Integrand size = 35, antiderivative size = 519 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {2 \left (a^2-b^2\right ) \left (675 a^4 A+285 a^2 A b^2+40 A b^4+1254 a^3 b B-110 a b^3 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{3465 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3465 a^3 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a^2 d}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3465 a d}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{693 d}+\frac {2 a (14 A b+11 a B) \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{99 d}+\frac {2 a A \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{11 d} \] Output:
2/3465*(a^2-b^2)*(675*A*a^4+285*A*a^2*b^2+40*A*b^4+1254*B*a^3*b-110*B*a*b^ 3)*((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*(3705* A*a^4*b+255*A*a^2*b^3+40*A*b^5+1617*B*a^5+3069*B*a^3*b^2-110*B*a*b^4)*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*se c(d*x+c))^(1/2)/a^3/d/((b+a*cos(d*x+c))/(a+b))^(1/2)+2/3465*(675*A*a^4+102 5*A*a^2*b^2-20*A*b^4+1793*B*a^3*b+55*B*a*b^3)*cos(d*x+c)^(1/2)*(a+b*sec(d* x+c))^(1/2)*sin(d*x+c)/a^2/d+2/3465*(1145*A*a^2*b+15*A*b^3+539*B*a^3+825*B *a*b^2)*cos(d*x+c)^(3/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a/d+2/693*(81*A *a^2+113*A*b^2+209*B*a*b)*cos(d*x+c)^(5/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+ c)/d+2/99*a*(14*A*b+11*B*a)*cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(1/2)*sin(d* x+c)/d+2/11*a*A*cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(3/2)*sin(d*x+c)/d
Result contains complex when optimal does not.
Time = 19.73 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.21 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\frac {\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {\left (6525 a^4 A+9330 a^2 A b^2-160 A b^4+16434 a^3 b B+440 a b^3 B\right ) \sin (c+d x)}{13860 a^2}+\frac {\left (3095 a^2 A b+30 A b^3+1463 a^3 B+1650 a b^2 B\right ) \sin (2 (c+d x))}{6930 a}+\frac {\left (513 a^2 A+452 A b^2+836 a b B\right ) \sin (3 (c+d x))}{5544}+\frac {1}{396} a (23 A b+11 a B) \sin (4 (c+d x))+\frac {1}{88} a^2 A \sin (5 (c+d x))\right )}{d (b+a \cos (c+d x))^2}-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{5/2} \left (-i (a+b) \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) E\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+i a (a+b) \left (40 A b^4-10 a b^3 (3 A+11 B)+15 a^2 b^2 (19 A+121 B)+6 a^3 b (505 A+209 B)+3 a^4 (225 A+539 B)\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3465 a^3 d (b+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \] Input:
Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d* x]),x]
Output:
(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((6525*a^4*A + 9330*a^2*A* b^2 - 160*A*b^4 + 16434*a^3*b*B + 440*a*b^3*B)*Sin[c + d*x])/(13860*a^2) + ((3095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B)*Sin[2*(c + d*x)])/ (6930*a) + ((513*a^2*A + 452*A*b^2 + 836*a*b*B)*Sin[3*(c + d*x)])/5544 + ( a*(23*A*b + 11*a*B)*Sin[4*(c + d*x)])/396 + (a^2*A*Sin[5*(c + d*x)])/88))/ (d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec [c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(a + b)*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*Elli pticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sq rt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A* b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (- a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x )/2]^2)/(a + b)] - (3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^( 3/2)*Tan[(c + d*x)/2]))/(3465*a^3*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5 /2))
Time = 4.64 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.07, number of steps used = 30, number of rules used = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3434, 3042, 4513, 27, 3042, 4582, 27, 3042, 4592, 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.
| \(\displaystyle \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{11/2} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 3434 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {2}{11} \int -\frac {\sqrt {a+b \sec (c+d x)} \left (b (6 a A+11 b B) \sec ^2(c+d x)+\left (9 A a^2+22 b B a+11 A b^2\right ) \sec (c+d x)+a (14 A b+11 a B)\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\sqrt {a+b \sec (c+d x)} \left (b (6 a A+11 b B) \sec ^2(c+d x)+\left (9 A a^2+22 b B a+11 A b^2\right ) \sec (c+d x)+a (14 A b+11 a B)\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (b (6 a A+11 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (9 A a^2+22 b B a+11 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (14 A b+11 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {3 b \left (22 B a^2+46 A b a+33 b^2 B\right ) \sec ^2(c+d x)+\left (77 B a^3+233 A b a^2+297 b^2 B a+99 A b^3\right ) \sec (c+d x)+a \left (81 A a^2+209 b B a+113 A b^2\right )}{2 \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {3 b \left (22 B a^2+46 A b a+33 b^2 B\right ) \sec ^2(c+d x)+\left (77 B a^3+233 A b a^2+297 b^2 B a+99 A b^3\right ) \sec (c+d x)+a \left (81 A a^2+209 b B a+113 A b^2\right )}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {3 b \left (22 B a^2+46 A b a+33 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 B a^3+233 A b a^2+297 b^2 B a+99 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (81 A a^2+209 b B a+113 A b^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2 \int -\frac {4 a b \left (81 A a^2+209 b B a+113 A b^2\right ) \sec ^2(c+d x)+a \left (405 A a^3+1507 b B a^2+1531 A b^2 a+693 b^3 B\right ) \sec (c+d x)+a \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right )}{2 \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{7 a}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {4 a b \left (81 A a^2+209 b B a+113 A b^2\right ) \sec ^2(c+d x)+a \left (405 A a^3+1507 b B a^2+1531 A b^2 a+693 b^3 B\right ) \sec (c+d x)+a \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right )}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\int \frac {4 a b \left (81 A a^2+209 b B a+113 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+a \left (405 A a^3+1507 b B a^2+1531 A b^2 a+693 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \int -\frac {\left (1617 B a^3+5055 A b a^2+6655 b^2 B a+2305 A b^3\right ) \sec (c+d x) a^2+2 b \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right ) \sec ^2(c+d x) a+3 \left (675 A a^4+1793 b B a^3+1025 A b^2 a^2+55 b^3 B a-20 A b^4\right ) a}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (1617 