Integrand size = 33, antiderivative size = 356 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^3 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a \left (63 A b^2+8 a^2 C+39 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^2 d}+\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{315 b^2 d}-\frac {8 a C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos (c+d x) (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{9 b d} \]
2/315*(8*a^2*C+7*b^2*(9*A+7*C))*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b^2/d-8/ 63*a*C*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b^2/d+2/9*C*cos(d*x+c)*(a+b*cos(d *x+c))^(5/2)*sin(d*x+c)/b/d+2/315*a*(63*A*b^2+8*C*a^2+39*C*b^2)*sin(d*x+c) *(a+b*cos(d*x+c))^(1/2)/b^2/d+2/315*(8*a^4*C+21*b^4*(9*A+7*C)+3*a^2*b^2*(2 1*A+11*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^3/d/((a+b* cos(d*x+c))/(a+b))^(1/2)-2/315*a*(a^2-b^2)*(63*A*b^2+8*C*a^2+39*C*b^2)*(co s(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 = 2.53 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.76 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (2 a b^2 \left (126 A b^2+\left (a^2+93 b^2\right ) C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (8 a^4 C+21 b^4 (9 A+7 C)+3 a^2 b^2 (21 A+11 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (-4 a \left (-252 A b^2+8 a^2 C-201 b^2 C\right ) \sin (c+d x)+b \left (2 \left (126 A b^2+6 a^2 C+133 b^2 C\right ) \sin (2 (c+d x))+5 b C (20 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right )\right )}{1260 b^3 d \sqrt {a+b \cos (c+d x)}} \]
(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(2*a*b^2*(126*A*b^2 + (a^2 + 93*b^2) *C)*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*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 *(-252*A*b^2 + 8*a^2*C - 201*b^2*C)*Sin[c + d*x] + b*(2*(126*A*b^2 + 6*a^2 *C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(20*a*Sin[3*(c + d*x)] + 7*b*Sin[ 4*(c + d*x)]))))/(1260*b^3*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.92 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.04, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3529, 27, 3042, 3502, 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.
| \(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (-4 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{3/2} \left (-4 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-4 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a C\right )dx}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2 \int -\frac {1}{2} (a+b \cos (c+d x))^{3/2} \left (6 a b C-\left (8 C a^2+7 b^2 (9 A+7 C)\right ) \cos (c+d x)\right )dx}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int (a+b \cos (c+d x))^{3/2} \left (6 a b C-\left (8 C a^2+7 b^2 (9 A+7 C)\right ) \cos (c+d x)\right )dx}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (6 a b C+\left (-8 C a^2-7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {\frac {2}{5} \int -\frac {3}{2} \sqrt {a+b \cos (c+d x)} \left (b \left (-2 C a^2+63 A b^2+49 b^2 C\right )+a \left (8 C a^2+63 A b^2+39 b^2 C\right ) \cos (c+d x)\right )dx-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \int \sqrt {a+b \cos (c+d x)} \left (b \left (-2 C a^2+63 A b^2+49 b^2 C\right )+a \left (8 C a^2+63 A b^2+39 b^2 C\right ) \cos (c+d x)\right )dx-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b \left (-2 C a^2+63 A b^2+49 b^2 C\right )+a \left (8 C a^2+63 A b^2+39 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {2}{3} \int \frac {2 a b \left (126 A b^2+\left (a^2+93 b^2\right ) C\right )+\left (8 C a^4+3 b^2 (21 A+11 C) a^2+21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \int \frac {2 a b \left (126 A b^2+\left (a^2+93 b^2\right ) C\right )+\left (8 C a^4+3 b^2 (21 A+11 C) a^2+21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \int \frac {2 a b \left (126 A b^2+\left (a^2+93 b^2\right ) C\right )+\left (8 C a^4+3 b^2 (21 A+11 C) a^2+21 b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {\left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {3}{5} \left (\frac {1}{3} \left (\frac {2 \left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {-\frac {2 \left (8 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}-\frac {3}{5} \left (\frac {2 a \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (a^2-b^2\right ) \left (8 a^2 C+63 A b^2+39 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}\right )\right )}{7 b}-\frac {8 a C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos (c+d x) (a+b \cos (c+d x))^{5/2}}{9 b d}\) |
(2*C*Cos[c + d*x]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(9*b*d) + ((-8* a*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d) - ((-2*(8*a^2*C + 7*b ^2*(9*A + 7*C))*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) - (3*(((2*( 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*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*(a^2 - b^2)*(63*A*b^2 + 8*a^2*C + 39*b^2*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*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b* Cos[c + d*x]]*Sin[c + d*x])/(3*d)))/5)/(7*b))/(9*b)
3.7.31.3.1 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. \(1526\) vs. \(2(386)=772\).
