Integrand size = 43, antiderivative size = 379 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \left (60 a^3 b B+36 a b^3 B-15 a^4 (A-C)+18 a^2 b^2 (5 A+3 C)+b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {2 b \left (117 a^2 b B+15 b^3 B-a^3 (126 A-62 C)+12 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{63 d}+\frac {2 b^2 \left (162 a b B-a^2 (315 A-123 C)+7 b^2 (9 A+7 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d}-\frac {2 b (21 a A-3 b B-5 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d}-\frac {2 b (9 A-C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 A (a+b \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \] Output:
2/15*(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 ))*EllipticE(sin(1/2*d*x+1/2*c),2^(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))*InverseJacobiAM(1/2*d*x+1/2*c,2^( 1/2))/d+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))* cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/315*b^2*(162*B*a*b-a^2*(315*A-123*C)+7*b^2 *(9*A+7*C))*cos(d*x+c)^(3/2)*sin(d*x+c)/d-2/21*b*(21*A*a-3*B*b-5*C*a)*cos( d*x+c)^(1/2)*(a+b*cos(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 = 8.34 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.73 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\frac {-14 \left (-60 a^3 b B-36 a b^3 B+15 a^4 (A-C)-18 a^2 b^2 (5 A+3 C)-b^4 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (21 a^4 B+42 a^2 b^2 B+5 b^4 B+28 a^3 b (3 A+C)+4 a b^3 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (30 b \left (168 a^2 b B+23 b^3 B+112 a^3 C+4 a b^2 (28 A+23 C)\right ) \sin (c+d x)+14 b^2 \left (18 A b^2+72 a b B+108 a^2 C+19 b^2 C\right ) \sin (2 (c+d x))+90 b^3 (b B+4 a C) \sin (3 (c+d x))+35 \left (b^4 C \sin (4 (c+d x))+72 a^4 A \tan (c+d x)\right )\right )}{105 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(3/2),x]
Output:
(-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] + 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] + (Sqrt[Cos[c + d*x]]*(30*b*(168*a^2*b*B + 23*b^3*B + 112*a^3*C + 4* a*b^2*(28*A + 23*C))*Sin[c + d*x] + 14*b^2*(18*A*b^2 + 72*a*b*B + 108*a^2* C + 19*b^2*C)*Sin[2*(c + d*x)] + 90*b^3*(b*B + 4*a*C)*Sin[3*(c + d*x)] + 3 5*(b^4*C*Sin[4*(c + d*x)] + 72*a^4*A*Tan[c + d*x])))/12)/(105*d)
Time = 2.52 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle 2 \int \frac {(a+b \cos (c+d x))^3 \left (-b (9 A-C) \cos ^2(c+d x)+(b B-a (A-C)) \cos (c+d x)+8 A b+a B\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (-b (9 A-C) \cos ^2(c+d x)+(b B-a (A-C)) \cos (c+d x)+8 A b+a B\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-b (9 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(b B-a (A-C)) \sin \left (c+d x+\frac {\pi }{2}\right )+8 A b+a B\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (-3 b (21 a A-3 b B-5 a C) \cos ^2(c+d x)+\left (-9 (A-C) a^2+18 b B a+b^2 (9 A+7 C)\right ) \cos (c+d x)+a (63 A b+C b+9 a B)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (-3 b (21 a A-3 b B-5 a C) \cos ^2(c+d x)+\left (-9 (A-C) a^2+18 b B a+b^2 (9 A+7 C)\right ) \cos (c+d x)+a (63 A b+C b+9 a B)\right )}{\sqrt {\cos (c+d x)}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (-3 b (21 a A-3 b B-5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-9 (A-C) a^2+18 b B a+b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (63 A b+C b+9 a B)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (b \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)+\left (-63 (A-C) a^3+189 b B a^2+b^2 (189 A+131 C) a+45 b^3 B\right ) \cos (c+d x)+a \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right )\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (b \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)+\left (-63 (A-C) a^3+189 b B a^2+b^2 (189 A+131 C) a+45 b^3 B\right ) \cos (c+d x)+a \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right )\right )}{\sqrt {\cos (c+d x)}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (-\left ((315 A-123 C) a^2\right )+162 b B a+7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (-63 (A-C) a^3+189 b B a^2+b^2 (189 A+131 C) a+45 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right ) a^2+15 b \left (-\left ((126 A-62 C) a^3\right )+117 b B a^2+12 b^2 (7 A+5 C) a+15 b^3 B\right ) \cos ^2(c+d x)+21 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right ) a^2+15 b \left (-\left ((126 A-62 C) a^3\right )+117 b B a^2+12 b^2 (7 A+5 C) a+15 b^3 B\right ) \cos ^2(c+d x)+21 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left (63 B a^2+378 A b a+22 b C a+9 b^2 B\right ) a^2+15 b \left (-\left ((126 A-62 C) a^3\right )+117 b B a^2+12 b^2 (7 A+5 C) a+15 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+21 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )+7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )+7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (21 B a^4+28 b (3 A+C) a^3+42 b^2 B a^2+4 b^3 (7 A+5 C) a+5 b^4 B\right )+7 \left (-15 (A-C) a^4+60 b B a^3+18 b^2 (5 A+3 C) a^2+36 b^3 B a+b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )+\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}\right )+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b^2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-\left (a^2 (315 A-123 C)\right )+162 a b B+7 b^2 (9 A+7 C)\right )}{5 d}+\frac {1}{5} \left (\frac {10 b \sin (c+d x) \sqrt {\cos (c+d x)} \left (-\left (a^3 (126 A-62 C)\right )+117 a^2 b B+12 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right )}{d}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right )}{d}\right )\right )\right )-\frac {6 b \sin (c+d x) \sqrt {\cos (c+d x)} (21 a A-5 a C-3 b B) (a+b \cos (c+d x))^2}{7 d}\right )-\frac {2 b (9 A-C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}+\frac {2 A \sin (c+d x) (a+b \cos (c+d x))^4}{d \sqrt {\cos (c+d x)}}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + 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*b*(21*a*A - 3*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*S in[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))*Elli pticE[(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
