Integrand size = 43, antiderivative size = 404 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{693 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d}+\frac {2 (11 b B+8 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d} \] Output:
2/15*(15*B*a^4+54*B*a^2*b^2+7*B*b^4+12*a^3*b*(5*A+3*C)+4*a*b^3*(9*A+7*C))* EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/231*(308*B*a^3*b+220*B*a*b^3+77* a^4*(3*A+C)+66*a^2*b^2*(7*A+5*C)+5*b^4*(11*A+9*C))*InverseJacobiAM(1/2*d*x +1/2*c,2^(1/2))/d+2/693*(682*B*a^3*b+660*B*a*b^3+64*a^4*C+15*b^4*(11*A+9*C )+9*a^2*b^2*(143*A+101*C))*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/3465*b*(1353*B* a^2*b+539*B*b^3+192*a^3*C+2*a*b^2*(891*A+673*C))*cos(d*x+c)^(3/2)*sin(d*x+ c)/d+2/231*(33*A*b^2+55*B*a*b+16*C*a^2+27*C*b^2)*cos(d*x+c)^(1/2)*(a+b*cos (d*x+c))^2*sin(d*x+c)/d+2/99*(11*B*b+8*C*a)*cos(d*x+c)^(1/2)*(a+b*cos(d*x+ c))^3*sin(d*x+c)/d+2/11*C*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^4*sin(d*x+c)/d
Time = 6.12 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.79 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {154 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+10 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {1}{12} \sqrt {\cos (c+d x)} \left (154 b \left (216 a^2 b B+43 b^3 B+144 a^3 C+4 a b^2 (36 A+43 C)\right ) \cos (c+d x)+5 \left (7392 a^3 b B+6864 a b^3 B+1848 a^4 C+792 a^2 b^2 (14 A+13 C)+3 b^4 (572 A+531 C)+36 b^2 \left (11 A b^2+44 a b B+66 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 b^3 (b B+4 a C) \cos (3 (c+d x))+63 b^4 C \cos (4 (c+d x))\right )\right ) \sin (c+d x)}{1155 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Sqrt[Cos[c + d*x]],x]
Output:
(154*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*( 9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 10*(308*a^3*b*B + 220*a*b^3*B + 77 *a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*b*(216*a^2*b*B + 43*b^3*B + 144*a ^3*C + 4*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(7392*a^3*b*B + 6864*a*b^3* B + 1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*(572*A + 531*C) + 36*b^ 2*(11*A*b^2 + 44*a*b*B + 66*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 154*b^3*( b*B + 4*a*C)*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[c + d*x])/ 12)/(1155*d)
Time = 2.53 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.04, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 3528, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left ((11 b B+8 a C) \cos ^2(c+d x)+(11 A b+9 C b+11 a B) \cos (c+d x)+a (11 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \int \frac {(a+b \cos (c+d x))^3 \left ((11 b B+8 a C) \cos ^2(c+d x)+(11 A b+9 C b+11 a B) \cos (c+d x)+a (11 A+C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((11 b B+8 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(11 A b+9 C b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \cos ^2(c+d x)+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \cos (c+d x)+a (99 a A+11 b B+17 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \cos ^2(c+d x)+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \cos (c+d x)+a (99 a A+11 b B+17 a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (99 a A+11 b B+17 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \cos ^2(c+d x)+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \cos (c+d x)+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \cos ^2(c+d x)+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \cos (c+d x)+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \cos ^2(c+d x)+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \cos ^2(c+d x)+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (77 (3 A+C) a^4+308 b B a^3+66 b^2 (7 A+5 C) a^2+220 b^3 B a+5 b^4 (11 A+9 C)\right )+77 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (77 (3 A+C) a^4+308 b B a^3+66 b^2 (7 A+5 C) a^2+220 b^3 B a+5 b^4 (11 A+9 C)\right )+77 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (77 (3 A+C) a^4+308 b B a^3+66 b^2 (7 A+5 C) a^2+220 b^3 B a+5 b^4 (11 A+9 C)\right )+77 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+77 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{d}\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}+\frac {1}{7} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{d}+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{d}\right )\right )\right )\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\) |
Input:
Int[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[ Cos[c + d*x]],x]
Output:
(2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(11*d) + ((2* (11*b*B + 8*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/( 9*d) + ((6*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sqrt[Cos[c + d*x]]* (a + b*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + ((2*b*(1353*a^2*b*B + 539*b^3 *B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x]) /(5*d) + (3*((154*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C ) + 4*a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/d + (10*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9 *C))*EllipticF[(c + d*x)/2, 2])/d) + (10*(682*a^3*b*B + 660*a*b^3*B + 64*a ^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]] *Sin[c + d*x])/d)/5)/7)/9)/11
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)*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. \(1264\) vs. \(2(383)=766\).
Time = 26.59 (sec) , antiderivative size = 1265, normalized size of antiderivative = 3.13
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1265\) |
| default | \(\text {Expression too large to display}\) | \(1273\) |
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)^(1/2),x, method=_RETURNVERBOSE)
Output:
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)*Elliptic E(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-2/45*(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)*(160 *cos(1/2*d*x+1/2*c)^11-480*cos(1/2*d*x+1/2*c)^9+616*cos(1/2*d*x+1/2*c)^7-4 32*cos(1/2*d*x+1/2*c)^5+160*cos(1/2*d*x+1/2*c)^3-21*(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^(1 /2))-24*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-2/21*(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)*(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*(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)*(-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))...
