Integrand size = 35, antiderivative size = 523 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {4 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{45045 b^4 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{45045 b^3 d}-\frac {2 \left (240 a^4 C-539 b^4 (13 A+11 C)+10 a^2 b^2 (143 A+124 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{45045 b^3 d}-\frac {4 a \left (143 A b^2+24 a^2 C+166 b^2 C\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9009 b^3 d}+\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{1287 b^3 d}-\frac {12 a C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{143 b^2 d}+\frac {2 C \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{13 b d} \]
-2/45045*(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-4/9009*a*(143*A*b^2+24*C*a^2+166*C*b^2)*(a +b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^3/d+2/1287*(24*a^2*C+11*b^2*(13*A+11*C)) *(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b^3/d-12/143*a*C*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)*si n(d*x+c)/b/d-4/45045*a*(120*a^4*C+5*a^2*b^2*(143*A+94*C)-3*b^4*(2717*A+217 4*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^3/d-2/45045*(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)+4/45045*a*(a^2-b^2)*(120*a^4*C+5*a^2*b^2*(143*A+94*C)-3*b^4*(2717*A+2 174*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 = 4.13 (sec) , antiderivative size = 395, normalized size of antiderivative = 0.76 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {32 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (a b^2 \left (-60 a^4 C+5 a^2 b^2 (4433 A+3337 C)+3 b^4 (12441 A+10277 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )-\left (240 a^6 C-1617 b^6 (13 A+11 C)+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (4 a \left (960 a^4 C+10 a^2 b^2 (572 A+331 C)+3 b^4 (71214 A+60793 C)\right ) \sin (c+d x)+b \left (\left (-1440 a^4 C+120 a^2 b^2 (1430 A+1457 C)+77 b^4 (1976 A+1897 C)\right ) \sin (2 (c+d x))+5 b \left (2 a \left (10868 A b^2+60 a^2 C+13939 b^2 C\right ) \sin (3 (c+d x))+7 b \left (\left (572 A b^2+636 a^2 C+880 b^2 C\right ) \sin (4 (c+d x))+9 b C (54 a \sin (5 (c+d x))+11 b \sin (6 (c+d x)))\right )\right )\right )\right )}{720720 b^4 d \sqrt {a+b \cos (c+d x)}} \]
(32*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(a*b^2*(-60*a^4*C + 5*a^2*b^2*(4433 *A + 3337*C) + 3*b^4*(12441*A + 10277*C))*EllipticF[(c + d*x)/2, (2*b)/(a + 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*a*(960*a^4*C + 10*a^2*b^2*(572*A + 331*C) + 3*b^4*(71214*A + 60793*C))*S in[c + d*x] + b*((-1440*a^4*C + 120*a^2*b^2*(1430*A + 1457*C) + 77*b^4*(19 76*A + 1897*C))*Sin[2*(c + d*x)] + 5*b*(2*a*(10868*A*b^2 + 60*a^2*C + 1393 9*b^2*C)*Sin[3*(c + d*x)] + 7*b*((572*A*b^2 + 636*a^2*C + 880*b^2*C)*Sin[4 *(c + d*x)] + 9*b*C*(54*a*Sin[5*(c + d*x)] + 11*b*Sin[6*(c + d*x)]))))))/( 720720*b^4*d*Sqrt[a + b*Cos[c + d*x]])
Time = 3.35 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.05, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.771, Rules used = {3042, 3529, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (-6 a C \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+4 a C\right )dx}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (-6 a C \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+4 a C\right )dx}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (-6 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (13 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a C\right )dx}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int -\frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (12 C a^2+10 b C \cos (c+d x) a-\left (24 C a^2+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int (a+b \cos (c+d x))^{5/2} \left (12 C a^2+10 b C \cos (c+d x) a-\left (24 C a^2+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (12 C a^2+10 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-24 C a^2-11 b^2 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {-\frac {\frac {2 \int -\frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (b \left (60 C a^2+1001 A b^2+847 b^2 C\right )-2 a \left (24 C a^2+143 A b^2+166 b^2 C\right ) \cos (c+d x)\right )dx}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int (a+b \cos (c+d x))^{5/2} \left (b \left (60 C a^2+1001 A b^2+847 b^2 C\right )-2 a \left (24 C a^2+143 A b^2+166 b^2 C\right ) \cos (c+d x)\right )dx}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (b \left (60 C a^2+1001 A b^2+847 b^2 C\right )-2 a \left (24 C a^2+143 A b^2+166 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (3 a b \left (60 C a^2+1859 A b^2+1423 b^2 C\right )-\left (240 C a^4+10 b^2 (143 A+124 C) a^2-539 b^4 (13 A+11 C)\right ) \cos (c+d x)\right )dx-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left (3 a b \left (60 C a^2+1859 A b^2+1423 b^2 C\right )-\left (240 C a^4+10 b^2 (143 A+124 C) a^2-539 b^4 (13 A+11 C)\right ) \cos (c+d x)\right )dx-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 a b \left (60 C a^2+1859 A b^2+1423 b^2 C\right )+\left (-240 C a^4-10 b^2 (143 A+124 C) a^2+539 b^4 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (60 C a^4+5 b^2 (1573 A+1175 C) a^2+539 b^4 (13 A+11 C)\right )-2 a \left (120 C a^4+5 b^2 (143 A+94 C) a^2-3 b^4 (2717 A+2174 C)\right ) \cos (c+d x)\right )dx-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (60 C a^4+5 b^2 (1573 A+1175 C) a^2+539 b^4 (13 A+11 C)\right )-2 a \left (120 C a^4+5 b^2 (143 A+94 C) a^2-3 b^4 (2717 A+2174 C)\right ) \cos (c+d x)\right )dx-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (60 C a^4+5 b^2 (1573 A+1175 C) a^2+539 b^4 (13 A+11 C)\right )-2 a \left (120 C a^4+5 b^2 (143 A+94 C) a^2-3 b^4 (2717 A+2174 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int -\frac {a b \left (60 C a^4-5 b^2 (4433 A+3337 C) a^2-3 b^4 (12441 A+10277 C)\right )+\left (240 C a^6+10 b^2 (143 A+76 C) a^4-3 b^4 (13299 A+10223 C) a^2-1617 b^6 (13 A+11 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (-\frac {1}{3} \int \frac {a b \left (60 C a^4-5 b^2 (4433 A+3337 C) a^2-3 b^4 (12441 A+10277 C)\right )+\left (240 C a^6+10 b^2 (143 A+76 C) a^4-3 b^4 (13299 A+10223 C) a^2-1617 b^6 (13 A+11 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (-\frac {1}{3} \int \frac {a b \left (60 C a^4-5 b^2 (4433 A+3337 C) a^2-3 b^4 (12441 A+10277 C)\right )+\left (240 C a^6+10 b^2 (143 A+76 C) a^4-3 b^4 (13299 A+10223 C) a^2-1617 b^6 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {\left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {-\frac {2 \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}-\frac {\frac {1}{7} \left (\frac {3}{5} \left (\frac {1}{3} \left (\frac {4 a \left (a^2-b^2\right ) \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {4 a \left (120 a^4 C+5 a^2 b^2 (143 A+94 C)-3 b^4 (2717 A+2174 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (240 a^4 C+10 a^2 b^2 (143 A+124 C)-539 b^4 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {4 a \left (24 a^2 C+143 A b^2+166 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}}{11 b}-\frac {12 a C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}}{13 b}+\frac {2 C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^{7/2}}{13 b d}\) |
(2*C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(13*b*d) + (( -12*a*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d) - ( (-2*(24*a^2*C + 11*b^2*(13*A + 11*C))*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d *x])/(9*b*d) - ((-4*a*(143*A*b^2 + 24*a^2*C + 166*b^2*C)*(a + b*Cos[c + d* x])^(5/2)*Sin[c + d*x])/(7*d) + ((-2*(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*(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)]) + (4*a*( a^2 - b^2)*(120*a^4*C + 5*a^2*b^2*(143*A + 94*C) - 3*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 - (4*a*(120*a^4*C + 5*a^2*b^2*(143*A + 94*C) - 3*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)
3.7.38.3.1 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])))
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : > Simp[(-C)*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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, 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. \(2222\) vs. \(2(545)=1090\).
Time = 53.59 (sec) , antiderivative size = 2223, normalized size of antiderivative = 4.25
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2223\) |
| parts | \(\text {Expression too large to display}\) | \(2364\) |
-2/45045*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-210 21*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))*b^7-17787*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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^7+240*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)*El lipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^7-240*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)*EllipticE( cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^7+13044*C*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))+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+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+700*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))*a^5*b^2+240*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)*EllipticE(cos(1/2*d*x+1/2*c),(-2* b/(a-b))^(1/2))*a^6*b+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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.43 \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
1/135135*(sqrt(2)*(-480*I*C*a^7 - 20*I*(143*A + 67*C)*a^5*b^2 + 3*I*(4433* A + 3761*C)*a^3*b^4 - 3*I*(23309*A + 18973*C)*a*b^6)*sqrt(b)*weierstrassPI nverse(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 + 20*I*(143 *A + 67*C)*a^5*b^2 - 3*I*(4433*A + 3761*C)*a^3*b^4 + 3*I*(23309*A + 18973* C)*a*b^6)*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 *sqrt(2)*(240*I*C*a^6*b + 10*I*(143*A + 76*C)*a^4*b^3 - 3*I*(13299*A + 102 23*C)*a^2*b^5 - 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 - 10*I*(143*A + 76*C )*a^4*b^3 + 3*I*(13299*A + 10223*C)*a^2*b^5 + 1617*I*(13*A + 11*C)*b^7)*sq rt(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*co s(d*x + c)^5 + 8505*C*a*b^6*cos(d*x + c)^4 + 120*C*a^5*b^2 + 5*(143*A + 79 *C)*a^3*b^4 + (23309*A + 18973*C)*a*b^6 + 35*(159*C*a^2*b^5 + 11*(13*A + 1 1*C)*b^7)*cos(d*x + c)^3 + 5*(15*C*a^3*b^4 + (2717*A + 2209*C)*a*b^6)*cos( d*x + c)^2 - (90*C*a^4*b^3 - 15*(715*A + 543*C)*a^2*b^5 - 539*(13*A + 1...
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
\[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
Timed out. \[ \int \cos ^2(c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]