Integrand size = 43, antiderivative size = 419 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}-\frac {2 b (21 a A-3 b B-5 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}-\frac {2 b (9 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {2 A (a+b \cos (c+d x))^4 \sqrt {\sec (c+d x)} \sin (c+d x)}{d} \]
2/315*b^2*(162*B*a*b-a^2*(315*A-123*C)+7*b^2*(9*A+7*C))*sin(d*x+c)/d/sec(d *x+c)^(3/2)+2/63*b*(117*B*a^2*b+15*B*b^3-a^3*(126*A-62*C)+12*a*b^2*(7*A+5* C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)-2/21*b*(21*A*a-3*B*b-5*C*a)*(a+b*cos(d*x +c))^2*sin(d*x+c)/d/sec(d*x+c)^(1/2)-2/9*b*(9*A-C)*(a+b*cos(d*x+c))^3*sin( d*x+c)/d/sec(d*x+c)^(1/2)+2*A*(a+b*cos(d*x+c))^4*sin(d*x+c)*sec(d*x+c)^(1/ 2)/d+2/15*(60*B*a^3*b+36*B*a*b^3-15*a^4*(A-C)+18*a^2*b^2*(5*A+3*C)+b^4*(9* A+7*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2* d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/21*(21*B*a^4+42* B*a^2*b^2+5*B*b^4+28*a^3*b*(3*A+C)+4*a*b^3*(7*A+5*C))*(cos(1/2*d*x+1/2*c)^ 2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+ c)^(1/2)*sec(d*x+c)^(1/2)/d
Time = 5.88 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.78 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\frac {\sqrt {\sec (c+d x)} \left (-336 \left (-60 a^3 b B-36 a b^3 B+15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (2520 a^4 A+252 A b^4+1008 a b^3 B+1512 a^2 b^2 C+301 b^4 C+30 b \left (168 a^2 b B+29 b^3 B+112 a^3 C+4 a b^2 (28 A+29 C)\right ) \cos (c+d x)+84 b^2 \left (3 A b^2+12 a b B+18 a^2 C+4 b^2 C\right ) \cos (2 (c+d x))+90 b^4 B \cos (3 (c+d x))+360 a b^3 C \cos (3 (c+d x))+35 b^4 C \cos (4 (c+d x))\right ) \sin (c+d x)\right )}{2520 d} \]
Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S ec[c + d*x]^(3/2),x]
(Sqrt[Sec[c + d*x]]*(-336*(-60*a^3*b*B - 36*a*b^3*B + 15*a^4*(A - C) - 18* a^2*b^2*(5*A + 3*C) - b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d *x)/2, 2] + 240*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(252 0*a^4*A + 252*A*b^4 + 1008*a*b^3*B + 1512*a^2*b^2*C + 301*b^4*C + 30*b*(16 8*a^2*b*B + 29*b^3*B + 112*a^3*C + 4*a*b^2*(28*A + 29*C))*Cos[c + d*x] + 8 4*b^2*(3*A*b^2 + 12*a*b*B + 18*a^2*C + 4*b^2*C)*Cos[2*(c + d*x)] + 90*b^4* B*Cos[3*(c + d*x)] + 360*a*b^3*C*Cos[3*(c + d*x)] + 35*b^4*C*Cos[4*(c + d* x)])*Sin[c + d*x]))/(2520*d)
Time = 2.66 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.98, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4709, 3042, 3526, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sec (c+d x)^{3/2} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (2 \int \frac {(a+b \cos (c+d x))^3 \left (-b (9 A-C) \cos ^2(c+d x)+(b B-a (A-C)) \cos (c+d x)+8 A b+a B\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\int \frac {(a+b \cos (c+d x))^3 \left (-b (9 A-C) \cos ^2(c+d x)+(b B-a (A-C)) \cos (c+d x)+8 A b+a B\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (9 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(b B-a (A-C)) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+a B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (-3 b (21 a A-3 b B-5 a C) \cos ^2(c+d x)+\left (-9 (A-C) a^2+18 b B a+b^2 (9 A+7 C)\right ) \cos (c+d x)+a (63 A b+C b+9 a B)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (-3 b (21 a A-3 b B-5 a C) \cos ^2(c+d x)+\left (-9 (A-C) a^2+18 b B a+b^2 (9 A+7 C)\right ) \cos (c+d x)+a (63 A b+C b+9 a B)\right )}{\sqrt {\cos (c+d x)}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-3 b (21 a A-3 b B-5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-9 (A-C) a^2+18 b B a+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (63 A b+C b+9 a B)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (b \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)+\left (-63 (A-C) a^3+189 b B a^2+b^2 (189 A+131 C) a+45 b^3 B\right ) \cos (c+d x)+a \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right )\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (b \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)+\left (-63 (A-C) a^3+189 b B a^2+b^2 (189 A+131 C) a+45 b^3 B\right ) \cos (c+d x)+a \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right )\right )}{\sqrt {\cos (c+d x)}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-63 (A-C) a^3+189 b B a^2+b^2 (189 A+131 C) a+45 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right ) a^2+15 b \left (-\left ((126 A-62 C) a^3\right )+117 b B a^2+12 b^2 (7 A+5 C) a+15 b^3 B\right ) \cos ^2(c+d x)+21 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right ) a^2+15 b \left (-\left ((126 A-62 C) a^3\right )+117 b B a^2+12 b^2 (7 A+5 C) a+15 b^3 B\right ) \cos ^2(c+d x)+21 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right ) a^2+15 b \left (-\left ((126 A-62 C) a^3\right )+117 b B a^2+12 b^2 (7 A+5 C) a+15 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+21 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )+7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )+7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )+7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}+\frac {1}{5} \left (\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )\right )\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((-2*b*(9*A - C)*Sqrt[Cos[c + d*x]]* (a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*A*(a + b*Cos[c + d*x])^4*S in[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ((-6*b*(21*a*A - 3*b*B - 5*a*C)*Sqrt [Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2*b^2*(162*a *b*B - a^2*(315*A - 123*C) + 7*b^2*(9*A + 7*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((14*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2* b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(21* a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C)) *EllipticF[(c + d*x)/2, 2])/d) + (10*b*(117*a^2*b*B + 15*b^3*B - a^3*(126* A - 62*C) + 12*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/5)/7 )/9)
3.15.79.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^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_.) + (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[(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. \(1336\) vs. \(2(441)=882\).