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2 \, {\left (315 \, C b^{4} \cos \left (d x + c\right )^{4} + 1155 \, C a^{4} + 4620 \, B a^{3} b + 990 \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 3300 \, B a b^{3} + 75 \, {\left (11 \, A + 9 \, C\right )} b^{4} + 385 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 45 \, {\left (66 \, C a^{2} b^{2} + 44 \, B a b^{3} + {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 77 \, {\left (36 \, C a^{3} b + 54 \, B a^{2} b^{2} + 4 \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 15 \, \sqrt {2} {\left (77 i \, {\left (3 \, A + C\right )} a^{4} + 308 i \, B a^{3} b + 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 i \, B a b^{3} + 5 i \, {\left (11 \, A + 9 \, C\right )} 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 (-77 i \, {\left (3 \, A + C\right )} a^{4} - 308 i \, B a^{3} b - 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} - 220 i \, B a b^{3} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 \, \sqrt {2} {\left (-15 i \, B a^{4} - 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} - 7 i \, B 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 ) - 231 \, \sqrt {2} {\left (15 i \, B a^{4} + 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 i \, B 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 )}{3465 \, d} \] 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)^(1 /2),x, algorithm="fricas")
Output:
1/3465*(2*(315*C*b^4*cos(d*x + c)^4 + 1155*C*a^4 + 4620*B*a^3*b + 990*(7*A + 5*C)*a^2*b^2 + 3300*B*a*b^3 + 75*(11*A + 9*C)*b^4 + 385*(4*C*a*b^3 + B* b^4)*cos(d*x + c)^3 + 45*(66*C*a^2*b^2 + 44*B*a*b^3 + (11*A + 9*C)*b^4)*co s(d*x + c)^2 + 77*(36*C*a^3*b + 54*B*a^2*b^2 + 4*(9*A + 7*C)*a*b^3 + 7*B*b ^4)*cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - 15*sqrt(2)*(77*I*(3*A + C)*a^4 + 308*I*B*a^3*b + 66*I*(7*A + 5*C)*a^2*b^2 + 220*I*B*a*b^3 + 5*I* (11*A + 9*C)*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c) ) - 15*sqrt(2)*(-77*I*(3*A + C)*a^4 - 308*I*B*a^3*b - 66*I*(7*A + 5*C)*a^2 *b^2 - 220*I*B*a*b^3 - 5*I*(11*A + 9*C)*b^4)*weierstrassPInverse(-4, 0, co s(d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)*(-15*I*B*a^4 - 12*I*(5*A + 3*C) *a^3*b - 54*I*B*a^2*b^2 - 4*I*(9*A + 7*C)*a*b^3 - 7*I*B*b^4)*weierstrassZe ta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231 *sqrt(2)*(15*I*B*a^4 + 12*I*(5*A + 3*C)*a^3*b + 54*I*B*a^2*b^2 + 4*I*(9*A + 7*C)*a*b^3 + 7*I*B*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0 , cos(d*x + c) - I*sin(d*x + c))))/d
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \] 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)* *(1/2),x)
Output:
Timed out
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \] 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)^(1 /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/s qrt(cos(d*x + c)), x)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \] 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)^(1 /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/s qrt(cos(d*x + c)), x)
Time = 1.92 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.49 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx =\text {Too large to display} \] 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)^(1/2),x)
Output:
(2*(A*a^4*ellipticF(c/2 + (d*x)/2, 2) + 4*A*a^3*b*ellipticE(c/2 + (d*x)/2, 2) + 2*A*a^2*b^2*ellipticF(c/2 + (d*x)/2, 2) + 2*A*a^2*b^2*cos(c + d*x)^( 1/2)*sin(c + d*x)))/d + (C*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2 *ellipticF(c/2 + (d*x)/2, 2))/3))/d + (2*B*a^4*ellipticE(c/2 + (d*x)/2, 2) )/d + (4*B*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 - (2*A*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*(sin(c + d*x)^2)^(1/2)) - (2*B*b^ 4*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d* x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*C*b^4*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d*x)^2 )^(1/2)) - (8*A*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*B*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)) - (8*C*a^3*b*cos(c + d*x)^(7/2)*sin(c + d*x)*hype rgeom([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)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos (c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (12*B*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)) - (4*C*a^2*b^2*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom ([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2))
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a^{5}+5 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{4} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{4} c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{3} b^{2}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) b^{5}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{3} b c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{2} b^{3} \] 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)^(1/2),x)
Output:
int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a**5 + 5*int(sqrt(cos(c + d*x)),x)* a**4*b + int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**4*c + 10*int(sqrt(cos(c + d*x))*cos(c + d*x),x)*a**3*b**2 + int(sqrt(cos(c + d*x))*cos(c + d*x)** 5,x)*b**4*c + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*a*b**3*c + int(s qrt(cos(c + d*x))*cos(c + d*x)**4,x)*b**5 + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*a* b**4 + 4*int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**3*b*c + 10*int(sqrt( cos(c + d*x))*cos(c + d*x)**2,x)*a**2*b**3