x + c)^2 + 15*(45*A*a^3 + 165*B*a^2*b + 165*A*a*b^2 + 77*B*b^3 )*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*sec(d*x+c))**3*(A+B*sec(d*x+c))/sec(d*x+c)**(11/2),x)
Output:
Timed out
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorit hm="maxima")
Output:
Timed out
\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x, algorit hm="giac")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^3/sec(d*x + c)^(11/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \] Input:
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^3)/(1/cos(c + d*x))^(11/2), x)
Output:
int(((A + B/cos(c + d*x))*(a + b/cos(c + d*x))^3)/(1/cos(c + d*x))^(11/2), x)
\[ \int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{6}}d x \right ) a^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a^{3} b +6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{2} b^{2}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a \,b^{3}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{4} \] Input:
int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c))/sec(d*x+c)^(11/2),x)
Output:
int(sqrt(sec(c + d*x))/sec(c + d*x)**6,x)*a**4 + 4*int(sqrt(sec(c + d*x))/ sec(c + d*x)**5,x)*a**3*b + 6*int(sqrt(sec(c + d*x))/sec(c + d*x)**4,x)*a* *2*b**2 + 4*int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x)*a*b**3 + int(sqrt(se c(c + d*x))/sec(c + d*x)**2,x)*b**4