Integrand size = 45, antiderivative size = 565 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \left (a^2-b^2\right ) \left (40 A b^4+1254 a^3 b B-110 a b^3 B+75 a^4 (9 A+11 C)+15 a^2 b^2 (19 A+33 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3465 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B+15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3465 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}-\frac {2 \left (20 A b^4-1793 a^3 b B-55 a b^3 B-75 a^4 (9 A+11 C)-5 a^2 b^2 (205 A+297 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (15 A b^3+539 a^3 B+825 a b^2 B+5 a^2 b (229 A+297 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac {2 \left (5 A b^2+44 a b B+3 a^2 (9 A+11 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{231 d}+\frac {2 (5 A b+11 a B) \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{99 d}+\frac {2 A \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{11 d} \] Output:
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))*((b+a*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x +1/2*c,2^(1/2)*(a/(a+b))^(1/2))/a^3/d/cos(d*x+c)^(1/2)/(a+b*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(d*x+c)^(1/2)*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)-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))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c )/a^2/d+2/3465*(15*A*b^3+539*B*a^3+825*B*a*b^2+5*a^2*b*(229*A+297*C))*cos( d*x+c)^(3/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a/d+2/231*(5*A*b^2+44*B*a*b +3*a^2*(9*A+11*C))*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/d+2/ 99*(5*A*b+11*B*a)*cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d+2/1 1*A*cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*sin(d*x+c)/d
Result contains complex when optimal does not.
Time = 34.81 (sec) , antiderivative size = 4170, normalized size of antiderivative = 7.38 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:
Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d* x] + C*Sec[c + d*x]^2),x]
Output:
(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec [c + d*x]^2)*(((6525*a^4*A + 9330*a^2*A*b^2 - 160*A*b^4 + 16434*a^3*b*B + 440*a*b^3*B + 7590*a^4*C + 11880*a^2*b^2*C)*Sin[c + d*x])/(6930*a^2) + ((3 095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B + 2970*a^2*b*C)*Sin[2*( c + d*x)])/(3465*a) + ((513*a^2*A + 452*A*b^2 + 836*a*b*B + 396*a^2*C)*Sin [3*(c + d*x)])/2772 + (a*(23*A*b + 11*a*B)*Sin[4*(c + d*x)])/198 + (a^2*A* Sin[5*(c + d*x)])/44))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d* x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((494*a^2*A*b*Sqrt[Cos[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (34*A*b^3*Sq rt[Cos[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16* A*b^5*Sqrt[Cos[c + d*x]])/(693*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d *x]]) + (14*a^3*B*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Se c[c + d*x]]) + (62*a*b^2*B*Sqrt[Cos[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x] ]*Sqrt[Sec[c + d*x]]) - (4*b^4*B*Sqrt[Cos[c + d*x]])/(63*a*Sqrt[b + a*Cos[ c + d*x]]*Sqrt[Sec[c + d*x]]) + (58*a^2*b*C*Sqrt[Cos[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*C*Sqrt[Cos[c + d*x]])/(7*S qrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (30*a^3*A*Sqrt[Cos[c + d*x]] *Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) + (442*a*A*b^2*Sqrt[Cos [c + d*x]]*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^4*S qrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(693*a*Sqrt[b + a*Cos[c + d*x]]) ...
Time = 5.27 (sec) , antiderivative size = 609, normalized size of antiderivative = 1.08, number of steps used = 30, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4753, 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.