Time = 12.05 (sec) , antiderivative size = 1337, normalized size of antiderivative = 3.19
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1337\) |
| default | \(\text {Expression too large to display}\) | \(1652\) |
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3/2),x, method=_RETURNVERBOSE)
-2*A*a^4*(-2*(-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)*sin(1/2*d*x+1/2*c)^2+(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d* x+1/2*c)^2-1)^(1/2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*E llipticE(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)/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d-2* (4*A*a^3*b+B*a^4)*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2) *(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(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))/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d-2/21*(B*b^4+ 4*C*a*b^3)*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(48*co s(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/2*c)^5-72*co s(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 +1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*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)/sin(1/2*d*x+1/2*c)/(-1+2 *cos(1/2*d*x+1/2*c)^2)^(1/2)/d-2/5*(A*b^4+4*B*a*b^3+6*C*a^2*b^2)*((-1+2*co s(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-8*cos(1/2*d*x+1/2*c)*sin (1/2*d*x+1/2*c)^6+8*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-2*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)/sin(1/2*d*x+1/2*c)/(-1+2*cos(1/2*d*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.05 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=-\frac {15 \, \sqrt {2} {\left (21 i \, B a^{4} + 28 i \, {\left (3 \, A + C\right )} a^{3} b + 42 i \, B a^{2} b^{2} + 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-21 i \, B a^{4} - 28 i \, {\left (3 \, A + C\right )} a^{3} b - 42 i \, B a^{2} b^{2} - 4 i \, {\left (7 \, A + 5 \, C\right )} a b^{3} - 5 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, {\left (A - C\right )} a^{4} - 60 i \, B a^{3} b - 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} - 36 i \, B a b^{3} - i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, {\left (A - C\right )} a^{4} + 60 i \, B a^{3} b + 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + 36 i \, B a b^{3} + i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (35 \, C b^{4} \cos \left (d x + c\right )^{4} + 315 \, A a^{4} + 45 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (54 \, C a^{2} b^{2} + 36 \, B a b^{3} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (28 \, C a^{3} b + 42 \, B a^{2} b^{2} + 4 \, {\left (7 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3 /2),x, algorithm="fricas")
-1/315*(15*sqrt(2)*(21*I*B*a^4 + 28*I*(3*A + C)*a^3*b + 42*I*B*a^2*b^2 + 4 *I*(7*A + 5*C)*a*b^3 + 5*I*B*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-21*I*B*a^4 - 28*I*(3*A + C)*a^3*b - 42*I* B*a^2*b^2 - 4*I*(7*A + 5*C)*a*b^3 - 5*I*B*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(15*I*(A - C)*a^4 - 60*I*B*a^3 *b - 18*I*(5*A + 3*C)*a^2*b^2 - 36*I*B*a*b^3 - I*(9*A + 7*C)*b^4)*weierstr assZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-15*I*(A - C)*a^4 + 60*I*B*a^3*b + 18*I*(5*A + 3*C)*a^2*b^2 + 36*I*B*a*b^3 + I*(9*A + 7*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*C*b^4*cos(d*x + c)^4 + 315*A*a^4 + 45*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^3 + 7*(54*C*a^2*b^2 + 36* B*a*b^3 + (9*A + 7*C)*b^4)*cos(d*x + c)^2 + 15*(28*C*a^3*b + 42*B*a^2*b^2 + 4*(7*A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3 /2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s ec(d*x + c)^(3/2), x)
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(3 /2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s ec(d*x + c)^(3/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac {3}{2}}(c+d x) \, dx=\int {\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]