Integrand size = 43, antiderivative size = 629 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (520 a^5 b B+3315 a^3 b^3 B+48165 a b^5 B-240 a^6 C+1617 b^6 (13 A+11 C)-10 a^4 b^2 (143 A+76 C)+3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (520 a^4 b B+3705 a^2 b^3 B+8775 b^5 B-240 a^5 C-10 a^3 b^2 (143 A+94 C)+6 a b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (520 a^3 b B+4355 a b^3 B-240 a^4 C+539 b^4 (13 A+11 C)-10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}+\frac {2 \left (104 a^2 b B+1053 b^3 B-48 a^3 C-2 a b^2 (143 A+166 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (143 A b^2-52 a b B+24 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}+\frac {2 (13 b B-6 a C) \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d} \]
2/45045*(520*B*a^3*b+4355*B*a*b^3-240*a^4*C+539*b^4*(13*A+11*C)-10*a^2*b^2 *(143*A+124*C))*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^3/d+2/9009*(104*B*a^2* b+1053*B*b^3-48*a^3*C-2*a*b^2*(143*A+166*C))*(a+b*cos(d*x+c))^(5/2)*sin(d* x+c)/b^3/d+2/1287*(143*A*b^2-52*B*a*b+24*C*a^2+121*C*b^2)*(a+b*cos(d*x+c)) ^(7/2)*sin(d*x+c)/b^3/d+2/143*(13*B*b-6*C*a)*cos(d*x+c)*(a+b*cos(d*x+c))^( 7/2)*sin(d*x+c)/b^2/d+2/13*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c )/b/d+2/45045*(520*B*a^4*b+3705*B*a^2*b^3+8775*B*b^5-240*C*a^5-10*a^3*b^2* (143*A+94*C)+6*a*b^4*(2717*A+2174*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^ 3/d+2/45045*(520*B*a^5*b+3315*B*a^3*b^3+48165*B*a*b^5-240*a^6*C+1617*b^6*( 13*A+11*C)-10*a^4*b^2*(143*A+76*C)+3*a^2*b^4*(13299*A+10223*C))*(cos(1/2*d *x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2) *(b/(a+b))^(1/2))*(a+b*cos(d*x+c))^(1/2)/b^4/d/((a+b*cos(d*x+c))/(a+b))^(1 /2)-2/45045*(a^2-b^2)*(520*B*a^4*b+3705*B*a^2*b^3+8775*B*b^5-240*C*a^5-10* a^3*b^2*(143*A+94*C)+6*a*b^4*(2717*A+2174*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2) /cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))* ((a+b*cos(d*x+c))/(a+b))^(1/2)/b^4/d/(a+b*cos(d*x+c))^(1/2)
Time = 5.49 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.80 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {32 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (130 a^4 b B+43095 a^2 b^3 B+8775 b^5 B-60 a^5 C+5 a^3 b^2 (4433 A+3337 C)+3 a b^4 (12441 A+10277 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (-520 a^5 b B-3315 a^3 b^3 B-48165 a b^5 B+240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (4 \left (-2080 a^4 b B+121290 a^2 b^3 B+84825 b^5 B+960 a^5 C+10 a^3 b^2 (572 A+331 C)+3 a b^4 (71214 A+60793 C)\right ) \sin (c+d x)+b \left (\left (3120 a^3 b B+321880 a b^3 B-1440 a^4 C+120 a^2 b^2 (1430 A+1457 C)+77 b^4 (1976 A+1897 C)\right ) \sin (2 (c+d x))+5 b \left (2 \left (5876 a^2 b B+6669 b^3 B+60 a^3 C+a b^2 (10868 A+13939 C)\right ) \sin (3 (c+d x))+7 b \left (4 \left (143 A b^2+299 a b B+159 a^2 C+220 b^2 C\right ) \sin (4 (c+d x))+9 b ((26 b B+54 a C) \sin (5 (c+d x))+11 b C \sin (6 (c+d x)))\right )\right )\right )\right )}{720720 b^4 d \sqrt {a+b \cos (c+d x)}} \]
Integrate[Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
(32*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(130*a^4*b*B + 43095*a^2*b^3*B + 8775*b^5*B - 60*a^5*C + 5*a^3*b^2*(4433*A + 3337*C) + 3*a*b^4*(12441*A + 10277*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] - (-520*a^5*b*B - 3315*a ^3*b^3*B - 48165*a*b^5*B + 240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2 *(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*((a + b)*EllipticE[(c + d *x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b*Cos[c + d*x])*(4*(-2080*a^4*b*B + 121290*a^2*b^3*B + 84825*b^5*B + 960* a^5*C + 10*a^3*b^2*(572*A + 331*C) + 3*a*b^4*(71214*A + 60793*C))*Sin[c + d*x] + b*((3120*a^3*b*B + 321880*a*b^3*B - 1440*a^4*C + 120*a^2*b^2*(1430* A + 1457*C) + 77*b^4*(1976*A + 1897*C))*Sin[2*(c + d*x)] + 5*b*(2*(5876*a^ 2*b*B + 6669*b^3*B + 60*a^3*C + a*b^2*(10868*A + 13939*C))*Sin[3*(c + d*x) ] + 7*b*(4*(143*A*b^2 + 299*a*b*B + 159*a^2*C + 220*b^2*C)*Sin[4*(c + d*x) ] + 9*b*((26*b*B + 54*a*C)*Sin[5*(c + d*x)] + 11*b*C*Sin[6*(c + d*x)]))))) )/(720720*b^4*d*Sqrt[a + b*Cos[c + d*x]])
