Integrand size = 42, antiderivative size = 461 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-35 a^5 C+65 a^3 b^2 C-24 a b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-105 a^6 C+223 a^4 b^2 C-128 a^2 b^4 C-8 b^6 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 b^5 \left (a^2-b^2\right )^2 d}+\frac {a^2 \left (15 a^4 b B-38 a^2 b^3 B+35 b^5 B-35 a^5 C+86 a^3 b^2 C-63 a b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^5 (a+b)^3 d}-\frac {\left (15 a^3 b B-33 a b^3 B-35 a^4 C+61 a^2 b^2 C-8 b^4 C\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d}+\frac {a (b B-a C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {a \left (3 a^2 b B-9 b^3 B-7 a^3 C+13 a b^2 C\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
1/4*(15*B*a^4*b-29*B*a^2*b^3+8*B*b^5-35*C*a^5+65*C*a^3*b^2-24*C*a*b^4)*(co s(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/(a^2-b^2)^2/d-1/12*(45*B*a^5*b-99*B*a^3*b^3+72*B*a*b^5-105*C* a^6+223*C*a^4*b^2-128*C*a^2*b^4-8*C*b^6)*(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/(a^2-b^2)^2/d+1/4 *a^2*(15*B*a^4*b-38*B*a^2*b^3+35*B*b^5-35*C*a^5+86*C*a^3*b^2-63*C*a*b^4)*( 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))/(a-b)^2/b^5/(a+b)^3/d+1/2*a*(B*b-C*a)*cos(d*x+c)^(5/ 2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/4*a*(3*B*a^2*b-9*B*b^3-7* C*a^3+13*C*a*b^2)*cos(d*x+c)^(3/2)*sin(d*x+c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d *x+c))-1/12*(15*B*a^3*b-33*B*a*b^3-35*C*a^4+61*C*a^2*b^2-8*C*b^4)*sin(d*x+ c)*cos(d*x+c)^(1/2)/b^3/(a^2-b^2)^2/d
Time = 5.68 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {4 \sqrt {\cos (c+d x)} \left (-15 a^5 b B+33 a^3 b^3 B+35 a^6 C-57 a^4 b^2 C+4 b^6 C+a b \left (-21 a^3 b B+39 a b^3 B+49 a^4 C-83 a^2 b^2 C+16 b^4 C\right ) \cos (c+d x)+4 \left (-a^2 b+b^3\right )^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {\frac {2 \left (-15 a^4 b B+21 a^2 b^3 B-24 b^5 B+35 a^5 C-73 a^3 b^2 C+56 a b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {16 \left (-3 a^3 b B+12 a b^3 B+7 a^4 C-14 a^2 b^2 C-2 b^4 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {6 \left (-15 a^4 b B+29 a^2 b^3 B-8 b^5 B+35 a^5 C-65 a^3 b^2 C+24 a b^4 C\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 b^3 d} \]
((4*Sqrt[Cos[c + d*x]]*(-15*a^5*b*B + 33*a^3*b^3*B + 35*a^6*C - 57*a^4*b^2 *C + 4*b^6*C + a*b*(-21*a^3*b*B + 39*a*b^3*B + 49*a^4*C - 83*a^2*b^2*C + 1 6*b^4*C)*Cos[c + d*x] + 4*(-(a^2*b) + b^3)^2*C*Cos[2*(c + d*x)])*Sin[c + d *x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) - ((2*(-15*a^4*b*B + 21*a^2*b^ 3*B - 24*b^5*B + 35*a^5*C - 73*a^3*b^2*C + 56*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(-3*a^3*b*B + 12*a*b^3*B + 7*a^4*C - 14*a^2*b^2*C - 2*b^4*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi [(2*b)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*(-2*a*b*EllipticE[ArcSi n[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]))/((a - b)^2*(a + b)^2))/(48 *b^3*d)
Time = 3.03 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.00, 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, 3468, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3538, 25, 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.
