Integrand size = 45, antiderivative size = 565 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3465 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3465 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d \sqrt {\sec (c+d x)}}+\frac {2 (5 A b+11 a B) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
2/99*(5*A*b+11*B*a)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2 /11*A*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/3465*(a^2-b^2 )*(40*A*b^4+1254*B*a^3*b-110*B*a*b^3+75*a^4*(9*A+11*C)+15*a^2*b^2*(19*A+33 *C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x +1/2*c),2^(1/2)*(a/(a+b))^(1/2))*((b+a*cos(d*x+c))/(a+b))^(1/2)*sec(d*x+c) ^(1/2)/a^3/d/(a+b*sec(d*x+c))^(1/2)+2/231*(5*A*b^2+44*B*a*b+3*a^2*(9*A+11* C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/d/sec(d*x+c)^(5/2)+2/3465*(15*A*b^3+ 539*B*a^3+825*B*a*b^2+5*a^2*b*(229*A+297*C))*sin(d*x+c)*(a+b*sec(d*x+c))^( 1/2)/a/d/sec(d*x+c)^(3/2)-2/3465*(20*A*b^4-1793*B*a^3*b-55*B*a*b^3-75*a^4* (9*A+11*C)-5*a^2*b^2*(205*A+297*C))*sin(d*x+c)*(a+b*sec(d*x+c))^(1/2)/a^2/ d/sec(d*x+c)^(1/2)+2/3465*(40*A*b^5+1617*a^5*B+3069*a^3*b^2*B-110*a*b^4*B+ 15*a^2*b^3*(17*A+33*C)+15*a^4*b*(247*A+319*C))*(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)*(a/(a+b))^(1/2)) *(a+b*sec(d*x+c))^(1/2)/a^3/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1 /2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 13.58 (sec) , antiderivative size = 7479, normalized size of antiderivative = 13.24 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Result too large to show} \]
Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x] ^2))/Sec[c + d*x]^(11/2),x]
Time = 4.98 (sec) , antiderivative size = 588, normalized size of antiderivative = 1.04, number of steps used = 28, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.622, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {2}{11} \int \frac {(a+b \sec (c+d x))^{3/2} \left (b (4 A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+5 A b+11 a B\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \sec (c+d x))^{3/2} \left (b (4 A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+5 A b+11 a B\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (4 A+11 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b+11 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (56 A b+99 C b+44 a B) \sec ^2(c+d x)+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (56 A b+99 C b+44 a B) \sec ^2(c+d x)+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (56 A b+99 C b+44 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \sec ^2(c+d x)+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \sec (c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \sec ^2(c+d x)+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \sec (c+d x)+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \sec (c+d x)+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \sec (c+d x) a+40 A b^5\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \sec (c+d x) a+40 A b^5}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+40 A b^5}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\) |
Int[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/S ec[c + d*x]^(11/2),x]
(2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + ((2*(5*A*b + 11*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((6*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Se c[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*(15*A*b^3 + 539*a ^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin [c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (-(((2*(a^2 - b^2)*(40*A*b^4 + 125 4*a^3*b*B - 110*a*b^3*B + 75*a^4*(9*A + 11*C) + 15*a^2*b^2*(19*A + 33*C))* Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*S qrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(40*A*b^5 + 1617*a^ 5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*( 247*A + 319*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d *x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a) + (2 *(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*( 205*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d*Sqrt[Sec[c + d *x]]))/(5*a))/7)/9)/11
3.11.51.3.1 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[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S qrt[b + a*Sin[e + f*x]]) Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a , b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S qrt[a + b*Csc[e + f*x]]) Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ {a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d _.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a I nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) /(a*d) Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ [{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
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 + 1)*((d *Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n) Int[(a + b*Csc[e + f*x])^m *(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d , e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(10940\) vs. \(2(577)=1154\).
Time = 19.22 (sec) , antiderivative size = 10941, normalized size of antiderivative = 19.36
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(10941\) |
| default | \(\text {Expression too large to display}\) | \(11011\) |
int((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(11/ 2),x,method=_RETURNVERBOSE)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.19 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.48 \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Too large to display} \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(11/2),x, algorithm="fricas")
1/10395*(sqrt(2)*(-225*I*(9*A + 11*C)*a^6 - 5379*I*B*a^5*b - 15*I*(169*A + 253*C)*a^4*b^2 + 1023*I*B*a^3*b^3 + 30*I*(16*A + 33*C)*a^2*b^4 - 220*I*B* a*b^5 + 80*I*A*b^6)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2 *b)/a) + sqrt(2)*(225*I*(9*A + 11*C)*a^6 + 5379*I*B*a^5*b + 15*I*(169*A + 253*C)*a^4*b^2 - 1023*I*B*a^3*b^3 - 30*I*(16*A + 33*C)*a^2*b^4 + 220*I*B*a *b^5 - 80*I*A*b^6)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8 /27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2* b)/a) - 3*sqrt(2)*(-1617*I*B*a^6 - 15*I*(247*A + 319*C)*a^5*b - 3069*I*B*a ^4*b^2 - 15*I*(17*A + 33*C)*a^3*b^3 + 110*I*B*a^2*b^4 - 40*I*A*a*b^5)*sqrt (a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(1617*I* B*a^6 + 15*I*(247*A + 319*C)*a^5*b + 3069*I*B*a^4*b^2 + 15*I*(17*A + 33*C) *a^3*b^3 - 110*I*B*a^2*b^4 + 40*I*A*a*b^5)*sqrt(a)*weierstrassZeta(-4/3*(3 *a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3 *a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I *a*sin(d*x + c) + 2*b)/a)) + 6*(315*A*a^6*cos(d*x + c)^5 + 35*(11*B*a^6 + 23*A*a^5*b)*cos(d*x + c)^4 + 5*(9*(9*A + 11*C)*a^6 + 209*B*a^5*b + 113*A*a ^4*b^2)*cos(d*x + c)^3 + (539*B*a^6 + 5*(229*A + 297*C)*a^5*b + 825*B*a...
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(11/2),x, algorithm="maxima")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sec(d*x + c)^(11/2), x)
\[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {11}{2}}} \,d x } \]
integrate((a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c )^(11/2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)/sec(d*x + c)^(11/2), x)
Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
int(((a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/( 1/cos(c + d*x))^(11/2),x)