Time = 3.67 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.04, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.628, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left ((13 b B-6 a C) \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+4 a C\right )dx}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left ((13 b B-6 a C) \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+4 a C\right )dx}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left ((13 b B-6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (13 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a C\right )dx}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (\left (24 C a^2-52 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+b (117 b B-10 a C) \cos (c+d x)+2 a (13 b B-6 a C)\right )dx}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int (a+b \cos (c+d x))^{5/2} \left (\left (24 C a^2-52 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+b (117 b B-10 a C) \cos (c+d x)+2 a (13 b B-6 a C)\right )dx}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (\left (24 C a^2-52 b B a+143 A b^2+121 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (117 b B-10 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a (13 b B-6 a C)\right )dx}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (b \left (60 C a^2-130 b B a+1001 A b^2+847 b^2 C\right )+\left (-48 C a^3+104 b B a^2-2 b^2 (143 A+166 C) a+1053 b^3 B\right ) \cos (c+d x)\right )dx}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int (a+b \cos (c+d x))^{5/2} \left (b \left (60 C a^2-130 b B a+1001 A b^2+847 b^2 C\right )+\left (-48 C a^3+104 b B a^2-2 b^2 (143 A+166 C) a+1053 b^3 B\right ) \cos (c+d x)\right )dx}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (b \left (60 C a^2-130 b B a+1001 A b^2+847 b^2 C\right )+\left (-48 C a^3+104 b B a^2-2 b^2 (143 A+166 C) a+1053 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {\frac {2}{7} \int -\frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 b \left (-60 C a^3+130 b B a^2-b^2 (1859 A+1423 C) a-1755 b^3 B\right )-\left (-240 C a^4+520 b B a^3-10 b^2 (143 A+124 C) a^2+4355 b^3 B a+539 b^4 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}-\frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (3 b \left (-60 C a^3+130 b B a^2-b^2 (1859 A+1423 C) a-1755 b^3 B\right )-\left (-240 C a^4+520 b B a^3-10 b^2 (143 A+124 C) a^2+4355 b^3 B a+539 b^4 (13 A+11 C)\right ) \cos (c+d x)\right )dx}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}-\frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (-60 C a^3+130 b B a^2-b^2 (1859 A+1423 C) a-1755 b^3 B\right )+\left (240 C a^4-520 b B a^3+10 b^2 (143 A+124 C) a^2-4355 b^3 B a-539 b^4 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-60 C a^4+130 b B a^3-5 b^2 (1573 A+1175 C) a^2-13130 b^3 B a-539 b^4 (13 A+11 C)\right )-\left (-240 C a^5+520 b B a^4-10 b^2 (143 A+94 C) a^3+3705 b^3 B a^2+6 b^4 (2717 A+2174 C) a+8775 b^5 B\right ) \cos (c+d x)\right )dx\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (-60 C a^4+130 b B a^3-5 b^2 (1573 A+1175 C) a^2-13130 b^3 B a-539 b^4 (13 A+11 C)\right )-\left (-240 C a^5+520 b B a^4-10 b^2 (143 A+94 C) a^3+3705 b^3 B a^2+6 b^4 (2717 A+2174 