Integrand size = 35, antiderivative size = 293 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 b \left (7 b^2 (A+3 C)+3 a^2 (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 a \left (24 A b^2+7 a^2 (7 A+9 C)\right ) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (8 A b^2+9 a^2 (5 A+7 C)\right ) \sin (c+d x)}{63 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {4 A b (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)} \] Output:
-2/15*a*(9*b^2*(3*A+5*C)+a^2*(7*A+9*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/ 2))/d+2/21*b*(7*b^2*(A+3*C)+3*a^2*(5*A+7*C))*InverseJacobiAM(1/2*d*x+1/2*c ,2^(1/2))/d+2/315*a*(24*A*b^2+7*a^2*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(5/ 2)+2/63*b*(8*A*b^2+9*a^2*(5*A+7*C))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/15*a*( 9*b^2*(3*A+5*C)+a^2*(7*A+9*C))*sin(d*x+c)/d/cos(d*x+c)^(1/2)+4/21*A*b*(a+b *cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/9*A*(a+b*cos(d*x+c))^3*sin( d*x+c)/d/cos(d*x+c)^(9/2)
Time = 9.15 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {-14 \left (9 a b^2 (3 A+5 C)+a^3 (7 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (7 b^3 (A+3 C)+3 a^2 b (5 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {70 a^3 A \sin (c+d x)}{3 \cos ^{\frac {9}{2}}(c+d x)}+\frac {90 a^2 A b \sin (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}+\frac {14 a \left (27 A b^2+a^2 (7 A+9 C)\right ) \sin (c+d x)}{3 \cos ^{\frac {5}{2}}(c+d x)}+\frac {10 b \left (7 A b^2+3 a^2 (5 A+7 C)\right ) \sin (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {14 a \left (9 b^2 (3 A+5 C)+a^2 (7 A+9 C)\right ) \sin (c+d x)}{\sqrt {\cos (c+d x)}}}{105 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11 /2),x]
Output:
(-14*(9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 1 0*(7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + (70* a^3*A*Sin[c + d*x])/(3*Cos[c + d*x]^(9/2)) + (90*a^2*A*b*Sin[c + d*x])/Cos [c + d*x]^(7/2) + (14*a*(27*A*b^2 + a^2*(7*A + 9*C))*Sin[c + d*x])/(3*Cos[ c + d*x]^(5/2)) + (10*b*(7*A*b^2 + 3*a^2*(5*A + 7*C))*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (14*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sin[c + d*x])/S qrt[Cos[c + d*x]])/(105*d)
Time = 1.77 (sec) , antiderivative size = 281, normalized size of antiderivative = 0.96, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 3527, 27, 3042, 3526, 27, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 3116, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (A+9 C) \cos ^2(c+d x)+a (7 A+9 C) \cos (c+d x)+6 A b\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (b (A+9 C) \cos ^2(c+d x)+a (7 A+9 C) \cos (c+d x)+6 A b\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b (A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (7 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (7 (7 A+9 C) a^2+2 b (43 A+63 C) \cos (c+d x) a+24 A b^2+b^2 (13 A+63 C) \cos ^2(c+d x)\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (7 (7 A+9 C) a^2+2 b (43 A+63 C) \cos (c+d x) a+24 A b^2+b^2 (13 A+63 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (7 (7 A+9 C) a^2+2 b (43 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+24 A b^2+b^2 (13 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {2}{5} \int -\frac {5 (13 A+63 C) \cos ^2(c+d x) b^3+15 \left (9 (5 A+7 C) a^2+8 A b^2\right ) b+21 a \left ((7 A+9 C) a^2+9 b^2 (3 A+5 C)\right ) \cos (c+d x)}{2 \cos ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (13 A+63 C) \cos ^2(c+d x) b^3+15 \left (9 (5 A+7 C) a^2+8 A b^2\right ) b+21 a \left ((7 A+9 C) a^2+9 b^2 (3 A+5 C)\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 (13 A+63 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^3+15 \left (9 (5 A+7 C) a^2+8 A b^2\right ) b+21 a \left ((7 A+9 C) a^2+9 b^2 (3 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (7 a \left ((7 A+9 C) a^2+9 b^2 (3 A+5 C)\right )+5 b \left (3 (5 A+7 C) a^2+7 b^2 (A+3 C)\right ) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {7 a \left ((7 A+9 C) a^2+9 b^2 (3 A+5 C)\right )+5 b \left (3 (5 A+7 C) a^2+7 b^2 (A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {7 a \left ((7 A+9 C) a^2+9 b^2 (3 A+5 C)\right )+5 b \left (3 (5 A+7 C) a^2+7 b^2 (A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (7 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (7 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 a \left (7 a^2 (7 A+9 C)+24 A b^2\right ) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {10 b \left (9 a^2 (5 A+7 C)+8 A b^2\right ) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+3 \left (\frac {10 b \left (3 a^2 (5 A+7 C)+7 b^2 (A+3 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+7 a \left (a^2 (7 A+9 C)+9 b^2 (3 A+5 C)\right ) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {12 A b \sin (c+d x) (a+b \cos (c+d x))^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
Input:
Int[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(11/2),x]
Output:
(2*A*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((12* A*b*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a* (24*A*b^2 + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (( 10*b*(8*A*b^2 + 9*a^2*(5*A + 7*C))*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + 3*((10*b*(7*b^2*(A + 3*C) + 3*a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/ d + 7*a*(9*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*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[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[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[(-(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. \(1242\) vs. \(2(272)=544\).
