Integrand size = 43, antiderivative size = 285 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (5 a^2 b B+3 b^3 B-5 a^3 C-a b^2 (5 A+3 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d}-\frac {2 \left (21 a^3 b B+7 a b^3 B-21 a^4 C-7 a^2 b^2 (3 A+C)-b^4 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^5 d}-\frac {2 a^3 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^5 (a+b) d}+\frac {2 \left (7 A b^2-7 a b B+7 a^2 C+5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac {2 (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b^2 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b d} \]
2/5*(5*B*a^2*b+3*B*b^3-5*a^3*C-a*b^2*(5*A+3*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^4/d-2/21*(21 *B*a^3*b+7*B*a*b^3-21*a^4*C-7*a^2*b^2*(3*A+C)-b^4*(7*A+5*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 ^5/d-2*a^3*(A*b^2-a*(B*b-C*a))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/ 2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2))/b^5/(a+b)/d+2/5*(B*b -C*a)*cos(d*x+c)^(3/2)*sin(d*x+c)/b^2/d+2/7*C*cos(d*x+c)^(5/2)*sin(d*x+c)/ b/d+2/21*(7*A*b^2-7*B*a*b+7*C*a^2+5*C*b^2)*sin(d*x+c)*cos(d*x+c)^(1/2)/b^3 /d
Time = 4.72 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {-\frac {2 \left (-35 a^2 b B-63 b^3 B+35 a^3 C+a b^2 (35 A+13 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 \left (35 A b^2+28 a b B-28 a^2 C+25 b^2 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+2 \sqrt {\cos (c+d x)} \left (70 A b^2-70 a b B+70 a^2 C+65 b^2 C+42 b (b B-a C) \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)-\frac {42 \left (-5 a^2 b B-3 b^3 B+5 a^3 C+a b^2 (5 A+3 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}}{210 b^3 d} \]
Integrate[(Cos[c + d*x]^(5/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x]),x]
((-2*(-35*a^2*b*B - 63*b^3*B + 35*a^3*C + a*b^2*(35*A + 13*C))*EllipticPi[ (2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(35*A*b^2 + 28*a*b*B - 28*a^2 *C + 25*b^2*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + 2*Sqrt[Cos[c + d*x]]*(70*A*b^2 - 70*a*b* B + 70*a^2*C + 65*b^2*C + 42*b*(b*B - a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*( c + d*x)])*Sin[c + d*x] - (42*(-5*a^2*b*B - 3*b^3*B + 5*a^3*C + a*b^2*(5*A + 3*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*E llipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (-2*a^2 + b^2)*EllipticPi[-(b/a ), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b^2*Sqrt[Sin[c + d*x] ^2]))/(210*b^3*d)
Time = 2.33 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 27, 3042, 3119, 3481, 3042, 3120, 3284}
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.
| \(\displaystyle \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {2 \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (7 (b B-a C) \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+5 a C\right )}{2 (a+b \cos (c+d x))}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (7 (b B-a C) \cos ^2(c+d x)+b (7 A+5 C) \cos (c+d x)+5 a C\right )}{a+b \cos (c+d x)}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (7 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (5 \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right ) \cos ^2(c+d x)+b (21 b B+4 a C) \cos (c+d x)+21 a (b B-a C)\right )}{2 (a+b \cos (c+d x))}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (5 \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right ) \cos ^2(c+d x)+b (21 b B+4 a C) \cos (c+d x)+21 a (b B-a C)\right )}{a+b \cos (c+d x)}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (5 \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (21 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+21 a (b B-a C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \int \frac {21 \left (-5 C a^3+5 b B a^2-b^2 (5 A+3 C) a+3 b^3 B\right ) \cos ^2(c+d x)+b \left (-28 C a^2+28 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)+5 a \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {21 \left (-5 C a^3+5 b B a^2-b^2 (5 A+3 C) a+3 b^3 B\right ) \cos ^2(c+d x)+b \left (-28 C a^2+28 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)+5 a \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {21 \left (-5 C a^3+5 b B a^2-b^2 (5 A+3 C) a+3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b \left (-28 C a^2+28 b B a+35 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {\frac {\frac {21 \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )-\left (-21 C a^4+21 b B a^3-7 b^2 (3 A+C) a^2+7 b^3 B a-b^4 (7 A+5 C)\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {21 \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}+\frac {5 \int \frac {a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )-\left (-21 C a^4+21 b B a^3-7 b^2 (3 