Integrand size = 35, antiderivative size = 378 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{315 b^5 d \sqrt {a+b \cos (c+d x)}}-\frac {4 a \left (42 A b^2+32 a^2 C+31 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^4 d}+\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{315 b^3 d}-\frac {16 a C \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{63 b^2 d}+\frac {2 C \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{9 b d} \]
-4/315*a*(42*A*b^2+32*C*a^2+31*C*b^2)*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^ 4/d+2/315*(48*a^2*C+7*b^2*(9*A+7*C))*cos(d*x+c)*sin(d*x+c)*(a+b*cos(d*x+c) )^(1/2)/b^3/d-16/63*a*C*cos(d*x+c)^2*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b^2 /d+2/9*C*cos(d*x+c)^3*sin(d*x+c)*(a+b*cos(d*x+c))^(1/2)/b/d+2/315*(128*a^4 *C+21*b^4*(9*A+7*C)+12*a^2*b^2*(14*A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(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^5/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/315*a*(128*a^4* C+4*a^2*b^2*(42*A+19*C)+3*b^4*(49*A+37*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/co s(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^5/d/(a+b*cos(d*x+c))^(1/2)
Time = 2.42 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.72 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {8 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (32 a^3 b C+6 a b^3 (7 A+6 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (128 a^4 C+21 b^4 (9 A+7 C)+12 a^2 b^2 (14 A+9 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )-b (a+b \cos (c+d x)) \left (32 a \left (21 A b^2+2 \left (8 a^2+9 b^2\right ) C\right ) \sin (c+d x)-b \left (2 \left (126 A b^2+96 a^2 C+133 b^2 C\right ) \sin (2 (c+d x))+5 b C (-16 a \sin (3 (c+d x))+7 b \sin (4 (c+d x)))\right )\right )}{1260 b^5 d \sqrt {a+b \cos (c+d x)}} \]
(8*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(32*a^3*b*C + 6*a*b^3*(7*A + 6*C) )*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (128*a^4*C + 21*b^4*(9*A + 7*C) + 12*a^2*b^2*(14*A + 9*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])*(32*a *(21*A*b^2 + 2*(8*a^2 + 9*b^2)*C)*Sin[c + d*x] - b*(2*(126*A*b^2 + 96*a^2* C + 133*b^2*C)*Sin[2*(c + d*x)] + 5*b*C*(-16*a*Sin[3*(c + d*x)] + 7*b*Sin[ 4*(c + d*x)]))))/(1260*b^5*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.32 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3529, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {2 \int \frac {\cos ^2(c+d x) \left (-8 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+6 a C\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) \left (-8 a C \cos ^2(c+d x)+b (9 A+7 C) \cos (c+d x)+6 a C\right )}{\sqrt {a+b \cos (c+d x)}}dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (-8 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (9 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 a C\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int -\frac {\cos (c+d x) \left (32 C a^2-2 b C \cos (c+d x) a-\left (48 C a^2+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {\cos (c+d x) \left (32 C a^2-2 b C \cos (c+d x) a-\left (48 C a^2+7 b^2 (9 A+7 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}}dx}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (32 C a^2-2 b C \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-48 C a^2-7 b^2 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {-\frac {\frac {2 \int -\frac {-6 a \left (32 C a^2+42 A b^2+31 b^2 C\right ) \cos ^2(c+d x)+b \left (-16 C a^2+189 A b^2+147 b^2 C\right ) \cos (c+d x)+2 a \left (48 C a^2+7 b^2 (9 A+7 C)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-6 a \left (32 C a^2+42 A b^2+31 b^2 C\right ) \cos ^2(c+d x)+b \left (-16 C a^2+189 A b^2+147 b^2 C\right ) \cos (c+d x)+2 a \left (48 C a^2+7 b^2 (9 A+7 C)\right )}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\int \frac {-6 a \left (32 C a^2+42 A b^2+31 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-16 C a^2+189 A b^2+147 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (48 C a^2+7 b^2 (9 A+7 C)\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {2 \int \frac {3 \left (2 a b \left (16 C a^2+3 b^2 (7 A+6 C)\right )+\left (128 C a^4+12 b^2 (14 A+9 C) a^2+21 b^4 (9 A+7 C)\right ) \cos (c+d x)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {2 a b \left (16 C a^2+3 b^2 (7 A+6 C)\right )+\left (128 C a^4+12 b^2 (14 A+9 C) a^2+21 b^4 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\int \frac {2 a b \left (16 C a^2+3 b^2 (7 A+6 C)\right )+\left (128 C a^4+12 b^2 (14 A+9 C) a^2+21 b^4 (9 A+7 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {\left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {2 \left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {2 \left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {-\frac {\frac {\frac {2 \left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {-\frac {-\frac {2 \left (48 a^2 C+7 b^2 (9 A+7 C)\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}-\frac {\frac {\frac {2 \left (128 a^4 C+12 a^2 b^2 (14 A+9 C)+21 b^4 (9 A+7 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 a \left (128 a^4 C+4 a^2 b^2 (42 A+19 C)+3 b^4 (49 A+37 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{b}-\frac {4 a \left (32 a^2 C+42 A b^2+31 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d}}{5 b}}{7 b}-\frac {16 a C \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{9 b}+\frac {2 C \sin (c+d x) \cos ^3(c+d x) \sqrt {a+b \cos (c+d x)}}{9 b d}\) |
(2*C*Cos[c + d*x]^3*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(9*b*d) + ((-16 *a*C*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d) - ((-2* (48*a^2*C + 7*b^2*(9*A + 7*C))*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) - (((2*(128*a^4*C + 21*b^4*(9*A + 7*C) + 12*a^2*b^2*(14*A + 9*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*(128*a^4*C + 4*a^2*b^2*(42* A + 19*C) + 3*b^4*(49*A + 37*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*Ellipt icF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/b - (4*a* (42*A*b^2 + 32*a^2*C + 31*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b *d))/(5*b))/(7*b))/(9*b)
3.7.48.3.1 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_.)*((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. \(1526\) vs. \(2(408)=816\).
Time = 19.95 (sec) , antiderivative size = 1527, normalized size of antiderivative = 4.04
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1527\) |
| parts | \(\text {Expression too large to display}\) | \(1662\) |
-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+(-80*C*a*b^4+2240*C*b^5)*si n(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-504*A*b^5-64*C*a^2*b^3+120*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)+(-84*A*a*b^4+504*A*b^ 5-64*C*a^3*b^2+64*C*a^2*b^3-112*C*a*b^4+952*C*b^5)*sin(1/2*d*x+1/2*c)^4*co s(1/2*d*x+1/2*c)+(168*A*a^2*b^3+42*A*a*b^4-126*A*b^5+128*C*a^4*b+32*C*a^3* b^2+108*C*a^2*b^3+36*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)-168*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^ 2-147*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+168* 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^2-168*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^5-128*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(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=-\frac {2 \, \sqrt {2} {\left (-128 i \, C a^{5} - 12 i \, {\left (14 \, A + 5 \, C\right )} a^{3} b^{2} - 3 i \, {\left (42 \, A + 31 \, 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 (128 i \, C a^{5} + 12 i \, {\left (14 \, A + 5 \, C\right )} a^{3} b^{2} + 3 i \, {\left (42 \, A + 31 \, 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 (-128 i \, C a^{4} b - 12 i \, {\left (14 \, A + 9 \, 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 (128 i \, C a^{4} b + 12 i \, {\left (14 \, A + 9 \, 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} - 40 \, C a b^{4} \cos \left (d x + c\right )^{2} - 64 \, C a^{3} b^{2} - 2 \, {\left (42 \, A + 31 \, C\right )} a b^{4} + {\left (48 \, 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^{6} d} \]
-1/945*(2*sqrt(2)*(-128*I*C*a^5 - 12*I*(14*A + 5*C)*a^3*b^2 - 3*I*(42*A + 31*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)*(128*I*C*a^5 + 12*I*(14*A + 5*C)*a^3*b^2 + 3*I*(42*A + 31*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*sqr t(2)*(-128*I*C*a^4*b - 12*I*(14*A + 9*C)*a^2*b^3 - 21*I*(9*A + 7*C)*b^5)*s qrt(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)*(128* I*C*a^4*b + 12*I*(14*A + 9*C)*a^2*b^3 + 21*I*(9*A + 7*C)*b^5)*sqrt(b)*weie rstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstr assPInverse(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 - 40*C*a*b^4*cos(d*x + c)^2 - 64*C*a^3*b^2 - 2*(42*A + 31*C)*a*b^4 + (48* 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^6*d)
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]