Integrand size = 35, antiderivative size = 369 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {8 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a b \left (891 A b^2+96 a^2 C+673 b^2 C\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt {\sec (c+d x)}} \] Output:
8/15*a*b*(3*a^2*(5*A+3*C)+b^2*(9*A+7*C))*cos(d*x+c)^(1/2)*EllipticE(sin(1/ 2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/231*(77*a^4*(3*A+C)+66*a^2*b^2* (7*A+5*C)+5*b^4*(11*A+9*C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c ,2^(1/2))*sec(d*x+c)^(1/2)/d+4/3465*a*b*(891*A*b^2+96*C*a^2+673*C*b^2)*sin (d*x+c)/d/sec(d*x+c)^(3/2)+2/693*(64*a^4*C+15*b^4*(11*A+9*C)+9*a^2*b^2*(14 3*A+101*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/231*(16*a^2*C+3*b^2*(11*A+9*C) )*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(1/2)+16/99*a*C*(a+b*cos(d*x+ c))^3*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/11*C*(a+b*cos(d*x+c))^4*sin(d*x+c)/d /sec(d*x+c)^(1/2)
Time = 6.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 0.72 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (29568 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+480 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (616 a b \left (36 A b^2+36 a^2 C+43 b^2 C\right ) \cos (c+d x)+5 \left (1848 a^4 C+792 a^2 b^2 (14 A+13 C)+3 b^4 (572 A+531 C)+36 \left (11 A b^4+66 a^2 b^2 C+16 b^4 C\right ) \cos (2 (c+d x))+616 a b^3 C \cos (3 (c+d x))+63 b^4 C \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{55440 d} \] Input:
Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]] ,x]
Output:
(Sqrt[Sec[c + d*x]]*(29568*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[ Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 480*(77*a^4*(3*A + C) + 66*a^2*b ^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x )/2, 2] + 2*(616*a*b*(36*A*b^2 + 36*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*(18 48*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*(572*A + 531*C) + 36*(11*A*b^ 4 + 66*a^2*b^2*C + 16*b^4*C)*Cos[2*(c + d*x)] + 616*a*b^3*C*Cos[3*(c + d*x )] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(55440*d)
Time = 2.43 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.99, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.629, Rules used = {3042, 4709, 3042, 3529, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+A\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left (8 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+a (11 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {(a+b \cos (c+d x))^3 \left (8 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+a (11 A+C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (8 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left ((99 A+17 C) a^2+2 b (99 A+73 C) \cos (c+d x) a+3 \left (16 C a^2+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left ((99 A+17 C) a^2+2 b (99 A+73 C) \cos (c+d x) a+3 \left (16 C a^2+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((99 A+17 C) a^2+2 b (99 A+73 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+3 \left (16 C a^2+3 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (2 a \left (96 C a^2+891 A b^2+673 b^2 C\right ) \cos ^2(c+d x)+b \left ((2079 A+1381 C) a^2+45 b^2 (11 A+9 C)\right ) \cos (c+d x)+a \left ((693 A+167 C) a^2+9 b^2 (11 A+9 C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (2 a \left (96 C a^2+891 A b^2+673 b^2 C\right ) \cos ^2(c+d x)+b \left ((2079 A+1381 C) a^2+45 b^2 (11 A+9 C)\right ) \cos (c+d x)+a \left ((693 A+167 C) a^2+9 b^2 (11 A+9 C)\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (2 a \left (96 C a^2+891 A b^2+673 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left ((2079 A+1381 C) a^2+45 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left ((693 A+167 C) a^2+9 b^2 (11 A+9 C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left ((693 A+167 C) a^2+9 b^2 (11 A+9 C)\right ) a^2+924 b \left (3 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x) a+15 \left (64 C a^4+9 b^2 (143 A+101 C) a^2+15 b^4 (11 A+9 C)\right ) \cos ^2(c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left ((693 A+167 C) a^2+9 b^2 (11 A+9 C)\right ) a^2+924 b \left (3 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x) a+15 \left (64 C a^4+9 b^2 (143 A+101 C) a^2+15 b^4 (11 A+9 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left ((693 A+167 C) a^2+9 b^2 (11 A+9 C)\right ) a^2+924 b \left (3 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+15 \left (64 C a^4+9 b^2 (143 A+101 C) a^2+15 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (77 (3 A+C) a^4+66 b^2 (7 A+5 C) a^2+5 b^4 (11 A+9 C)\right )+308 a b \left (3 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (77 (3 A+C) a^4+66 b^2 (7 A+5 C) a^2+5 b^4 (11 A+9 C)\right )+308 a b \left (3 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (77 (3 A+C) a^4+66 b^2 (7 A+5 C) a^2+5 b^4 (11 A+9 C)\right )+308 a b \left (3 (5 A+3 C) a^2+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (308 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx+5 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (308 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {616 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \left (16 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}+\frac {1}{7} \left (\frac {4 a b \left (96 a^2 C+891 A b^2+673 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {10 \left (64 a^4 C+9 a^2 b^2 (143 A+101 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}+3 \left (\frac {616 a b \left (3 a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {10 \left (77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}\right )\right )\right )\right )+\frac {16 a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
Input:
Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]],x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[ c + d*x])^4*Sin[c + d*x])/(11*d) + ((16*a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[ c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*(16*a^2*C + 3*b^2*(11*A + 9*C))*Sqrt [Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + ((4*a*b*(891*A *b^2 + 96*a^2*C + 673*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*( (616*a*b*(3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/ d + (10*(77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*E llipticF[(c + d*x)/2, 2])/d) + (10*(64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2 *b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/5)/7)/9)/11)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x ])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*( n + 1)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(923\) vs. \(2(340)=680\).