A+C) a^2+7 b^3 B a-b^4 (7 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {21 \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}+\frac {5 \int \frac {a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )+\left (21 C a^4-21 b B a^3+7 b^2 (3 A+C) a^2-7 b^3 B a+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {\frac {\frac {5 \int \frac {a b \left (7 C a^2-7 b B a+7 A b^2+5 b^2 C\right )+\left (21 C a^4-21 b B a^3+7 b^2 (3 A+C) a^2-7 b^3 B a+b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {\frac {\frac {5 \left (-\frac {21 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {\left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {5 \left (-\frac {21 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {\left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {\frac {\frac {5 \left (-\frac {21 a^3 \left (A b^2-a (b B-a C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right )}{b d}\right )}{b}+\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}}{3 b}+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (7 a^2 C-7 a b B+7 A b^2+5 b^2 C\right )}{3 b d}+\frac {\frac {42 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+5 a^2 b B-a b^2 (5 A+3 C)+3 b^3 B\right )}{b d}+\frac {5 \left (-\frac {42 a^3 \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}-\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 (3 A+C)+7 a b^3 B-b^4 (7 A+5 C)\right )}{b d}\right )}{b}}{3 b}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
(2*C*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*b*d) + ((14*(b*B - a*C)*Cos[c + d *x]^(3/2)*Sin[c + d*x])/(5*b*d) + (((42*(5*a^2*b*B + 3*b^3*B - 5*a^3*C - a *b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(b*d) + (5*((-2*(21*a^3*b*B + 7*a*b^3*B - 21*a^4*C - 7*a^2*b^2*(3*A + C) - b^4*(7*A + 5*C))*EllipticF[( c + d*x)/2, 2])/(b*d) - (42*a^3*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*b)/( a + b), (c + d*x)/2, 2])/(b*(a + b)*d)))/b)/(3*b) + (10*(7*A*b^2 - 7*a*b*B + 7*a^2*C + 5*b^2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d))/(5*b))/(7* b)
3.11.95.3.1 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[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(2/(f*(a + b)*Sqrt[c + d]))*EllipticPi[ 2*(b/(a + b)), (1/2)*(e - Pi/2 + f*x), 2*(d/(c + d))], x] /; FreeQ[{a, b, c , d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]
Int[(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[ B/d Int[(a + b*Sin[e + f*x])^m, x], x] - Simp[(B*c - A*d)/d Int[(a + b* Sin[e + f*x])^m/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
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_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1096\) vs. \(2(347)=694\).
Time = 13.23 (sec) , antiderivative size = 1097, normalized size of antiderivative = 3.85
int(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)),x,me thod=_RETURNVERBOSE)
-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(8/105*C/b*(60* cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8-258*cos(1/2*d*x+1/2*c)*sin(1/2*d*x +1/2*c)^6+448*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-167*sin(1/2*d*x+1/2* c)^2*cos(1/2*d*x+1/2*c)+85*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2 *c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-168*(sin(1/2*d*x+1/2* c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c), 2^(1/2)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)-4/5/b^2*(B* b-C*a-4*C*b)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*cos(1 /2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6-14*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2 *c)+6*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-5*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 9*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE( cos(1/2*d*x+1/2*c),2^(1/2)))+4*a^3*(A*b^2-B*a*b+C*a^2)/b^4/(-2*a*b+2*b^2)* (sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2 *d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c),-2 *b/(a-b),2^(1/2))+4/3/b^3*(A*b^2-B*a*b-3*B*b^2+C*a^2+3*C*a*b+6*C*b^2)*(2*s in(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/ 2*c)+2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Ellip ticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2 *d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-2*sin(1...
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) ),x, algorithm="fricas")
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) ),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(5/2)/(b*co s(d*x + c) + a), x)
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) ),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(5/2)/(b*co s(d*x + c) + a), x)
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]