Integrand size = 35, antiderivative size = 422 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \] Output:
2/195*(39*a^4*(5*A+3*C)+78*a^2*b^2*(9*A+7*C)+7*b^4*(13*A+11*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+8/231*a*b* (11*a^2*(7*A+5*C)+5*b^2*(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+4/9009*a*b*(1573*A*b^2+96*C*a^2+1259* C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/6435*(192*a^4*C+77*b^4*(13*A+11*C)+ 11*a^2*b^2*(637*A+491*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/1287*(48*a^2*C+1 1*b^2*(13*A+11*C))*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(3/2)+16/143 *a*C*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/13*C*(a+b*cos(d*x+ c))^4*sin(d*x+c)/d/sec(d*x+c)^(3/2)+8/231*a*b*(11*a^2*(7*A+5*C)+5*b^2*(11* A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)
Time = 7.63 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (14784 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+49920 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (154 \left (936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 b \left (312 a \left (44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right )+3744 a \left (11 A b^2+11 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+77 \left (52 A b^3+312 a^2 b C+89 b^3 C\right ) \cos (3 (c+d x))+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{1441440 d} \] 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]]*(14784*(39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 49920* a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF [(c + d*x)/2, 2] + 2*(154*(936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(11 18*A + 1171*C))*Cos[c + d*x] + 5*b*(312*a*(44*a^2*(14*A + 13*C) + b^2*(572 *A + 531*C)) + 3744*a*(11*A*b^2 + 11*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 77*(52*A*b^3 + 312*a^2*b*C + 89*b^3*C)*Cos[3*(c + d*x)] + 6552*a*b^2*C*Cos [4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(1441440*d )
Time = 2.68 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.95, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.686, Rules used = {3042, 4709, 3042, 3529, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos (c+d x)^2\right )}{\sqrt {\sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{13} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (8 a C \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (8 a C \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (8 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (13 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left ((143 A+57 C) a^2+2 b (143 A+113 C) \cos (c+d x) a+\left (48 C a^2+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left ((143 A+57 C) a^2+2 b (143 A+113 C) \cos (c+d x) a+\left (48 C a^2+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((143 A+57 C) a^2+2 b (143 A+113 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (48 C a^2+11 b^2 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (2 a \left (96 C a^2+1573 A b^2+1259 b^2 C\right ) \cos ^2(c+d x)+b \left (3 (1287 A+961 C) a^2+77 b^2 (13 A+11 C)\right ) \cos (c+d x)+3 a \left (3 (143 A+73 C) a^2+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (2 a \left (96 C a^2+1573 A b^2+1259 b^2 C\right ) \cos ^2(c+d x)+b \left (3 (1287 A+961 C) a^2+77 b^2 (13 A+11 C)\right ) \cos (c+d x)+3 a \left (3 (143 A+73 C) a^2+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (2 a \left (96 C a^2+1573 A b^2+1259 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (3 (1287 A+961 C) a^2+77 b^2 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (3 (143 A+73 C) a^2+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 \left (3 (143 A+73 C) a^2+11 b^2 (13 A+11 C)\right ) a^2+468 b \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x) a+7 \left (192 C a^4+11 b^2 (637 A+491 C) a^2+77 b^4 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 \left (3 (143 A+73 C) a^2+11 b^2 (13 A+11 C)\right ) a^2+468 b \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x) a+7 \left (192 C a^4+11 b^2 (637 A+491 C) a^2+77 b^4 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 \left (3 (143 A+73 C) a^2+11 b^2 (13 A+11 C)\right ) a^2+468 b \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+7 \left (192 C a^4+11 b^2 (637 A+491 C) a^2+77 b^4 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (77 \left (39 (5 A+3 C) a^4+78 b^2 (9 A+7 C) a^2+7 b^4 (13 A+11 C)\right )+780 a b \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (77 \left (39 (5 A+3 C) a^4+78 b^2 (9 A+7 C) a^2+7 b^4 (13 A+11 C)\right )+780 a b \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 \left (39 (5 A+3 C) a^4+78 b^2 (9 A+7 C) a^2+7 b^4 (13 A+11 C)\right )+780 a b \left (11 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (780 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (780 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+77 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (780 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (780 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (780 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{9 d}+\frac {1}{9} \left (\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {1}{7} \left (\frac {14 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (780 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {16 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
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*Cos[c + d*x]^(3/2)*(a + b*Cos[ c + d*x])^4*Sin[c + d*x])/(13*d) + ((16*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[ c + d*x])^3*Sin[c + d*x])/(11*d) + ((2*(48*a^2*C + 11*b^2*(13*A + 11*C))*C os[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + ((4*a*b*(15 73*A*b^2 + 96*a^2*C + 1259*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + ((14*(192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[ c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((154*(39*a^4*(5*A + 3*C) + 78*a^2 *b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2])/d + 780 *a*b*(11*a^2*(7*A + 5*C) + 5*b^2*(11*A + 9*C))*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d))))/5)/7)/9)/11)/13)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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. \(1016\) vs. \(2(389)=778\).