| \(\displaystyle \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \frac {\cos ^{\frac {7}{2}}(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\int -\frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-7 C a^2+3 b B a+4 b^2 C\right ) \cos ^2(c+d x)\right )-4 b (b B-a C) \cos (c+d x)+5 a (b B-a C)\right )}{2 (a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (-\left (\left (-7 C a^2+3 b B a+4 b^2 C\right ) \cos ^2(c+d x)\right )-4 b (b B-a C) \cos (c+d x)+5 a (b B-a C)\right )}{(a+b \cos (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\left (7 C a^2-3 b B a-4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 b (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )+5 a (b B-a C)\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\int -\frac {\sqrt {\cos (c+d x)} \left (-\left (\left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right ) \cos ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-4 b^2 C a+2 b^3 B\right ) \cos (c+d x)+3 a \left (-7 C a^3+3 b B a^2+13 b^2 C a-9 b^3 B\right )\right )}{2 (a+b \cos (c+d x))}dx}{b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\left (\left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right ) \cos ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-4 b^2 C a+2 b^3 B\right ) \cos (c+d x)+3 a \left (-7 C a^3+3 b B a^2+13 b^2 C a-9 b^3 B\right )\right )}{a+b \cos (c+d x)}dx}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (35 C a^4-15 b B a^3-61 b^2 C a^2+33 b^3 B a+8 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (C a^3+b B a^2-4 b^2 C a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (-7 C a^3+3 b B a^2+13 b^2 C a-9 b^3 B\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {-3 \left (-35 C a^5+15 b B a^4+65 b^2 C a^3-29 b^3 B a^2-24 b^4 C a+8 b^5 B\right ) \cos ^2(c+d x)-4 b \left (-7 C a^4+3 b B a^3+14 b^2 C a^2-12 b^3 B a+2 b^4 C\right ) \cos (c+d x)+a \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-3 \left (-35 C a^5+15 b B a^4+65 b^2 C a^3-29 b^3 B a^2-24 b^4 C a+8 b^5 B\right ) \cos ^2(c+d x)-4 b \left (-7 C a^4+3 b B a^3+14 b^2 C a^2-12 b^3 B a+2 b^4 C\right ) \cos (c+d x)+a \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\int \frac {-3 \left (-35 C a^5+15 b B a^4+65 b^2 C a^3-29 b^3 B a^2-24 b^4 C a+8 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 b \left (-7 C a^4+3 b B a^3+14 b^2 C a^2-12 b^3 B a+2 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle \frac {\frac {-\frac {-\frac {3 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {a b \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )+\left (-105 C a^6+45 b B a^5+223 b^2 C a^4-99 b^3 B a^3-128 b^4 C a^2+72 b^5 B a-8 b^6 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {a b \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )+\left (-105 C a^6+45 b B a^5+223 b^2 C a^4-99 b^3 B a^3-128 b^4 C a^2+72 b^5 B a-8 b^6 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {3 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {a b \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )+\left (-105 C a^6+45 b B a^5+223 b^2 C a^4-99 b^3 B a^3-128 b^4 C a^2+72 b^5 B a-8 b^6 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {3 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\int \frac {a b \left (-35 C a^4+15 b B a^3+61 b^2 C a^2-33 b^3 B a-8 b^4 C\right )+\left (-105 C a^6+45 b B a^5+223 b^2 C a^4-99 b^3 B a^3-128 b^4 C a^2+72 b^5 B a-8 b^6 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (-105 a^6 C+45 a^5 b B+223 a^4 b^2 C-99 a^3 b^3 B-128 a^2 b^4 C+72 a b^5 B-8 b^6 C\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}-\frac {3 a^2 \left (-35 a^5 C+15 a^4 b B+86 a^3 b^2 C-38 a^2 b^3 B-63 a b^4 C+35 b^5 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{b}-\frac {6 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {\left (-105 a^6 C+45 a^5 b B+223 a^4 b^2 C-99 a^3 b^3 B-128 a^2 b^4 C+72 a b^5 B-8 b^6 C\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}-\frac {3 a^2 \left (-35 a^5 C+15 a^4 b B+86 a^3 b^2 C-38 a^2 b^3 B-63 a b^4 C+35 b^5 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{b}-\frac {6 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {-\frac {\frac {\frac {2 \left (-105 a^6 C+45 a^5 b B+223 a^4 b^2 C-99 a^3 b^3 B-128 a^2 b^4 C+72 a b^5 B-8 b^6 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {3 a^2 \left (-35 a^5 C+15 a^4 b B+86 a^3 b^2 C-38 a^2 b^3 B-63 a b^4 C+35 b^5 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{b}-\frac {6 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}}{2 b \left (a^2-b^2\right )}+\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}+\frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle \frac {a (b B-a C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\frac {a \left (-7 a^3 C+3 a^2 b B+13 a b^2 C-9 b^3 B\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac {-\frac {2 \left (-35 a^4 C+15 a^3 b B+61 a^2 b^2 C-33 a b^3 B-8 b^4 C\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 b d}-\frac {\frac {\frac {2 \left (-105 a^6 C+45 a^5 b B+223 a^4 b^2 C-99 a^3 b^3 B-128 a^2 b^4 C+72 a b^5 B-8 b^6 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{b d}-\frac {6 a^2 \left (-35 a^5 C+15 a^4 b B+86 a^3 b^2 C-38 a^2 b^3 B-63 a b^4 C+35 b^5 B\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}}{b}-\frac {6 \left (-35 a^5 C+15 a^4 b B+65 a^3 b^2 C-29 a^2 b^3 B-24 a b^4 C+8 b^5 B\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\) |
(a*(b*B - a*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b* Cos[c + d*x])^2) + ((a*(3*a^2*b*B - 9*b^3*B - 7*a^3*C + 13*a*b^2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])) + (-1/3* ((-6*(15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - 35*a^5*C + 65*a^3*b^2*C - 24*a *b^4*C)*EllipticE[(c + d*x)/2, 2])/(b*d) + ((2*(45*a^5*b*B - 99*a^3*b^3*B + 72*a*b^5*B - 105*a^6*C + 223*a^4*b^2*C - 128*a^2*b^4*C - 8*b^6*C)*Ellipt icF[(c + d*x)/2, 2])/(b*d) - (6*a^2*(15*a^4*b*B - 38*a^2*b^3*B + 35*b^5*B - 35*a^5*C + 86*a^3*b^2*C - 63*a*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x )/2, 2])/(b*(a + b)*d))/b)/b - (2*(15*a^3*b*B - 33*a*b^3*B - 35*a^4*C + 61 *a^2*b^2*C - 8*b^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*b*d))/(2*b*(a^2 - b^2)))/(4*b*(a^2 - b^2))
3.9.91.3.1 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*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{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] && GtQ[m, 0] && LtQ[n, -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_)] + (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. \(2194\) vs. \(2(521)=1042\).
Time = 89.44 (sec) , antiderivative size = 2195, normalized size of antiderivative = 4.76
int(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+cos(d*x+c)*b)^3,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)*(2/b^5*a^4*(B*b -C*a)*(-1/2/a*b^2/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+si n(1/2*d*x+1/2*c)^2)^(1/2)/(2*b*cos(1/2*d*x+1/2*c)^2+a-b)^2-3/4*b^2*(3*a^2- b^2)/a^2/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d *x+1/2*c)^2)^(1/2)/(2*b*cos(1/2*d*x+1/2*c)^2+a-b)-7/8/(a+b)/(a^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2) )+1/4/(a+b)/(a^2-b^2)/a*(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)*Elliptic F(cos(1/2*d*x+1/2*c),2^(1/2))*b+3/8/(a+b)/(a^2-b^2)/a^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*b^2-9/8*b/(a ^2-b^2)^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)*EllipticF(cos(1/2*d*x+ 1/2*c),2^(1/2))+3/8*b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*c os(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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+9/8*b/(a^2-b^2)^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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3/8* b^3/a^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)...
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^ 3,x, algorithm="fricas")
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^ 3,x, algorithm="maxima")
\[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
integrate(cos(d*x+c)^(5/2)*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^ 3,x, algorithm="giac")
Timed out. \[ \int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{5/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]