| \(\displaystyle \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{11/2} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4753 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )}{\sec ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {(a+b \sec (c+d x))^{3/2} \left (b (4 A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+5 A b+11 a B\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {(a+b \sec (c+d x))^{3/2} \left (b (4 A+11 C) \sec ^2(c+d x)+(9 a A+11 b B+11 a C) \sec (c+d x)+5 A b+11 a B\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (4 A+11 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(9 a A+11 b B+11 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b+11 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (56 A b+99 C b+44 a B) \sec ^2(c+d x)+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (56 A b+99 C b+44 a B) \sec ^2(c+d x)+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \sec (c+d x)+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (56 A b+99 C b+44 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^2+152 A b a+198 b C a+99 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (3 (9 A+11 C) a^2+44 b B a+5 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \sec ^2(c+d x)+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \sec (c+d x)}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \sec ^2(c+d x)+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3+b \left (36 (9 A+11 C) a^2+836 b B a+b^2 (452 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 (9 A+11 C) a^3+1507 b B a^2+b^2 (1531 A+2079 C) a+693 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \sec (c+d x)+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \sec ^2(c+d x)-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \sec (c+d x)+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\int \frac {-2 b \left (539 B a^3+5 b (229 A+297 C) a^2+825 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2-a \left (1617 B a^3+15 b (337 A+429 C) a^2+6655 b^2 B a+5 b^3 (461 A+693 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-75 (9 A+11 C) a^4-1793 b B a^3-5 b^2 (205 A+297 C) a^2-55 b^3 B a+20 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \sec (c+d x) a+40 A b^5\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \sec (c+d x) a+40 A b^5}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {1617 B a^5+15 b (247 A+319 C) a^4+3069 b^2 B a^3+15 b^3 (17 A+33 C) a^2-110 b^4 B a+\left (75 (9 A+11 C) a^4+2871 b B a^3+15 b^2 (221 A+297 C) a^2+1705 b^3 B a+10 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+40 A b^5}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )+\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \left (3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{5 a d \sec ^{\frac {3}{2}}(c+d x)}-\frac {\frac {2 \sin (c+d x) \left (-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}\right )\right )+\frac {2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
Input:
Int[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C *Sec[c + d*x]^2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*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*Sec[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 + 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))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]* EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[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)
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]
Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a + b*x])^m*(c*Sec[a + b*x])^m Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSecantIntegrandQ[u, x ]
Timed out.
hanged
Input:
int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^ 2),x)
Output:
int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^ 2),x)
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 829, normalized size of antiderivative = 1.47 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d *x+c)^2),x, algorithm="fricas")
Output:
2/10395*(3*(315*A*a^6*cos(d*x + c)^4 + 75*(9*A + 11*C)*a^6 + 1793*B*a^5*b + 5*(205*A + 297*C)*a^4*b^2 + 55*B*a^3*b^3 - 20*A*a^2*b^4 + 35*(11*B*a^6 + 23*A*a^5*b)*cos(d*x + c)^3 + 5*(9*(9*A + 11*C)*a^6 + 209*B*a^5*b + 113*A* a^4*b^2)*cos(d*x + c)^2 + (539*B*a^6 + 5*(229*A + 297*C)*a^5*b + 825*B*a^4 *b^2 + 15*A*a^3*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) *sqrt(cos(d*x + c))*sin(d*x + c) - sqrt(1/2)*(225*I*(9*A + 11*C)*a^6 + 537 9*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(1/2)*(-225*I*(9*A + 11*C)*a^6 - 5 379*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)*weierstrassPInver se(-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(1/2)*(-1617*I*B*a^6 - 15*I*(2 47*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(1/2)*(1617*I*B*a^6 + 15*I*(247*A + 319*C)*a^5*b + 306 9*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...
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(11/2)*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec (d*x+c)**2),x)
Output:
Timed out
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d *x+c)^2),x, algorithm="maxima")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)*cos(d*x + c)^(11/2), x)
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d *x+c)^2),x, algorithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 2)*cos(d*x + c)^(11/2), x)
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{11/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:
int(cos(c + d*x)^(11/2)*(a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C /cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)^(11/2)*(a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C /cos(c + d*x)^2), x)
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) b^{2} c +2 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) a b c +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) b^{3}+\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) a^{2} c +3 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) a^{3} \] Input:
int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^ 2),x)
Output:
int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d* x)**4,x)*b**2*c + 2*int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**3,x)*a*b*c + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos (c + d*x))*cos(c + d*x)**5*sec(c + d*x)**3,x)*b**3 + int(sqrt(sec(c + d*x) *b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**2,x)*a**2*c + 3*i nt(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x )**2,x)*a*b**2 + 3*int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x),x)*a**2*b + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**5,x)*a**3