Time = 64.29 (sec) , antiderivative size = 1017, normalized size of antiderivative = 2.41
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1017\) |
| parts | \(\text {Expression too large to display}\) | \(1446\) |
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:
-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520* C*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(1048320*C*a*b^3+1330560*C* b^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^4-960960*C*a^2* b^2-2620800*C*a*b^3-1798720*C*b^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c )+(411840*A*a*b^3+320320*A*b^4+411840*C*a^3*b+1921920*C*a^2*b^2+2957760*C* a*b^3+1379840*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-432432*A*a^ 2*b^2-617760*A*a*b^3-296296*A*b^4-72072*C*a^4-617760*C*a^3*b-1777776*C*a^2 *b^2-1815840*C*a*b^3-666512*C*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c) +(240240*A*a^3*b+432432*A*a^2*b^2+480480*A*a*b^3+136136*A*b^4+72072*C*a^4+ 480480*C*a^3*b+816816*C*a^2*b^2+720720*C*a*b^3+198352*C*b^4)*sin(1/2*d*x+1 /2*c)^4*cos(1/2*d*x+1/2*c)+(-120120*A*a^3*b-108108*A*a^2*b^2-137280*A*a*b^ 3-24024*A*b^4-18018*C*a^4-137280*C*a^3*b-144144*C*a^2*b^2-145080*C*a*b^3-2 7258*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+60060*A*a^3*b*(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))+42900*a*A*b^3*(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))-45045*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^4-162162*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^2*b^2-21021*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(...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {780 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 780 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 5 i \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 i \, {\left (13 \, A + 11 \, C\right )} b^{4}\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 (3465 \, C b^{4} \cos \left (d x + c\right )^{6} + 16380 \, C a b^{3} \cos \left (d x + c\right )^{5} + 385 \, {\left (78 \, C a^{2} b^{2} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 2340 \, {\left (11 \, C a^{3} b + {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (117 \, C a^{4} + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 780 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 5 \, {\left (11 \, A + 9 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, 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:
-1/45045*(780*sqrt(2)*(11*I*(7*A + 5*C)*a^3*b + 5*I*(11*A + 9*C)*a*b^3)*we ierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 780*sqrt(2)*(-11 *I*(7*A + 5*C)*a^3*b - 5*I*(11*A + 9*C)*a*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-39*I*(5*A + 3*C)*a^4 - 78*I *(9*A + 7*C)*a^2*b^2 - 7*I*(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weier strassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(39*I* (5*A + 3*C)*a^4 + 78*I*(9*A + 7*C)*a^2*b^2 + 7*I*(13*A + 11*C)*b^4)*weiers trassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) ) - 2*(3465*C*b^4*cos(d*x + c)^6 + 16380*C*a*b^3*cos(d*x + c)^5 + 385*(78* C*a^2*b^2 + (13*A + 11*C)*b^4)*cos(d*x + c)^4 + 2340*(11*C*a^3*b + (11*A + 9*C)*a*b^3)*cos(d*x + c)^3 + 77*(117*C*a^4 + 78*(9*A + 7*C)*a^2*b^2 + 7*( 13*A + 11*C)*b^4)*cos(d*x + c)^2 + 780*(11*(7*A + 5*C)*a^3*b + 5*(11*A + 9 *C)*a*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \] 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
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] 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)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \] 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. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/(1/cos(c + d*x))^(1/2) ,x)
Output:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/(1/cos(c + d*x))^(1/2) , x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{5}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{6}}{\sec \left (d x +c \right )}d x \right ) b^{4} c +4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\sec \left (d x +c \right )}d x \right ) a \,b^{3} c +6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )}d x \right ) a^{2} b^{2} c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )}d x \right ) a \,b^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) a^{3} b c +4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )}d x \right ) a^{2} b^{3}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a^{4} c +6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )}d x \right ) a^{3} b^{2} \] 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))/sec(c + d*x),x)*a**5 + 4*int((sqrt(sec(c + d*x))*co s(c + d*x))/sec(c + d*x),x)*a**4*b + int((sqrt(sec(c + d*x))*cos(c + d*x)* *6)/sec(c + d*x),x)*b**4*c + 4*int((sqrt(sec(c + d*x))*cos(c + d*x)**5)/se c(c + d*x),x)*a*b**3*c + 6*int((sqrt(sec(c + d*x))*cos(c + d*x)**4)/sec(c + d*x),x)*a**2*b**2*c + int((sqrt(sec(c + d*x))*cos(c + d*x)**4)/sec(c + d *x),x)*a*b**4 + 4*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x),x) *a**3*b*c + 4*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x),x)*a** 2*b**3 + int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x),x)*a**4*c + 6*int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x),x)*a**3*b**2