Integrand size = 42, antiderivative size = 246 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (5 a^2+3 b^2\right ) (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b^4 d}-\frac {2 \left (21 a^3 b B+7 a b^3 B-21 a^4 C-7 a^2 b^2 C-5 b^4 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 b^5 d}+\frac {2 a^4 (b B-a C) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b^5 (a+b) d}-\frac {2 \left (7 a b B-7 a^2 C-5 b^2 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 b^3 d}+\frac {2 (b B-a C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 b^2 d}+\frac {2 C \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 b d} \]
2/5*(5*a^2+3*b^2)*(B*b-C*a)*(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^3-2 1*C*a^4-7*C*a^2*b^2-5*C*b^4)*(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^4*(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*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 = 2.99 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.24 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\frac {2 \left (35 a^2 b B+63 b^3 B-35 a^3 C-13 a b^2 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {4 \left (28 a b B-28 a^2 C+25 b^2 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+2 \sqrt {\cos (c+d x)} \left (-70 a b B+70 a^2 C+65 b^2 C+42 b (b B-a C) \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)-\frac {42 \left (5 a^2+3 b^2\right ) (-b B+a C) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}}{210 b^3 d} \]
((2*(35*a^2*b*B + 63*b^3*B - 35*a^3*C - 13*a*b^2*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(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*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 + 3*b^2)*(-(b*B) + a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*a*(a + b)*EllipticF[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 = 2.21 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3508, 3042, 3469, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x) (B+C \cos (c+d x))}{a+b \cos (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 3469 |
| \(\displaystyle \frac {2 \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (7 (b B-a C) \cos ^2(c+d x)+5 b C \cos (c+d x)+5 a C\right )}{2 (a+b \cos (c+d x))}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (7 (b B-a C) \cos ^2(c+d x)+5 b C \cos (c+d x)+5 a C\right )}{a+b \cos (c+d x)}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (7 (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+5 b C \sin \left (c+d x+\frac {\pi }{2}\right )+5 a C\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int \frac {\sqrt {\cos (c+d x)} \left (-5 \left (-7 C a^2+7 b B a-5 b^2 C\right ) \cos ^2(c+d x)+b (21 b B+4 a C) \cos (c+d x)+21 a (b B-a C)\right )}{2 (a+b \cos (c+d x))}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-5 \left (-7 C a^2+7 b B a-5 b^2 C\right ) \cos ^2(c+d x)+b (21 b B+4 a C) \cos (c+d x)+21 a (b B-a C)\right )}{a+b \cos (c+d x)}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (-5 \left (-7 C a^2+7 b B a-5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (21 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+21 a (b B-a C)\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {-21 \left (5 a^2+3 b^2\right ) (b B-a C) \cos ^2(c+d x)-b \left (-28 C a^2+28 b B a+25 b^2 C\right ) \cos (c+d x)+5 a \left (-7 C a^2+7 b B a-5 b^2 C\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-21 \left (5 a^2+3 b^2\right ) (b B-a C) \cos ^2(c+d x)-b \left (-28 C a^2+28 b B a+25 b^2 C\right ) \cos (c+d x)+5 a \left (-7 C a^2+7 b B a-5 b^2 C\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-21 \left (5 a^2+3 b^2\right ) (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-28 C a^2+28 b B a+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a \left (-7 C a^2+7 b B a-5 b^2 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {21 \left (5 a^2+3 b^2\right ) (b B-a C) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {5 \left (a b \left (-7 C a^2+7 b B a-5 b^2 C\right )+\left (-21 C a^4+21 b B a^3-7 b^2 C a^2+7 b^3 B a-5 b^4 C\right ) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\frac {5 \int \frac {a b \left (-7 C a^2+7 b B a-5 b^2 C\right )+\left (-21 C a^4+21 b B a^3-7 b^2 C a^2+7 b^3 B a-5 b^4 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {21 \left (5 a^2+3 b^2\right ) (b B-a C) \int \sqrt {\cos (c+d x)}dx}{b}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {5 \int \frac {a b \left (-7 C a^2+7 b B a-5 b^2 C\right )+\left (-21 C a^4+21 b B a^3-7 b^2 C a^2+7 b^3 B a-5 