C) a+8775 b^5 B\right ) \cos (c+d x)\right )dx\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-60 C a^4+130 b B a^3-5 b^2 (1573 A+1175 C) a^2-13130 b^3 B a-539 b^4 (13 A+11 C)\right )+\left (240 C a^5-520 b B a^4+10 b^2 (143 A+94 C) a^3-3705 b^3 B a^2-6 b^4 (2717 A+2174 C) a-8775 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {2}{3} \int -\frac {b \left (-60 C a^5+130 b B a^4+5 b^2 (4433 A+3337 C) a^3+43095 b^3 B a^2+3 b^4 (12441 A+10277 C) a+8775 b^5 B\right )+\left (-240 C a^6+520 b B a^5-10 b^2 (143 A+76 C) a^4+3315 b^3 B a^3+3 b^4 (13299 A+10223 C) a^2+48165 b^5 B a+1617 b^6 (13 A+11 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (-60 C a^5+130 b B a^4+5 b^2 (4433 A+3337 C) a^3+43095 b^3 B a^2+3 b^4 (12441 A+10277 C) a+8775 b^5 B\right )+\left (-240 C a^6+520 b B a^5-10 b^2 (143 A+76 C) a^4+3315 b^3 B a^3+3 b^4 (13299 A+10223 C) a^2+48165 b^5 B a+1617 b^6 (13 A+11 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (-60 C a^5+130 b B a^4+5 b^2 (4433 A+3337 C) a^3+43095 b^3 B a^2+3 b^4 (12441 A+10277 C) a+8775 b^5 B\right )+\left (-240 C a^6+520 b B a^5-10 b^2 (143 A+76 C) a^4+3315 b^3 B a^3+3 b^4 (13299 A+10223 C) a^2+48165 b^5 B a+1617 b^6 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )+\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}+\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 \sin (c+d x) \left (24 a^2 C-52 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^{7/2}}{9 b d}+\frac {\frac {2 \sin (c+d x) \left (-48 a^3 C+104 a^2 b B-2 a b^2 (143 A+166 C)+1053 b^3 B\right ) (a+b \cos (c+d x))^{5/2}}{7 d}+\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (-240 a^4 C+520 a^3 b B-10 a^2 b^2 (143 A+124 C)+4355 a b^3 B+539 b^4 (13 A+11 C)\right ) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (-240 a^6 C+520 a^5 b B-10 a^4 b^2 (143 A+76 C)+3315 a^3 b^3 B+3 a^2 b^4 (13299 A+10223 C)+48165 a b^5 B+1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \sin (c+d x) \left (-240 a^5 C+520 a^4 b B-10 a^3 b^2 (143 A+94 C)+3705 a^2 b^3 B+6 a b^4 (2717 A+2174 C)+8775 b^5 B\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )}{9 b}}{11 b}+\frac {2 (13 b B-6 a C) \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
(2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b*d) + (( 2*(13*b*B - 6*a*C)*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/( 11*b*d) + ((2*(143*A*b^2 - 52*a*b*B + 24*a^2*C + 121*b^2*C)*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d) + ((2*(104*a^2*b*B + 1053*b^3*B - 48*a^ 3*C - 2*a*b^2*(143*A + 166*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7 *d) + ((2*(520*a^3*b*B + 4355*a*b^3*B - 240*a^4*C + 539*b^4*(13*A + 11*C) - 10*a^2*b^2*(143*A + 124*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5* d) - (3*(((-2*(520*a^5*b*B + 3315*a^3*b^3*B + 48165*a*b^5*B - 240*a^6*C + 1617*b^6*(13*A + 11*C) - 10*a^4*b^2*(143*A + 76*C) + 3*a^2*b^4*(13299*A + 10223*C))*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/ (b*d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) + (2*(a^2 - b^2)*(520*a^4*b*B + 3 705*a^2*b^3*B + 8775*b^5*B - 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6*a*b ^4*(2717*A + 2174*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d* x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/3 - (2*(520*a^4*b*B + 3705*a^2*b^3*B + 8775*b^5*B - 240*a^5*C - 10*a^3*b^2*(143*A + 94*C) + 6* a*b^4*(2717*A + 2174*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5) /7)/(9*b))/(11*b))/(13*b)
3.11.29.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
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_)])^(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. \(3164\) vs. \(2(651)=1302\).
