Integrand size = 35, antiderivative size = 320 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {2 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}-\frac {2 a b (21 A-5 C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \] Output:
-2/15*(15*a^4*(A-C)-18*a^2*b^2*(5*A+3*C)-b^4*(9*A+7*C))*EllipticE(sin(1/2* d*x+1/2*c),2^(1/2))/d+8/21*a*b*(7*a^2*(3*A+C)+b^2*(7*A+5*C))*InverseJacobi AM(1/2*d*x+1/2*c,2^(1/2))/d-4/63*a*b*(a^2*(63*A-31*C)-6*b^2*(7*A+5*C))*cos (d*x+c)^(1/2)*sin(d*x+c)/d-2/315*b^2*(3*a^2*(105*A-41*C)-7*b^2*(9*A+7*C))* cos(d*x+c)^(3/2)*sin(d*x+c)/d-2/21*a*b*(21*A-5*C)*cos(d*x+c)^(1/2)*(a+b*co s(d*x+c))^2*sin(d*x+c)/d-2/9*b*(9*A-C)*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^3 *sin(d*x+c)/d+2*A*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(1/2)
Time = 6.49 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.68 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {-14 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (7 a^3 b (3 A+C)+a b^3 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (120 a b \left (28 A b^2+28 a^2 C+23 b^2 C\right ) \sin (c+d x)+14 \left (18 A b^4+108 a^2 b^2 C+19 b^4 C\right ) \sin (2 (c+d x))+5 \left (72 a b^3 C \sin (3 (c+d x))+7 b^4 C \sin (4 (c+d x))+504 a^4 A \tan (c+d x)\right )\right )}{105 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/ 2),x]
Output:
(-14*(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] + 40*(7*a^3*b*(3*A + C) + a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(120*a*b*(28*A*b^2 + 28*a^2*C + 23*b^2 *C)*Sin[c + d*x] + 14*(18*A*b^4 + 108*a^2*b^2*C + 19*b^4*C)*Sin[2*(c + d*x )] + 5*(72*a*b^3*C*Sin[3*(c + d*x)] + 7*b^4*C*Sin[4*(c + d*x)] + 504*a^4*A *Tan[c + d*x])))/12)/(105*d)
Time = 2.25 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.04, 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, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle 2 \int \frac {(a+b \cos (c+d x))^3 \left (-b (9 A-C) \cos ^2(c+d x)-a (A-C) \cos (c+d x)+8 A b\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (-b (9 A-C) \cos ^2(c+d x)-a (A-C) \cos (c+d x)+8 A b\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (9 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a (A-C) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (-3 a b (21 A-5 C) \cos ^2(c+d x)-\left (9 a^2 (A-C)-b^2 (9 A+7 C)\right ) \cos (c+d x)+a b (63 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (-3 a b (21 A-5 C) \cos ^2(c+d x)-\left (9 a^2 (A-C)-b^2 (9 A+7 C)\right ) \cos (c+d x)+a b (63 A+C)\right )}{\sqrt {\cos (c+d x)}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-3 a b (21 A-5 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (b^2 (9 A+7 C)-9 a^2 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a b (63 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (2 b (189 A+11 C) a^2-\left (63 a^2 (A-C)-b^2 (189 A+131 C)\right ) \cos (c+d x) a-b \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (2 b (189 A+11 C) a^2-\left (63 a^2 (A-C)-b^2 (189 A+131 C)\right ) \cos (c+d x) a-b \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (2 b (189 A+11 C) a^2-\left (63 a^2 (A-C)-b^2 (189 A+131 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a-b \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {10 b (189 A+11 C) a^3-30 b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \cos ^2(c+d x) a-21 \left (15 (A-C) a^4-18 b^2 (5 A+3 C) a^2-b^4 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {10 b (189 A+11 C) a^3-30 b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \cos ^2(c+d x) a-21 \left (15 (A-C) a^4-18 b^2 (5 A+3 C) a^2-b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {10 b (189 A+11 C) a^3-30 b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a-21 \left (15 (A-C) a^4-18 b^2 (5 A+3 C) a^2-b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (20 a b \left (7 (3 A+C) a^2+b^2 (7 A+5 C)\right )-7 \left (15 (A-C) a^4-18 b^2 (5 A+3 C) a^2-b^4 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {20 a b \left (7 (3 A+C) a^2+b^2 (7 A+5 C)\right )-7 \left (15 (A-C) a^4-18 b^2 (5 A+3 C) a^2-b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {20 a b \left (7 (3 A+C) a^2+b^2 (7 A+5 C)\right )-7 \left (15 (A-C) a^4-18 b^2 (5 A+3 C) a^2-b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (20 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-7 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (20 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-7 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (20 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {14 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (\frac {40 a b \left (7 a^2 (3 A+C)+b^2 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {14 \left (15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {20 a b \left (a^2 (63 A-31 C)-6 b^2 (7 A+5 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )-\frac {2 b^2 \left (3 a^2 (105 A-41 C)-7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )-\frac {6 a b (21 A-5 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2))/Cos[c + d*x]^(3/2),x]
Output:
(-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*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + ( (-6*a*b*(21*A - 5*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x ])/(7*d) + ((-2*b^2*(3*a^2*(105*A - 41*C) - 7*b^2*(9*A + 7*C))*Cos[c + d*x ]^(3/2)*Sin[c + d*x])/(5*d) + (3*((-14*(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 + (40*a*b*(7*a^2*(3* A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/d) - (20*a*b*(a^2*(63 *A - 31*C) - 6*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/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[(-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_.) + (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]
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])))
Leaf count of result is larger than twice the leaf count of optimal. \(1182\) vs. \(2(301)=602\).