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])))
Leaf count of result is larger than twice the leaf count of optimal. \(1221\) vs. \(2(360)=720\).
Time = 14.70 (sec) , antiderivative size = 1222, normalized size of antiderivative = 3.22
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1222\) |
| default | \(\text {Expression too large to display}\) | \(1652\) |
Input:
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x, method=_RETURNVERBOSE)
Output:
2*(4*A*a^3*b+B*a^4)/d*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))-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*cos (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*cos (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*cos (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*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4-2*cos(1/2*d*x+1 /2*c)*sin(1/2*d*x+1/2*c)^2-3*(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*x+1/ 2*c)^2)^(1/2)/d-2/3*(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)*((-1+2*cos(1/2*d*x+1 /2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2 *c)^4-2*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)*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)/( -1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d+2*(6*A*a^2*b^2+4*B*a^3*b+C*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.25 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\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 )} \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 ) + 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 )} \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} - 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 )} \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} + 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 )} \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} + 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 )} \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+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3 /2),x, algorithm="fricas")
Output:
-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)*cos(d*x + c)*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)*cos(d*x + c)*w eierstrassPInverse(-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)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstrassPInverse( -4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(-15*I*(A - C)*a^4 + 6 0*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) *cos(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 + 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))*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c))
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)* *(3/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3 /2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4/c os(d*x + c)^(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3 /2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4/c os(d*x + c)^(3/2), x)
Time = 1.98 (sec) , antiderivative size = 547, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(3/2),x)
Output:
(2*(B*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*B*a^3*b*ellipticE(c/2 + (d*x)/2, 2) + 2*B*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2) + 2*B*a^2*b^2*cos(c + d*x)^( 1/2)*sin(c + d*x)))/d + (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*ellipticF(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 + (1 2*A*a^2*b^2*ellipticE(c/2 + (d*x)/2, 2))/d + (2*A*a^4*sin(c + d*x)*hyperge om([-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*B*b^4*cos(c + d* x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(s in(c + d*x)^2)^(1/2)) - (2*C*b^4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeo m([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (8*B *a*b^3*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)) - (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))
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx=5 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) a^{5}+\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{4} c +10 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{3} b^{2}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{3} b c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{2} b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b^{5}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{4} \] Input:
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(3/2),x)
Output:
5*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 + 10*int(sqrt(co s(c + d*x)),x)*a**3*b**2 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**3*b *c + 10*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 + int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*b**5 + 6*int(sq rt(cos(c + d*x))*cos(c + d*x)**2,x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x)) *cos(c + d*x)**2,x)*a*b**4