Time = 23.26 (sec) , antiderivative size = 1527, normalized size of antiderivative = 4.29
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1527\) |
| parts | \(\text {Expression too large to display}\) | \(1660\) |
-2/315*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-1120* C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10*b^5+(1360*C*a*b^4+2240*C*b^5)*s in(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*b^5-424*C*a^2*b^3-2040*C*a* b^4-2072*C*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(756*A*a*b^4+504*A *b^5-4*C*a^3*b^2+424*C*a^2*b^3+1568*C*a*b^4+952*C*b^5)*sin(1/2*d*x+1/2*c)^ 4*cos(1/2*d*x+1/2*c)+(-252*A*a^2*b^3-378*A*a*b^4-126*A*b^5+8*C*a^4*b+2*C*a ^3*b^2-282*C*a^2*b^3-444*C*a*b^4-168*C*b^5)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d *x+1/2*c)-63*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^3 *b^2+63*a*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))*b^4+63 *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^2-63*A*(s in(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^3+189*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^4-189*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)*Ellip ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^5-8*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(co...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.67 \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {2 \, \sqrt {2} {\left (-8 i \, C a^{5} - 3 i \, {\left (21 \, A + 10 \, C\right )} a^{3} b^{2} + 3 i \, {\left (63 \, A + 44 \, C\right )} a b^{4}\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 ) + 2 \, \sqrt {2} {\left (8 i \, C a^{5} + 3 i \, {\left (21 \, A + 10 \, C\right )} a^{3} b^{2} - 3 i \, {\left (63 \, A + 44 \, C\right )} a b^{4}\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^{4} b - 3 i \, {\left (21 \, A + 11 \, C\right )} a^{2} b^{3} - 21 i \, {\left (9 \, A + 7 \, C\right )} 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^{4} b + 3 i \, {\left (21 \, A + 11 \, C\right )} a^{2} b^{3} + 21 i \, {\left (9 \, A + 7 \, C\right )} 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 (35 \, C b^{5} \cos \left (d x + c\right )^{3} + 50 \, C a b^{4} \cos \left (d x + c\right )^{2} - 4 \, C a^{3} b^{2} + 2 \, {\left (63 \, A + 44 \, C\right )} a b^{4} + {\left (3 \, C a^{2} b^{3} + 7 \, {\left (9 \, A + 7 \, C\right )} 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 )}{945 \, b^{4} d} \]
-1/945*(2*sqrt(2)*(-8*I*C*a^5 - 3*I*(21*A + 10*C)*a^3*b^2 + 3*I*(63*A + 44 *C)*a*b^4)*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) + 2*sqrt(2)*(8*I*C*a^5 + 3*I*(21*A + 10*C)*a^3*b^2 - 3*I*(63*A + 44*C)*a*b^4 )*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) *(-8*I*C*a^4*b - 3*I*(21*A + 11*C)*a^2*b^3 - 21*I*(9*A + 7*C)*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, wei erstrassPInverse(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^4* b + 3*I*(21*A + 11*C)*a^2*b^3 + 21*I*(9*A + 7*C)*b^5)*sqrt(b)*weierstrassZ eta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInve rse(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*(35*C*b^5*cos(d*x + c)^3 + 50*C* a*b^4*cos(d*x + c)^2 - 4*C*a^3*b^2 + 2*(63*A + 44*C)*a*b^4 + (3*C*a^2*b^3 + 7*(9*A + 7*C)*b^5)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/ (b^4*d)
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
\[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cos (c+d x) (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \cos \left (c+d\,x\right )\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]