Time = 20.49 (sec) , antiderivative size = 1243, normalized size of antiderivative = 4.24
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1243\) |
| parts | \(\text {Expression too large to display}\) | \(1437\) |
Input:
int((a+b*cos(d*x+c))^3*(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*C*b^3*(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(cos(1/2*d*x+1/2*c),2^(1/2))+2* A*a^3*(-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(cos(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(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*(3*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)*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)^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...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-3 i \, {\left (5 \, A + 7 \, C\right )} a^{2} b - 7 i \, {\left (A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (i \, {\left (7 \, A + 9 \, C\right )} a^{3} + 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{3} - 9 i \, {\left (3 \, A + 5 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (135 \, A a^{2} b \cos \left (d x + c\right ) + 21 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 9 \, {\left (3 \, A + 5 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{4} + 35 \, A a^{3} + 15 \, {\left (3 \, {\left (5 \, A + 7 \, C\right )} a^{2} b + 7 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{3} + 27 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{5}} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x, algor ithm="fricas")
Output:
-1/315*(15*sqrt(2)*(3*I*(5*A + 7*C)*a^2*b + 7*I*(A + 3*C)*b^3)*cos(d*x + c )^5*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2) *(-3*I*(5*A + 7*C)*a^2*b - 7*I*(A + 3*C)*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 ^3 + 9*I*(3*A + 5*C)*a*b^2)*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstr assPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-I*(7*A + 9*C)*a^3 - 9*I*(3*A + 5*C)*a*b^2)*cos(d*x + c)^5*weierstrassZeta(-4, 0, w eierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(135*A*a^2*b *cos(d*x + c) + 21*((7*A + 9*C)*a^3 + 9*(3*A + 5*C)*a*b^2)*cos(d*x + c)^4 + 35*A*a^3 + 15*(3*(5*A + 7*C)*a^2*b + 7*A*b^3)*cos(d*x + c)^3 + 7*((7*A + 9*C)*a^3 + 27*A*a*b^2)*cos(d*x + c)^2)*sqrt(cos(d*x + c))*sin(d*x + c))/( d*cos(d*x + c)^5)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(11/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(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)^3/cos(d*x + c)^(11/2 ), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(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)^3/cos(d*x + c)^(11/2 ), x)
Time = 2.75 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\frac {70\,A\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {9}{4},\frac {1}{2};\ -\frac {5}{4};\ {\cos \left (c+d\,x\right )}^2\right )+210\,A\,b^3\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+378\,A\,a\,b^2\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )+270\,A\,a^2\,b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{315\,d\,{\cos \left (c+d\,x\right )}^{9/2}\,\sqrt {1-{\cos \left (c+d\,x\right )}^2}}+\frac {2\,C\,b^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,C\,a^3\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {6\,C\,a\,b^2\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^2\,b\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3)/cos(c + d*x)^(11/2),x)
Output:
(70*A*a^3*sin(c + d*x)*hypergeom([-9/4, 1/2], -5/4, cos(c + d*x)^2) + 210* A*b^3*cos(c + d*x)^3*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x) ^2) + 378*A*a*b^2*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2) + 270*A*a^2*b*cos(c + d*x)*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(315*d*cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2 )^(1/2)) + (2*C*b^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*C*a^3*sin(c + d*x) *hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(si n(c + d*x)^2)^(1/2)) + (6*C*a*b^2*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (2*C*a^2 *b*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(d*cos(c + d* x)^(3/2)*(sin(c + d*x)^2)^(1/2))
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) b^{3} c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a^{4}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a^{3} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{3} c +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a^{2} b^{2}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a^{2} b c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a \,b^{3}+3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a \,b^{2} c \] Input:
int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(11/2),x)
Output:
int(sqrt(cos(c + d*x))/cos(c + d*x),x)*b**3*c + int(sqrt(cos(c + d*x))/cos (c + d*x)**6,x)*a**4 + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a**3*b + int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*a**3*c + 3*int(sqrt(cos(c + d* x))/cos(c + d*x)**4,x)*a**2*b**2 + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)** 3,x)*a**2*b*c + int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*a*b**3 + 3*int(s qrt(cos(c + d*x))/cos(c + d*x)**2,x)*a*b**2*c