Time = 63.82 (sec) , antiderivative size = 3165, normalized size of antiderivative = 5.03
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3165\) |
| parts | \(\text {Expression too large to display}\) | \(3504\) |
int(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, method=_RETURNVERBOSE)
-2/45045*((2*cos(1/2*d*x+1/2*c)^2*b+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-143 0*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a -b))^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b^2-17732* A*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/ (a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4-13984*a^ 3*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a -b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^4+16302*a*A* b^6*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a -b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-760*C*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*E llipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b^2-520*B*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ellip ticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^6*b+21021*A*(sin(1/2*d*x+1/2 *c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE (cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^6-39897*A*(sin(1/2*d*x+1/2*c)^ 2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos (1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^5+1430*A*(sin(1/2*d*x+1/2*c)^2)^ (1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/ 2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^5*b^2+48165*B*(sin(1/2*d*x+1/2*c)^2)^(1 /2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.32 (sec) , antiderivative size = 929, normalized size of antiderivative = 1.48 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="fricas")
1/135135*(sqrt(2)*(-480*I*C*a^7 + 1040*I*B*a^6*b - 20*I*(143*A + 67*C)*a^5 *b^2 + 6240*I*B*a^4*b^3 + 3*I*(4433*A + 3761*C)*a^3*b^4 - 32955*I*B*a^2*b^ 5 - 3*I*(23309*A + 18973*C)*a*b^6 - 26325*I*B*b^7)*sqrt(b)*weierstrassPInv erse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d* x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(2)*(480*I*C*a^7 - 1040*I*B*a^ 6*b + 20*I*(143*A + 67*C)*a^5*b^2 - 6240*I*B*a^4*b^3 - 3*I*(4433*A + 3761* C)*a^3*b^4 + 32955*I*B*a^2*b^5 + 3*I*(23309*A + 18973*C)*a*b^6 + 26325*I*B *b^7)*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3*sqr t(2)*(240*I*C*a^6*b - 520*I*B*a^5*b^2 + 10*I*(143*A + 76*C)*a^4*b^3 - 3315 *I*B*a^3*b^4 - 3*I*(13299*A + 10223*C)*a^2*b^5 - 48165*I*B*a*b^6 - 1617*I* (13*A + 11*C)*b^7)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27* (8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27* (8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b )) - 3*sqrt(2)*(-240*I*C*a^6*b + 520*I*B*a^5*b^2 - 10*I*(143*A + 76*C)*a^4 *b^3 + 3315*I*B*a^3*b^4 + 3*I*(13299*A + 10223*C)*a^2*b^5 + 48165*I*B*a*b^ 6 + 1617*I*(13*A + 11*C)*b^7)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/ b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/ b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) + 6*(3465*C*b^7*cos(d*x + c)^5 + 120*C*a^5*b^2 - 260*B*a^...
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*cos(d*x + c)^2, x)
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
integrate(cos(d*x+c)^2*(a+b*cos(d*x+c))^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c) ^2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(5/ 2)*cos(d*x + c)^2, x)
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]