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {21 \left (5 a^2+3 b^2\right ) (b B-a C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {-\frac {\frac {5 \int \frac {a b \left (-7 C a^2+7 b B a-5 b^2 C\right )+\left (-21 C a^4+21 b B a^3-7 b^2 C a^2+7 b^3 B a-5 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {42 \left (5 a^2+3 b^2\right ) (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {-\frac {\frac {5 \left (\frac {\left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 C+7 a b^3 B-5 b^4 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}-\frac {21 a^4 (b B-a C) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}\right )}{b}-\frac {42 \left (5 a^2+3 b^2\right ) (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {5 \left (\frac {\left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 C+7 a b^3 B-5 b^4 C\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {21 a^4 (b B-a C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}\right )}{b}-\frac {42 \left (5 a^2+3 b^2\right ) (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {-\frac {\frac {5 \left (\frac {2 \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 C+7 a b^3 B-5 b^4 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {21 a^4 (b B-a C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}\right )}{b}-\frac {42 \left (5 a^2+3 b^2\right ) (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {\frac {-\frac {10 \left (-7 a^2 C+7 a b B-5 b^2 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}-\frac {\frac {5 \left (\frac {2 \left (-21 a^4 C+21 a^3 b B-7 a^2 b^2 C+7 a b^3 B-5 b^4 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {42 a^4 (b B-a C) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}\right )}{b}-\frac {42 \left (5 a^2+3 b^2\right ) (b B-a C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}}{5 b}+\frac {14 (b B-a C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 b d}}{7 b}+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 b d}\) |
(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) + (-1/3*((-42*(5*a^2 + 3*b^2)*(b*B - a*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*C - 5*b^4*C)*EllipticF[(c + d*x)/2, 2])/(b*d) - (42*a^4*(b* B - a*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d)))/b)/b - (10*(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)
3.9.76.3.1 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_)])^(n_), x_Symbol] :> Si mp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^( n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a^2*A*d*(m + n + 1) + b*B*(b*c*( m - 1) + a*d*(n + 1)) + (a*d*(2*A*b + a*B)*(m + n + 1) - b*B*(a*c - b*d*(m + n)))*Sin[e + f*x] + b*(A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin [e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && !(IGt Q[n, 1] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 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. \(1374\) vs. \(2(308)=616\).
Time = 16.99 (sec) , antiderivative size = 1375, normalized size of antiderivative = 5.59
-2/105*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*((240*C*a*b ^4-240*C*b^5)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-168*B*a*b^4+168*B* b^5+168*C*a^2*b^3-528*C*a*b^4+360*C*b^5)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+ 1/2*c)+(-140*B*a^2*b^3+308*B*a*b^4-168*B*b^5+140*C*a^3*b^2-308*C*a^2*b^3+4 48*C*a*b^4-280*C*b^5)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(70*B*a^2*b^ 3-112*B*a*b^4+42*B*b^5-70*C*a^3*b^2+112*C*a^2*b^3-122*C*a*b^4+80*C*b^5)*si n(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-105*B*(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))*a^4* b+105*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elli pticF(cos(1/2*d*x+1/2*c),2^(1/2))*a^3*b^2-35*B*(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))*a ^2*b^3+35*B*(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))*a*b^4-105*B*(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) )*a^3*b^2+105*B*(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))*a^2*b^3-63*B*(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))*a*b^4+63*B*(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))*b^5+105*B*(sin(1/2*d*x+1/2*c)^ 2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c)...
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)), x, algorithm="fricas")
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)), x, algorithm="maxima")
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c)), x, algorithm="giac")
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]