Time = 48.71 (sec) , antiderivative size = 924, normalized size of antiderivative = 2.50
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(924\) |
| parts | \(\text {Expression too large to display}\) | \(1341\) |
Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x,method=_RETUR NVERBOSE)
Output:
-2/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C*b ^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-49280*C*a*b^3-50400*C*b^4)*s in(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^4+47520*C*a^2*b^2+98560* C*a*b^3+56880*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-22176*A*a*b ^3-11880*A*b^4-22176*C*a^3*b-71280*C*a^2*b^2-91168*C*a*b^3-34920*C*b^4)*si n(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(27720*A*a^2*b^2+22176*A*a*b^3+9240* A*b^4+4620*C*a^4+22176*C*a^3*b+55440*C*a^2*b^2+41888*C*a*b^3+13860*C*b^4)* sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-13860*A*a^2*b^2-5544*A*a*b^3-264 0*A*b^4-2310*C*a^4-5544*C*a^3*b-15840*C*a^2*b^2-7392*C*a*b^3-2790*C*b^4)*s in(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+3465*A*a^4*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2) )+6930*A*a^2*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^( 1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+825*A*b^4*(sin(1/2*d*x+1/2*c)^2 )^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1 /2))-13860*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2) *EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*b-8316*A*(sin(1/2*d*x+1/2*c)^2) ^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/ 2))*a*b^3+1155*a^4*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+4950*C*a^2*b^2*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.04 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (77 i \, {\left (3 \, A + C\right )} a^{4} + 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-77 i \, {\left (3 \, A + C\right )} a^{4} - 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 924 \, \sqrt {2} {\left (-3 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b - i \, {\left (9 \, A + 7 \, C\right )} a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 924 \, \sqrt {2} {\left (3 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b + i \, {\left (9 \, A + 7 \, C\right )} a b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, C b^{4} \cos \left (d x + c\right )^{5} + 1540 \, C a b^{3} \cos \left (d x + c\right )^{4} + 45 \, {\left (66 \, C a^{2} b^{2} + {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} + 308 \, {\left (9 \, C a^{3} b + {\left (9 \, A + 7 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (77 \, C a^{4} + 66 \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 5 \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algori thm="fricas")
Output:
-1/3465*(15*sqrt(2)*(77*I*(3*A + C)*a^4 + 66*I*(7*A + 5*C)*a^2*b^2 + 5*I*( 11*A + 9*C)*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-77*I*(3*A + C)*a^4 - 66*I*(7*A + 5*C)*a^2*b^2 - 5*I*(11*A + 9*C)*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 92 4*sqrt(2)*(-3*I*(5*A + 3*C)*a^3*b - I*(9*A + 7*C)*a*b^3)*weierstrassZeta(- 4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 924*sqr t(2)*(3*I*(5*A + 3*C)*a^3*b + I*(9*A + 7*C)*a*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(315*C*b^4* cos(d*x + c)^5 + 1540*C*a*b^3*cos(d*x + c)^4 + 45*(66*C*a^2*b^2 + (11*A + 9*C)*b^4)*cos(d*x + c)^3 + 308*(9*C*a^3*b + (9*A + 7*C)*a*b^3)*cos(d*x + c )^2 + 15*(77*C*a^4 + 66*(7*A + 5*C)*a^2*b^2 + 5*(11*A + 9*C)*b^4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**(1/2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4*sqrt(sec(d*x + c)) , x)
\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4*sqrt(sec(d*x + c)) , x)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \] Input:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^4,x )
Output:
int((A + C*cos(c + d*x)^2)*(1/cos(c + d*x))^(1/2)*(a + b*cos(c + d*x))^4, x)
\[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a^{5}+4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{4} b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{6}d x \right ) b^{4} c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) a \,b^{3} c +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a^{2} b^{2} c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{3} b c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} b^{3}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{4} c +6 \left (\int \sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{3} b^{2} \] Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x)
Output:
int(sqrt(sec(c + d*x)),x)*a**5 + 4*int(sqrt(sec(c + d*x))*cos(c + d*x),x)* a**4*b + int(sqrt(sec(c + d*x))*cos(c + d*x)**6,x)*b**4*c + 4*int(sqrt(sec (c + d*x))*cos(c + d*x)**5,x)*a*b**3*c + 6*int(sqrt(sec(c + d*x))*cos(c + d*x)**4,x)*a**2*b**2*c + int(sqrt(sec(c + d*x))*cos(c + d*x)**4,x)*a*b**4 + 4*int(sqrt(sec(c + d*x))*cos(c + d*x)**3,x)*a**3*b*c + 4*int(sqrt(sec(c + d*x))*cos(c + d*x)**3,x)*a**2*b**3 + int(sqrt(sec(c + d*x))*cos(c + d*x) **2,x)*a**4*c + 6*int(sqrt(sec(c + d*x))*cos(c + d*x)**2,x)*a**3*b**2