Time = 57.06 (sec) , antiderivative size = 1183, normalized size of antiderivative = 3.70
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1183\) |
| default | \(\text {Expression too large to display}\) | \(1209\) |
Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x,method=_RETUR NVERBOSE)
Output:
-2/5*(A*b^4+6*C*a^2*b^2)*((2*cos(1/2*d*x+1/2*c)^2-1)*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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d-2/3*(4*A*a*b^3+4*C*a^3*b) *((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*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)+(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)))/(-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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d+2*(6*A*a^2*b^2+C*a^4 )*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*(1-2*cos(1/2*d*x+1/2*c)^2)^(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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d-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)*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)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d-2/45*C*b^4*((2*cos(1/2*d*x+1/2*c...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {60 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} b + i \, {\left (7 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 60 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} b - i \, {\left (7 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, {\left (A - C\right )} a^{4} - 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} - i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (-15 i \, {\left (A - C\right )} a^{4} + 18 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b^{2} + i \, {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, {\left (35 \, C b^{4} \cos \left (d x + c\right )^{4} + 180 \, C a b^{3} \cos \left (d x + c\right )^{3} + 315 \, A a^{4} + 7 \, {\left (54 \, C a^{2} b^{2} + {\left (9 \, A + 7 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (7 \, C a^{3} b + {\left (7 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )} \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algori thm="fricas")
Output:
-1/315*(60*sqrt(2)*(7*I*(3*A + C)*a^3*b + I*(7*A + 5*C)*a*b^3)*cos(d*x + c )*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 60*sqrt(2)*( -7*I*(3*A + C)*a^3*b - I*(7*A + 5*C)*a*b^3)*cos(d*x + c)*weierstrassPInver se(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(15*I*(A - C)*a^4 - 18*I*(5*A + 3*C)*a^2*b^2 - I*(9*A + 7*C)*b^4)*cos(d*x + c)*weierstrassZeta (-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sq rt(2)*(-15*I*(A - C)*a^4 + 18*I*(5*A + 3*C)*a^2*b^2 + I*(9*A + 7*C)*b^4)*c os(d*x + c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*C*b^4*cos(d*x + c)^4 + 180*C*a*b^3*cos(d*x + c )^3 + 315*A*a^4 + 7*(54*C*a^2*b^2 + (9*A + 7*C)*b^4)*cos(d*x + c)^2 + 60*( 7*C*a^3*b + (7*A + 5*C)*a*b^3)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)/cos(d*x+c)**(3/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(3/2) , x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^4/cos(d*x + c)^(3/2) , x)
Time = 1.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,A\,a^3\,b\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,A\,a\,b^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {4\,C\,a^3\,b\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,A\,a^2\,b^2\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,b^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,b^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,C\,a\,b^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {12\,C\,a^2\,b^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/cos(c + d*x)^(3/2),x)
Output:
(2*C*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (8*A*a^3*b*ellipticF(c/2 + (d*x) /2, 2))/d + (4*A*a*b^3*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipti cF(c/2 + (d*x)/2, 2))/3))/d + (4*C*a^3*b*((2*cos(c + d*x)^(1/2)*sin(c + d* x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (12*A*a^2*b^2*ellipticE(c/ 2 + (d*x)/2, 2))/d + (2*A*a^4*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*A*b^4*cos (c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/ (7*d*(sin(c + d*x)^2)^(1/2)) - (2*C*b^4*cos(c + d*x)^(11/2)*sin(c + d*x)*h ypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (8*C*a*b^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (12*C*a^2*b^2*cos(c + d*x) ^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin (c + d*x)^2)^(1/2))
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=4 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{5}+\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{4} c +6 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{3} b^{2}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{3} b c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{2} b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{3} c +6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{4} \] Input:
int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x)
Output:
4*int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a**4*b + int(sqrt(cos(c + d*x))/c os(c + d*x)**2,x)*a**5 + int(sqrt(cos(c + d*x)),x)*a**4*c + 6*int(sqrt(cos (c + d*x)),x)*a**3*b**2 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**3*b* c + 4*int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**2*b**3 + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*b**4*c + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**3 ,x)*a*b**3*c + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**2*b**2*c + i nt(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a*b**4