B a^3+5055 A b a^2+6655 b^2 B a+2305 A b^3\right ) \sec (c+d x) a^2+2 b \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right ) \sec ^2(c+d x) a+3 \left (675 A a^4+1793 b B a^3+1025 A b^2 a^2+55 b^3 B a-20 A b^4\right ) a}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\int \frac {\left (1617 B a^3+5055 A b a^2+6655 b^2 B a+2305 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2+2 b \left (539 B a^3+1145 A b a^2+825 b^2 B a+15 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2 a+3 \left (675 A a^4+1793 b B a^3+1025 A b^2 a^2+55 b^3 B a-20 A b^4\right ) a}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4592 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}-\frac {2 \int -\frac {3 \left (\left (675 A a^4+2871 b B a^3+3315 A b^2 a^2+1705 b^3 B a+10 A b^4\right ) \sec (c+d x) a^2+\left (1617 B a^5+3705 A b a^4+3069 b^2 B a^3+255 A b^3 a^2-110 b^4 B a+40 A b^5\right ) a\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\left (675 A a^4+2871 b B a^3+3315 A b^2 a^2+1705 b^3 B a+10 A b^4\right ) \sec (c+d x) a^2+\left (1617 B a^5+3705 A b a^4+3069 b^2 B a^3+255 A b^3 a^2-110 b^4 B a+40 A b^5\right ) a}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\int \frac {\left (675 A a^4+2871 b B a^3+3315 A b^2 a^2+1705 b^3 B a+10 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2+\left (1617 B a^5+3705 A b a^4+3069 b^2 B a^3+255 A b^3 a^2-110 b^4 B a+40 A b^5\right ) a}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4523 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx+\left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4343 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{\sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {\left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 4345 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {\frac {\frac {\frac {\left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}}{5 a}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}}{7 a}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{d \sqrt {\sec (c+d x)}}+\frac {\frac {2 \left (a^2-b^2\right ) \left (675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}}{5 a}}{7 a}\right )+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \sec (c+d x)}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )\) |
Input:
Int[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*A*(a + b*Sec[c + d*x])^(3/2)*S in[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + ((2*a*(14*A*b + 11*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*(81*a^2*A + 1 13*A*b^2 + 209*a*b*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*(1145*a^2*A*b + 15*A*b^3 + 539*a^3*B + 825*a*b^2*B)*Sqrt [a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (((2*(a^2 - b^2)*(675*a^4*A + 285*a^2*A*b^2 + 40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B)*S qrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sq rt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3705*a^4*A*b + 255*a^ 2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*EllipticE[ (c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a + (2*(675*a^4*A + 1025*a^2*A*b^2 - 20*A*b^4 + 1793*a^3*b*B + 55*a*b^3*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d *x])/(d*Sqrt[Sec[c + d*x]]))/(5*a))/(7*a))/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[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]* (d_.) + (c_))^(n_.)*((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_.), x_Symbol] :> Sim p[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p Int[(a + b*Csc[e + f*x])^m*((c + d*Csc[e + f*x])^n/(g*Csc[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g , m, n, p}, x] && NeQ[b*c - a*d, 0] && !IntegerQ[p] && !(IntegerQ[m] && I ntegerQ[n])
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_)]*(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_.) + (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]
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)),x)
Output:
int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x)
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.44 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, alg orithm="fricas")
Output:
2/10395*(3*(315*A*a^6*cos(d*x + c)^4 + 675*A*a^6 + 1793*B*a^5*b + 1025*A*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*(81*A*a^6 + 209*B*a^5*b + 113*A*a^4*b^2)*cos(d*x + c)^2 + (539* B*a^6 + 1145*A*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)*(2025*I*A*a^6 + 5379*I*B*a^5*b + 2535*I*A*a^4*b^2 - 1023*I*B*a^3*b^3 - 480*I*A*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)*(-2025*I*A*a^6 - 5379*I*B*a^5 *b - 2535*I*A*a^4*b^2 + 1023*I*B*a^3*b^3 + 480*I*A*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(1/2)*(-1617*I*B*a^6 - 3705*I*A*a^5*b - 3069*I*B*a^4*b^2 - 255*I *A*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 + 3705*I*A*a^5*b + 3069*I*B*a^4*b^2 + 255*I*A*a^3*b^3 - 110*I*B*a^2*b^4 + 40*I*A*a*b^5)*sqr t(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...
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(11/2)*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)),x)
Output:
Timed out
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, alg orithm="maxima")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^(11 /2), x)
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int { {\left (B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x, alg orithm="giac")
Output:
integrate((B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/2)*cos(d*x + c)^(11 /2), x)
Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\int {\cos \left (c+d\,x\right )}^{11/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \] Input:
int(cos(c + d*x)^(11/2)*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(5/2),x)
Output:
int(cos(c + d*x)^(11/2)*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^(5/2), x )
\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx=\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) b^{3}+3 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) a \,b^{2}+3 \left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )d x \right ) a^{2} b +\left (\int \sqrt {\sec \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) a^{3} \] Input:
int(cos(d*x+c)^(11/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)),x)
Output:
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 + 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*b**2 + 3*int(sqrt(sec(c + d*x)*b + a)*sqrt(co s(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