Integrand size = 33, antiderivative size = 435 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{693 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (8 a^4 C+15 b^4 (11 A+9 C)+3 a^2 b^2 (33 A+19 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 b^2 d}+\frac {2 a \left (99 A b^2+8 a^2 C+67 b^2 C\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{693 b^2 d}+\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{693 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{99 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{11 b d} \]
2/693*a*(99*A*b^2+8*C*a^2+67*C*b^2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/ d+2/693*(8*a^2*C+9*b^2*(11*A+9*C))*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^2/d -8/99*a*C*(a+b*cos(d*x+c))^(7/2)*sin(d*x+c)/b^2/d+2/11*C*cos(d*x+c)*(a+b*c os(d*x+c))^(7/2)*sin(d*x+c)/b/d+2/693*(8*a^4*C+15*b^4*(11*A+9*C)+3*a^2*b^2 *(33*A+19*C))*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2/d+2/693*a*(8*a^4*C+3*a ^2*b^2*(33*A+17*C)+3*b^4*(319*A+247*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*c os(d*x+c))^(1/2)/b^3/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/693*(a^2-b^2)*(8*a ^4*C+15*b^4*(11*A+9*C)+3*a^2*b^2*(33*A+19*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^3/d/(a+b*cos(d*x+c))^(1/2)
Time = 2.94 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.75 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {16 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (2 a^4 b C+15 b^5 (11 A+9 C)+3 a^2 b^3 (297 A+221 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (-64 a^4 C+12 a^2 b^2 (396 A+311 C)+6 b^4 (506 A+435 C)\right ) \sin (c+d x)+b \left (4 a \left (594 A b^2+6 a^2 C+619 b^2 C\right ) \sin (2 (c+d x))+b \left (\left (396 A b^2+452 a^2 C+513 b^2 C\right ) \sin (3 (c+d x))+7 b C (46 a \sin (4 (c+d x))+9 b \sin (5 (c+d x)))\right )\right )\right )}{5544 b^3 d \sqrt {a+b \cos (c+d x)}} \]
(16*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^4*b*C + 15*b^5*(11*A + 9*C) + 3*a^2*b^3*(297*A + 221*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + a*(8 *a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*((a + b)*Ellipti cE[(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])*((-64*a^4*C + 12*a^2*b^2*(396*A + 311*C) + 6*b^4 *(506*A + 435*C))*Sin[c + d*x] + b*(4*a*(594*A*b^2 + 6*a^2*C + 619*b^2*C)* Sin[2*(c + d*x)] + b*((396*A*b^2 + 452*a^2*C + 513*b^2*C)*Sin[3*(c + d*x)] + 7*b*C*(46*a*Sin[4*(c + d*x)] + 9*b*Sin[5*(c + d*x)])))))/(5544*b^3*d*Sq rt[a + b*Cos[c + d*x]])
Time = 2.55 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.04, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {3042, 3529, 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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (-4 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+2 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{5/2} \left (-4 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+2 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (-4 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int -\frac {1}{2} (a+b \cos (c+d x))^{5/2} \left (10 a b C-\left (8 C a^2+9 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int (a+b \cos (c+d x))^{5/2} \left (10 a b C-\left (8 C a^2+9 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (10 a b C+\left (-8 C a^2-9 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {\frac {2}{7} \int -\frac {5}{2} (a+b \cos (c+d x))^{3/2} \left (3 b \left (-2 C a^2+33 A b^2+27 b^2 C\right )+a \left (8 C a^2+99 A b^2+67 b^2 C\right ) \cos (c+d x)\right )dx-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \int (a+b \cos (c+d x))^{3/2} \left (3 b \left (-2 C a^2+33 A b^2+27 b^2 C\right )+a \left (8 C a^2+99 A b^2+67 b^2 C\right ) \cos (c+d x)\right )dx-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (3 b \left (-2 C a^2+33 A b^2+27 b^2 C\right )+a \left (8 C a^2+99 A b^2+67 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2}{5} \int -\frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (2 a b \left (a^2 C-b^2 (132 A+101 C)\right )-\left (8 C a^4+3 b^2 (33 A+19 C) a^2+15 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (2 a b \left (a^2 C-b^2 (132 A+101 C)\right )-\left (8 C a^4+3 b^2 (33 A+19 C) a^2+15 b^4 (11 A+9 C)\right ) \cos (c+d x)\right )dx\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (2 a b \left (a^2 C-b^2 (132 A+101 C)\right )+\left (-8 C a^4-3 b^2 (33 A+19 C) a^2-15 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {2}{3} \int -\frac {b \left (2 C a^4+3 b^2 (297 A+221 C) a^2+15 b^4 (11 A+9 C)\right )+a \left (8 C a^4+3 b^2 (33 A+17 C) a^2+3 b^4 (319 A+247 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (2 C a^4+3 b^2 (297 A+221 C) a^2+15 b^4 (11 A+9 C)\right )+a \left (8 C a^4+3 b^2 (33 A+17 C) a^2+3 b^4 (319 A+247 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (-\frac {1}{3} \int \frac {b \left (2 C a^4+3 b^2 (297 A+221 C) a^2+15 b^4 (11 A+9 C)\right )+a \left (8 C a^4+3 b^2 (33 A+17 C) a^2+3 b^4 (319 A+247 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {-\frac {2 \left (8 a^2 C+9 b^2 (11 A+9 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}-\frac {5}{7} \left (\frac {2 a \left (8 a^2 C+99 A b^2+67 b^2 C\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (a^2-b^2\right ) \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}-\frac {2 a \left (8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\right )-\frac {2 \left (8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )\right )}{9 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{9 b d}}{11 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{7/2}}{11 b d}\) |
(2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(11*b*d) + ((-8 *a*C*(a + b*Cos[c + d*x])^(7/2)*Sin[c + d*x])/(9*b*d) - ((-2*(8*a^2*C + 9* b^2*(11*A + 9*C))*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) - (5*((2* a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x]) /(5*d) - (3*(((-2*a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 24 7*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)*(8*a^4*C + 15*b^4*(1 1*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*E llipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/3 - (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Co s[c + d*x]]*Sin[c + d*x])/(3*d)))/5))/7)/(9*b))/(11*b)
3.7.39.3.1 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_.) + (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. \(1790\) vs. \(2(461)=922\).
