Integrand size = 43, antiderivative size = 477 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{6435 d}+\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d}+\frac {2 (13 b B+8 a C) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d}+\frac {2 C \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d} \] Output:
2/195*(468*B*a^3*b+364*B*a*b^3+39*a^4*(5*A+3*C)+78*a^2*b^2*(9*A+7*C)+7*b^4 *(13*A+11*C))*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/231*(77*B*a^4+330* B*a^2*b^2+45*B*b^4+44*a^3*b*(7*A+5*C)+20*a*b^3*(11*A+9*C))*InverseJacobiAM (1/2*d*x+1/2*c,2^(1/2))/d+2/231*(77*B*a^4+330*B*a^2*b^2+45*B*b^4+44*a^3*b* (7*A+5*C)+20*a*b^3*(11*A+9*C))*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/6435*(3458* B*a^3*b+4004*B*a*b^3+192*a^4*C+77*b^4*(13*A+11*C)+11*a^2*b^2*(637*A+491*C) )*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/9009*b*(2171*B*a^2*b+1053*B*b^3+192*a^3* C+2*a*b^2*(1573*A+1259*C))*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/1287*(143*A*b^2 +221*B*a*b+48*C*a^2+121*C*b^2)*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^2*sin(d*x +c)/d+2/143*(13*B*b+8*C*a)*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^3*sin(d*x+c)/ d+2/13*C*cos(d*x+c)^(3/2)*(a+b*cos(d*x+c))^4*sin(d*x+c)/d
Time = 5.57 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.80 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {48 \left (77 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+65 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )\right )+\sqrt {\cos (c+d x)} \left (154 \left (3744 a^3 b B+4472 a b^3 B+936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 \left (78 \left (616 a^4 B+3432 a^2 b^2 B+531 b^4 B+176 a^3 b (14 A+13 C)+4 a b^3 (572 A+531 C)\right )+1872 b \left (33 a^2 b B+8 b^3 B+22 a^3 C+2 a b^2 (11 A+16 C)\right ) \cos (2 (c+d x))+77 b^2 \left (52 A b^2+208 a b B+312 a^2 C+89 b^2 C\right ) \cos (3 (c+d x))+1638 b^3 (b B+4 a C) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right ) \sin (c+d x)}{360360 d} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(48*(77*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2] + 65*(77*a^4*B + 3 30*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*El lipticF[(c + d*x)/2, 2]) + Sqrt[Cos[c + d*x]]*(154*(3744*a^3*b*B + 4472*a* b^3*B + 936*a^4*C + 156*a^2*b^2*(36*A + 43*C) + b^4*(1118*A + 1171*C))*Cos [c + d*x] + 5*(78*(616*a^4*B + 3432*a^2*b^2*B + 531*b^4*B + 176*a^3*b*(14* A + 13*C) + 4*a*b^3*(572*A + 531*C)) + 1872*b*(33*a^2*b*B + 8*b^3*B + 22*a ^3*C + 2*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b^2*(52*A*b^2 + 208*a* b*B + 312*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^3*(b*B + 4*a*C)*Cos[ 4*(c + d*x)] + 693*b^4*C*Cos[5*(c + d*x)]))*Sin[c + d*x])/(360360*d)
Time = 2.77 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2}{13} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left ((13 b B+8 a C) \cos ^2(c+d x)+(13 A b+11 C b+13 a B) \cos (c+d x)+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left ((13 b B+8 a C) \cos ^2(c+d x)+(13 A b+11 C b+13 a B) \cos (c+d x)+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((13 b B+8 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(13 A b+11 C b+13 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\left (48 C a^2+221 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+\left (143 B a^2+286 A b a+226 b C a+117 b^2 B\right ) \cos (c+d x)+a (143 a A+39 b B+57 a C)\right )dx+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\left (48 C a^2+221 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+\left (143 B a^2+286 A b a+226 b C a+117 b^2 B\right ) \cos (c+d x)+a (143 a A+39 b B+57 a C)\right )dx+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (\left (48 C a^2+221 b B a+143 A b^2+121 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (143 B a^2+286 A b a+226 b C a+117 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (143 a A+39 b B+57 a C)\right )dx+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\left (192 C a^3+2171 b B a^2+2 b^2 (1573 A+1259 C) a+1053 b^3 B\right ) \cos ^2(c+d x)+\left (1287 B a^3+3 b (1287 A+961 C) a^2+2951 b^2 B a+77 b^3 (13 A+11 C)\right ) \cos (c+d x)+3 a \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\left (192 C a^3+2171 b B a^2+2 b^2 (1573 A+1259 C) a+1053 b^3 B\right ) \cos ^2(c+d x)+\left (1287 B a^3+3 b (1287 A+961 C) a^2+2951 b^2 B a+77 b^3 (13 A+11 C)\right ) \cos (c+d x)+3 a \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (192 C a^3+2171 b B a^2+2 b^2 (1573 A+1259 C) a+1053 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (1287 B a^3+3 b (1287 A+961 C) a^2+2951 b^2 B a+77 b^3 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right ) a^2+7 \left (192 C a^4+3458 b B a^3+11 b^2 (637 A+491 C) a^2+4004 b^3 B a+77 b^4 (13 A+11 C)\right ) \cos ^2(c+d x)+117 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right ) a^2+7 \left (192 C a^4+3458 b B a^3+11 b^2 (637 A+491 C) a^2+4004 b^3 B a+77 b^4 (13 A+11 C)\right ) \cos ^2(c+d x)+117 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right ) a^2+7 \left (192 C a^4+3458 b B a^3+11 b^2 (637 A+491 C) a^2+4004 b^3 B a+77 b^4 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+117 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (77 \left (39 (5 A+3 C) a^4+468 b B a^3+78 b^2 (9 A+7 C) a^2+364 b^3 B a+7 b^4 (13 A+11 C)\right )+195 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (77 \left (39 (5 A+3 C) a^4+468 b B a^3+78 b^2 (9 A+7 C) a^2+364 b^3 B a+7 b^4 (13 A+11 C)\right )+195 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 \left (39 (5 A+3 C) a^4+468 b B a^3+78 b^2 (9 A+7 C) a^2+364 b^3 B a+7 b^4 (13 A+11 C)\right )+195 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{d}\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}+\frac {1}{9} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}+\frac {1}{7} \left (\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}+\frac {3}{5} \left (\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{d}+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[ c + d*x]^2),x]
Output:
(2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(13*d) + ((2* (13*b*B + 8*a*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/( 11*d) + ((2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3 /2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + ((2*b*(2171*a^2*b*B + 105 3*b^3*B + 192*a^3*C + 2*a*b^2*(1573*A + 1259*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + ((14*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13 *A + 11*C) + 11*a^2*b^2*(637*A + 491*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/ (5*d) + (3*((154*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2* b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*EllipticE[(c + d*x)/2, 2])/d + 195* (77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11 *A + 9*C))*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Si n[c + d*x])/(3*d))))/5)/7)/9)/11)/13
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-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. \(1406\) vs. \(2(452)=904\).
Time = 18.70 (sec) , antiderivative size = 1407, normalized size of antiderivative = 2.95
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1407\) |
| parts | \(\text {Expression too large to display}\) | \(1485\) |
Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_RETURNVERBOSE)
Output:
-2/45045*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520 *C*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(262080*B*b^4+1048320*C*a* b^3+1330560*C*b^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^4 -640640*B*a*b^3-655200*B*b^4-960960*C*a^2*b^2-2620800*C*a*b^3-1798720*C*b^ 4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(411840*A*a*b^3+320320*A*b^4+6 17760*B*a^2*b^2+1281280*B*a*b^3+739440*B*b^4+411840*C*a^3*b+1921920*C*a^2* b^2+2957760*C*a*b^3+1379840*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c) +(-432432*A*a^2*b^2-617760*A*a*b^3-296296*A*b^4-288288*B*a^3*b-926640*B*a^ 2*b^2-1185184*B*a*b^3-453960*B*b^4-72072*C*a^4-617760*C*a^3*b-1777776*C*a^ 2*b^2-1815840*C*a*b^3-666512*C*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c )+(240240*A*a^3*b+432432*A*a^2*b^2+480480*A*a*b^3+136136*A*b^4+60060*B*a^4 +288288*B*a^3*b+720720*B*a^2*b^2+544544*B*a*b^3+180180*B*b^4+72072*C*a^4+4 80480*C*a^3*b+816816*C*a^2*b^2+720720*C*a*b^3+198352*C*b^4)*sin(1/2*d*x+1/ 2*c)^4*cos(1/2*d*x+1/2*c)+(-120120*A*a^3*b-108108*A*a^2*b^2-137280*A*a*b^3 -24024*A*b^4-30030*B*a^4-72072*B*a^3*b-205920*B*a^2*b^2-96096*B*a*b^3-3627 0*B*b^4-18018*C*a^4-137280*C*a^3*b-144144*C*a^2*b^2-145080*C*a*b^3-27258*C *b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+60060*A*a^3*b*(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))+42900*a*A*b^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-45045*A*(sin(1/2*d*...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.15 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
