Integrand size = 35, antiderivative size = 325 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \left (15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a b \left (7 b^2 (A+3 C)+a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a b \left (32 A b^2+a^2 (101 A+147 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (192 A b^4+21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)\right ) \sin (c+d x)}{315 d \sqrt {\cos (c+d x)}}+\frac {2 \left (48 A b^2+7 a^2 (7 A+9 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {16 A b (a+b \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \] Output:
-2/15*(15*b^4*(A-C)+18*a^2*b^2*(3*A+5*C)+a^4*(7*A+9*C))*EllipticE(sin(1/2* d*x+1/2*c),2^(1/2))/d+8/21*a*b*(7*b^2*(A+3*C)+a^2*(5*A+7*C))*InverseJacobi AM(1/2*d*x+1/2*c,2^(1/2))/d+4/315*a*b*(32*A*b^2+a^2*(101*A+147*C))*sin(d*x +c)/d/cos(d*x+c)^(3/2)+2/315*(192*A*b^4+21*a^4*(7*A+9*C)+7*a^2*b^2*(155*A+ 261*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+2/315*(48*A*b^2+7*a^2*(7*A+9*C))*(a+ b*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+16/63*A*b*(a+b*cos(d*x+c))^3 *sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/9*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d *x+c)^(9/2)
Time = 10.48 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {2 \left (-21 \left (15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+60 \left (7 a b^3 (A+3 C)+a^3 b (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {35 a^4 A \sin (c+d x)}{\cos ^{\frac {9}{2}}(c+d x)}+\frac {180 a^3 A b \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}+\frac {7 a^2 \left (54 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {60 a b \left (7 A b^2+a^2 (5 A+7 C)\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {21 \left (15 A b^4+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}\right )}{315 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11 /2),x]
Output:
(2*(-21*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Ellipt icE[(c + d*x)/2, 2] + 60*(7*a*b^3*(A + 3*C) + a^3*b*(5*A + 7*C))*EllipticF [(c + d*x)/2, 2] + (35*a^4*A*Sin[c + d*x])/Cos[c + d*x]^(9/2) + (180*a^3*A *b*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (7*a^2*(54*A*b^2 + a^2*(7*A + 9*C))* Sin[c + d*x])/Cos[c + d*x]^(5/2) + (60*a*b*(7*A*b^2 + a^2*(5*A + 7*C))*Sin [c + d*x])/Cos[c + d*x]^(3/2) + (21*(15*A*b^4 + 18*a^2*b^2*(3*A + 5*C) + a ^4*(7*A + 9*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(315*d)
Time = 2.29 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3527, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3510, 27, 3042, 3500, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^3 \left (-b (A-9 C) \cos ^2(c+d x)+a (7 A+9 C) \cos (c+d x)+8 A b\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^3 \left (-b (A-9 C) \cos ^2(c+d x)+a (7 A+9 C) \cos (c+d x)+8 A b\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (A-9 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (7 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x))^2 \left (7 (7 A+9 C) a^2+2 b (41 A+63 C) \cos (c+d x) a+48 A b^2-3 b^2 (5 A-21 C) \cos ^2(c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x))^2 \left (7 (7 A+9 C) a^2+2 b (41 A+63 C) \cos (c+d x) a+48 A b^2-3 b^2 (5 A-21 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (7 (7 A+9 C) a^2+2 b (41 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+48 A b^2-3 b^2 (5 A-21 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \cos (c+d x)) \left (-b \left (7 (7 A+9 C) a^2+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)+a \left (21 (7 A+9 C) a^2+b^2 (479 A+945 C)\right ) \cos (c+d x)+6 \left (32 A b^3+\frac {1}{3} a^2 (303 A b+441 C b)\right )\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \cos (c+d x)) \left (-b \left (7 (7 A+9 C) a^2+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)+a \left (21 (7 A+9 C) a^2+b^2 (479 A+945 C)\right ) \cos (c+d x)+6 b \left ((101 A+147 C) a^2+32 A b^2\right )\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (7 (7 A+9 C) a^2+3 b^2 (41 A-105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (21 (7 A+9 C) a^2+b^2 (479 A+945 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+6 b \left ((101 A+147 C) a^2+32 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}-\frac {2}{3} \int -\frac {3 \left (21 (7 A+9 C) a^4+7 b^2 (155 A+261 C) a^2+60 b \left ((5 A+7 C) a^2+7 b^2 (A+3 C)\right ) \cos (c+d x) a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+7 b^2 (155 A+261 C) a^2+60 b \left ((5 A+7 C) a^2+7 b^2 (A+3 C)\right ) \cos (c+d x) a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+3 b^2 (41 A-105 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+7 b^2 (155 A+261 C) a^2+60 b \left ((5 A+7 C) a^2+7 b^2 (A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+3 b^2 (41 A-105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (2 \int \frac {3 \left (20 a b \left ((5 A+7 C) a^2+7 b^2 (A+3 C)\right )-7 \left ((7 A+9 C) a^4+18 b^2 (3 A+5 C) a^2+15 b^4 (A-C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {20 a b \left ((5 A+7 C) a^2+7 b^2 (A+3 C)\right )-7 \left ((7 A+9 C) a^4+18 b^2 (3 A+5 C) a^2+15 b^4 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {20 a b \left ((5 A+7 C) a^2+7 b^2 (A+3 C)\right )-7 \left ((7 A+9 C) a^4+18 b^2 (3 A+5 C) a^2+15 b^4 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (20 a b \left (a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-7 \left (a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (20 a b \left (a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-7 \left (a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (20 a b \left (a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {14 \left (a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \left (7 a^2 (7 A+9 C)+48 A b^2\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {4 a b \left (a^2 (101 A+147 C)+32 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (21 a^4 (7 A+9 C)+7 a^2 b^2 (155 A+261 C)+192 A b^4\right ) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+3 \left (\frac {40 a b \left (a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {14 \left (a^4 (7 A+9 C)+18 a^2 b^2 (3 A+5 C)+15 b^4 (A-C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {16 A b \sin (c+d x) (a+b \cos (c+d x))^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((16* A*b*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*(4 8*A*b^2 + 7*a^2*(7*A + 9*C))*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos [c + d*x]^(5/2)) + (3*((-14*(15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4 *(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/d + (40*a*b*(7*b^2*(A + 3*C) + a^ 2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/d) + (4*a*b*(32*A*b^2 + a^2*(101 *A + 147*C))*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + (2*(192*A*b^4 + 21*a^4 *(7*A + 9*C) + 7*a^2*b^2*(155*A + 261*C))*Sin[c + d*x])/(d*Sqrt[Cos[c + d* x]]))/5)/7)/9
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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
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[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, 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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
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^2*C + A*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1423\) vs. \(2(304)=608\).