Time = 32.21 (sec) , antiderivative size = 1791, normalized size of antiderivative = 4.12
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1791\) |
| parts | \(\text {Expression too large to display}\) | \(1969\) |
-2/693*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4032*C *cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12*b^6+(-7168*C*a*b^5-10080*C*b^6)* sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(1584*A*b^6+4384*C*a^2*b^4+14336* C*a*b^5+11376*C*b^6)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-3168*A*a*b^ 5-2376*A*b^6-928*C*a^3*b^3-6576*C*a^2*b^4-13232*C*a*b^5-6984*C*b^6)*sin(1/ 2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(2376*A*a^2*b^4+3168*A*a*b^5+1848*A*b^6- 4*C*a^4*b^2+928*C*a^3*b^3+5024*C*a^2*b^4+6064*C*a*b^5+2772*C*b^6)*sin(1/2* d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-594*A*a^3*b^3-1188*A*a^2*b^4-1122*A*a*b^ 5-528*A*b^6+8*C*a^5*b+2*C*a^4*b^2-642*C*a^3*b^3-1416*C*a^2*b^4-1338*C*a*b^ 5-558*C*b^6)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-99*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^4*b^2-66*A*a^2*(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+165*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))+99*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^4*b^2-99*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^3*b^3+957*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.53 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} {\left (16 i \, C a^{6} + 6 i \, {\left (33 \, A + 16 \, C\right )} a^{4} b^{2} - 3 i \, {\left (253 \, A + 169 \, C\right )} a^{2} b^{4} - 45 i \, {\left (11 \, A + 9 \, C\right )} b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + \sqrt {2} {\left (-16 i \, C a^{6} - 6 i \, {\left (33 \, A + 16 \, C\right )} a^{4} b^{2} + 3 i \, {\left (253 \, A + 169 \, C\right )} a^{2} b^{4} + 45 i \, {\left (11 \, A + 9 \, C\right )} b^{6}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {2} {\left (-8 i \, C a^{5} b - 3 i \, {\left (33 \, A + 17 \, C\right )} a^{3} b^{3} - 3 i \, {\left (319 \, A + 247 \, C\right )} a b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {2} {\left (8 i \, C a^{5} b + 3 i \, {\left (33 \, A + 17 \, C\right )} a^{3} b^{3} + 3 i \, {\left (319 \, A + 247 \, C\right )} a b^{5}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) + 6 \, {\left (63 \, C b^{6} \cos \left (d x + c\right )^{4} + 161 \, C a b^{5} \cos \left (d x + c\right )^{3} - 4 \, C a^{4} b^{2} + {\left (297 \, A + 205 \, C\right )} a^{2} b^{4} + 15 \, {\left (11 \, A + 9 \, C\right )} b^{6} + {\left (113 \, C a^{2} b^{4} + 9 \, {\left (11 \, A + 9 \, C\right )} b^{6}\right )} \cos \left (d x + c\right )^{2} + {\left (3 \, C a^{3} b^{3} + {\left (297 \, A + 229 \, C\right )} a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{2079 \, b^{4} d} \]
1/2079*(sqrt(2)*(16*I*C*a^6 + 6*I*(33*A + 16*C)*a^4*b^2 - 3*I*(253*A + 169 *C)*a^2*b^4 - 45*I*(11*A + 9*C)*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) + sqrt(2)*(-16*I*C*a^6 - 6*I*(33*A + 16*C)*a^4*b^2 + 3*I*(253*A + 169*C)*a^2*b^4 + 45*I*(11*A + 9*C)*b^6)*sqrt(b)*weierstras sPInverse(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*c os(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) - 3*sqrt(2)*(-8*I*C*a^5*b - 3*I *(33*A + 17*C)*a^3*b^3 - 3*I*(319*A + 247*C)*a*b^5)*sqrt(b)*weierstrassZet a(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInvers e(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)*(8*I*C*a^5*b + 3*I*(33*A + 17*C)*a^3*b^3 + 3*I*(319*A + 247*C)*a*b^5)*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*(63*C*b^6*cos(d*x + c)^4 + 161*C*a*b^5*cos (d*x + c)^3 - 4*C*a^4*b^2 + (297*A + 205*C)*a^2*b^4 + 15*(11*A + 9*C)*b^6 + (113*C*a^2*b^4 + 9*(11*A + 9*C)*b^6)*cos(d*x + c)^2 + (3*C*a^3*b^3 + (29 7*A + 229*C)*a*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/( b^4*d)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2} \,d x \]