Output:
1/45045*(2*(3465*C*b^4*cos(d*x + c)^5 + 15015*B*a^4 + 8580*(7*A + 5*C)*a^3 *b + 64350*B*a^2*b^2 + 3900*(11*A + 9*C)*a*b^3 + 8775*B*b^4 + 4095*(4*C*a* b^3 + B*b^4)*cos(d*x + c)^4 + 385*(78*C*a^2*b^2 + 52*B*a*b^3 + (13*A + 11* C)*b^4)*cos(d*x + c)^3 + 585*(44*C*a^3*b + 66*B*a^2*b^2 + 4*(11*A + 9*C)*a *b^3 + 9*B*b^4)*cos(d*x + c)^2 + 77*(117*C*a^4 + 468*B*a^3*b + 78*(9*A + 7 *C)*a^2*b^2 + 364*B*a*b^3 + 7*(13*A + 11*C)*b^4)*cos(d*x + c))*sqrt(cos(d* x + c))*sin(d*x + c) - 195*sqrt(2)*(77*I*B*a^4 + 44*I*(7*A + 5*C)*a^3*b + 330*I*B*a^2*b^2 + 20*I*(11*A + 9*C)*a*b^3 + 45*I*B*b^4)*weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 195*sqrt(2)*(-77*I*B*a^4 - 44*I* (7*A + 5*C)*a^3*b - 330*I*B*a^2*b^2 - 20*I*(11*A + 9*C)*a*b^3 - 45*I*B*b^4 )*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*sqrt(2)* (-39*I*(5*A + 3*C)*a^4 - 468*I*B*a^3*b - 78*I*(9*A + 7*C)*a^2*b^2 - 364*I* B*a*b^3 - 7*I*(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*sqrt(2)*(39*I*(5*A + 3*C)*a ^4 + 468*I*B*a^3*b + 78*I*(9*A + 7*C)*a^2*b^2 + 364*I*B*a*b^3 + 7*I*(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c ) - I*sin(d*x + c))))/d
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+ c)**2),x)
Output:
Timed out
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c) ^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)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c) ^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 = 2.14 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.89 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos( c + d*x)^2),x)
Output:
(B*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2 , 2))/3))/d - (136*hypergeom([1/2, 15/4], 23/4, cos(c + d*x)^2)*((11*C*a^4 *cos(c + d*x)^(11/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (9*C*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (42*C*a^2*b^2*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2)))/(21945*d) - (2*hyperg eom([1/2, 15/4], 19/4, cos(c + d*x)^2)*((165*C*a^4*cos(c + d*x)^(7/2)*sin( c + d*x))/(sin(c + d*x)^2)^(1/2) - (52*C*a^4*cos(c + d*x)^(11/2)*sin(c + d *x))/(sin(c + d*x)^2)^(1/2) - (36*C*a^4*cos(c + d*x)^(15/2)*sin(c + d*x))/ (sin(c + d*x)^2)^(1/2) + (77*C*b^4*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin( c + d*x)^2)^(1/2) + (630*C*a^2*b^2*cos(c + d*x)^(11/2)*sin(c + d*x))/(sin( c + d*x)^2)^(1/2) - (168*C*a^2*b^2*cos(c + d*x)^(15/2)*sin(c + d*x))/(sin( c + d*x)^2)^(1/2)))/(1155*d) - (8*hypergeom([1/2, 13/4], 17/4, cos(c + d*x )^2)*((13*C*a^3*b*cos(c + d*x)^(9/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) + (9*C*a*b^3*cos(c + d*x)^(13/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2) - (4 *C*a^3*b*cos(c + d*x)^(13/2)*sin(c + d*x))/(sin(c + d*x)^2)^(1/2)))/(117*d ) + (2*A*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (4*A*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*c os(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*B*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...
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a^{5}+5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a^{4} b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6}d x \right ) b^{4} c +4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) a \,b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) b^{5}+6 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{3} b c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) a^{2} b^{3}+\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{4} c +10 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a^{3} b^{2} \] Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
int(sqrt(cos(c + d*x)),x)*a**5 + 5*int(sqrt(cos(c + d*x))*cos(c + d*x),x)* a**4*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**6,x)*b**4*c + 4*int(sqrt(cos (c + d*x))*cos(c + d*x)**5,x)*a*b**3*c + int(sqrt(cos(c + d*x))*cos(c + d* x)**5,x)*b**5 + 6*int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*a**2*b**2*c + 5*int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*a*b**4 + 4*int(sqrt(cos(c + d* x))*cos(c + d*x)**3,x)*a**3*b*c + 10*int(sqrt(cos(c + d*x))*cos(c + d*x)** 3,x)*a**2*b**3 + int(sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**4*c + 10*int (sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a**3*b**2