Time = 37.04 (sec) , antiderivative size = 1424, normalized size of antiderivative = 4.38
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1424\) |
| parts | \(\text {Expression too large to display}\) | \(1588\) |
Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x,method=_RETU RNVERBOSE)
Output:
-(-(1-2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*A*a^4*(-1/144 *cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/( cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c )^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(cos(1/2*d*x+1/2*c)^2-1/2)^3-14/15*sin(1/2 *d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(1-2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+ 1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-2*cos(1/2*d*x+1/2*c)^ 2)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(co s(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(1-2*cos(1/2*d *x+1/2*c)^2)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(E llipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)) ))+2/5*a^2*(6*A*b^2+C*a^2)/(8*sin(1/2*d*x+1/2*c)^6-12*sin(1/2*d*x+1/2*c)^4 +6*sin(1/2*d*x+1/2*c)^2-1)/sin(1/2*d*x+1/2*c)^2*(24*cos(1/2*d*x+1/2*c)*sin (1/2*d*x+1/2*c)^6-12*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4-24*sin (1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+12*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*E llipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2* d*x+1/2*c)^2+8*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-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)*EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2)))*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*b^2*(A*b^ 2+6*C*a^2)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {60 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{3} b + 7 i \, {\left (A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 60 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{3} b - 7 i \, {\left (A + 3 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{4} + 18 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} + 15 i \, {\left (A - C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{4} - 18 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} - 15 i \, {\left (A - C\right )} b^{4}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (180 \, A a^{3} b \cos \left (d x + c\right ) + 35 \, A a^{4} + 21 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{4} + 18 \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} + 15 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 60 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{3} b + 7 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{4} + 54 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algor ithm="fricas")
Output:
-1/315*(60*sqrt(2)*(I*(5*A + 7*C)*a^3*b + 7*I*(A + 3*C)*a*b^3)*cos(d*x + c )^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 60*sqrt(2) *(-I*(5*A + 7*C)*a^3*b - 7*I*(A + 3*C)*a*b^3)*cos(d*x + c)^5*weierstrassPI nverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(I*(7*A + 9*C)*a ^4 + 18*I*(3*A + 5*C)*a^2*b^2 + 15*I*(A - C)*b^4)*cos(d*x + c)^5*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-I*(7*A + 9*C)*a^4 - 18*I*(3*A + 5*C)*a^2*b^2 - 15*I*(A - C)* b^4)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( d*x + c) - I*sin(d*x + c))) - 2*(180*A*a^3*b*cos(d*x + c) + 35*A*a^4 + 21* ((7*A + 9*C)*a^4 + 18*(3*A + 5*C)*a^2*b^2 + 15*A*b^4)*cos(d*x + c)^4 + 60* ((5*A + 7*C)*a^3*b + 7*A*a*b^3)*cos(d*x + c)^3 + 7*((7*A + 9*C)*a^4 + 54*A *a^2*b^2)*cos(d*x + c)^2)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) ^5)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(11/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algor ithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(11/2 ), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algor ithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(11/2 ), x)
Time = 3.76 (sec) , antiderivative size = 658, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/cos(c + d*x)^(11/2),x)
Output:
(8*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2)*((7*A*a*b^3*sin(c + d*x))/( cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (4*A*a^3*b*sin(c + d*x))/(cos (c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*A*a^3*b*sin(c + d*x))/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))))/(21*d) - (8*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2)*((7*A*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (5*A*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^ 2)^(1/2)) + (54*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^ 2)^(1/2))))/(135*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((28* A*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (12*A*a^ 4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*A*a^4*sin (c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2)) + (45*A*b^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (216*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (54*A*a^2*b^2*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*C*b^4*el lipticE(c/2 + (d*x)/2, 2))/d + (8*C*a*b^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*C*a^4*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d* cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (32*A*a^3*b*sin(c + d*x)*hype rgeom([-3/4, 1/2], 5/4, cos(c + d*x)^2))/(21*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*C*a^3*b*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos( c + d*x)^2))/(3*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (12*C*a^...
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a \,b^{3} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a^{5}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{4} c +6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{3} b c +4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{2} b^{3}+6 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{2} b^{2} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a \,b^{4}+\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) b^{4} c \] Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x)
Output:
4*int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a*b**3*c + int(sqrt(cos(c + d*x)) /cos(c + d*x)**6,x)*a**5 + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a** 4*b + int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**4*c + 6*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**3*b**2 + 4*int(sqrt(cos(c + d*x))/cos(c + d* x)**3,x)*a**3*b*c + 4*int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a**2*b**3 + 6*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a**2*b**2*c + int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*a*b**4 + int(sqrt(cos(c + d*x)),x)*b**4*c