\(\int \frac {\cos ^{\frac {3}{2}}(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\) [1111]

Optimal result

Integrand size = 43, antiderivative size = 423 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=-\frac {\left (3 a^3 b B-9 a b^3 B+b^4 (5 A-8 C)-15 a^4 C+a^2 b^2 (A+29 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (3 a^4 b B-5 a^2 b^3 B+8 b^5 B-15 a^5 C-a b^4 (7 A+24 C)+a^3 b^2 (A+33 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (3 A b^6-3 a^5 b B+6 a^3 b^3 B-15 a b^5 B+15 a^6 C+5 a^2 b^4 (2 A+7 C)-a^4 b^2 (A+38 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 (a-b)^2 b^4 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac {\left (3 A b^4+a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (3 A+11 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:

-1/4*(3*B*a^3*b-9*B*a*b^3+b^4*(5*A-8*C)-15*a^4*C+a^2*b^2*(A+29*C))*Ellipti 
cE(sin(1/2*d*x+1/2*c),2^(1/2))/b^3/(a^2-b^2)^2/d+1/4*(3*B*a^4*b-5*B*a^2*b^ 
3+8*B*b^5-15*C*a^5-a*b^4*(7*A+24*C)+a^3*b^2*(A+33*C))*InverseJacobiAM(1/2* 
d*x+1/2*c,2^(1/2))/b^4/(a^2-b^2)^2/d+1/4*(3*A*b^6-3*B*a^5*b+6*B*a^3*b^3-15 
*B*a*b^5+15*a^6*C+5*a^2*b^4*(2*A+7*C)-a^4*b^2*(A+38*C))*EllipticPi(sin(1/2 
*d*x+1/2*c),2*b/(a+b),2^(1/2))/(a-b)^2/b^4/(a+b)^3/d-1/2*(A*b^2-a*(B*b-C*a 
))*cos(d*x+c)^(3/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2+1/4*(3*A*b 
^4+B*a^3*b-7*B*a*b^3-5*a^4*C+a^2*b^2*(3*A+11*C))*cos(d*x+c)^(1/2)*sin(d*x+ 
c)/b^2/(a^2-b^2)^2/d/(a+b*cos(d*x+c))
 

Mathematica [A] (warning: unable to verify)

Time = 7.96 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \left (5 a^2 A b^2+A b^4-a^3 b B-5 a b^3 B+5 a^4 C-7 a^2 b^2 C+8 b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (-24 a A b^3+8 a^2 b^2 B+16 b^4 B+8 a^3 b C-32 a b^3 C\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{b}+\frac {2 \left (-a^2 A b^2-5 A b^4-3 a^3 b B+9 a b^3 B+15 a^4 C-29 a^2 b^2 C+8 b^4 C\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{16 (a-b)^2 b^2 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (-\frac {a A b^2 \sin (c+d x)-a^2 b B \sin (c+d x)+a^3 C \sin (c+d x)}{2 b^2 \left (-a^2+b^2\right ) (a+b \cos (c+d x))^2}+\frac {a^2 A b^2 \sin (c+d x)+5 A b^4 \sin (c+d x)+3 a^3 b B \sin (c+d x)-9 a b^3 B \sin (c+d x)-7 a^4 C \sin (c+d x)+13 a^2 b^2 C \sin (c+d x)}{4 b^2 \left (-a^2+b^2\right )^2 (a+b \cos (c+d x))}\right )}{d} \] Input:

Integrate[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a 
+ b*Cos[c + d*x])^3,x]
 

Output:

((2*(5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B + 5*a^4*C - 7*a^2*b^2*C + 8 
*b^4*C)*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) + ((-24*a*A*b^3 
 + 8*a^2*b^2*B + 16*b^4*B + 8*a^3*b*C - 32*a*b^3*C)*(2*EllipticF[(c + d*x) 
/2, 2] - (2*a*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b)))/b + (2* 
(-(a^2*A*b^2) - 5*A*b^4 - 3*a^3*b*B + 9*a*b^3*B + 15*a^4*C - 29*a^2*b^2*C 
+ 8*b^4*C)*Cos[2*(c + d*x)]*(-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[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(16*(a - b)^2*b^2*(a 
+ b)^2*d) + (Sqrt[Cos[c + d*x]]*(-1/2*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[ 
c + d*x] + a^3*C*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(a + b*Cos[c + d*x])^2) + 
 (a^2*A*b^2*Sin[c + d*x] + 5*A*b^4*Sin[c + d*x] + 3*a^3*b*B*Sin[c + d*x] - 
 9*a*b^3*B*Sin[c + d*x] - 7*a^4*C*Sin[c + d*x] + 13*a^2*b^2*C*Sin[c + d*x] 
)/(4*b^2*(-a^2 + b^2)^2*(a + b*Cos[c + d*x]))))/d
 

Rubi [A] (verified)

Time = 2.72 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.01, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.349, Rules used = {3042, 3526, 27, 3042, 3526, 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.

\(\displaystyle \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\left (\left (5 C a^2-b B a+A b^2-4 b^2 C\right ) \cos ^2(c+d x)\right )+4 b (b B-a (A+C)) \cos (c+d x)+3 \left (A b^2-a (b B-a C)\right )\right )}{2 (a+b \cos (c+d x))^2}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\left (\left (5 C a^2-b B a+A b^2-4 b^2 C\right ) \cos ^2(c+d x)\right )+4 b (b B-a (A+C)) \cos (c+d x)+3 \left (A b^2-a (b B-a C)\right )\right )}{(a+b \cos (c+d x))^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (-5 C a^2+b B a-A b^2+4 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 b (b B-a (A+C)) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (A b^2-a (b B-a C)\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3526

\(\displaystyle -\frac {-\frac {\int \frac {-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4-\left (-15 C a^4+3 b B a^3+b^2 (A+29 C) a^2-9 b^3 B a+b^4 (5 A-8 C)\right ) \cos ^2(c+d x)+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4-\left (-15 C a^4+3 b B a^3+b^2 (A+29 C) a^2-9 b^3 B a+b^4 (5 A-8 C)\right ) \cos ^2(c+d x)+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\int \frac {-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4+\left (15 C a^4-3 b B a^3-b^2 (A+29 C) a^2+9 b^3 B a-b^4 (5 A-8 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3538

\(\displaystyle -\frac {-\frac {-\frac {\left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {b \left (-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4\right )+\left (-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2-b^4 (7 A+24 C) a+8 b^5 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\frac {\int \frac {b \left (-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4\right )+\left (-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2-b^4 (7 A+24 C) a+8 b^5 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {\left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\int \frac {b \left (-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4\right )+\left (-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2-b^4 (7 A+24 C) a+8 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {\left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3119

\(\displaystyle -\frac {-\frac {\frac {\int \frac {b \left (-5 C a^4+b B a^3+b^2 (3 A+11 C) a^2-7 b^3 B a+3 A b^4\right )+\left (-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2-b^4 (7 A+24 C) a+8 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {-\frac {\frac {\frac {\left (-15 a^5 C+3 a^4 b B+a^3 b^2 (A+33 C)-5 a^2 b^3 B-a b^4 (7 A+24 C)+8 b^5 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}+\frac {\left (15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{b}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\frac {\left (-15 a^5 C+3 a^4 b B+a^3 b^2 (A+33 C)-5 a^2 b^3 B-a b^4 (7 A+24 C)+8 b^5 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {\left (15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{b}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle -\frac {-\frac {\frac {\frac {\left (15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-15 a^5 C+3 a^4 b B+a^3 b^2 (A+33 C)-5 a^2 b^3 B-a b^4 (7 A+24 C)+8 b^5 B\right )}{b d}}{b}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-15 a^5 C+3 a^4 b B+a^3 b^2 (A+33 C)-5 a^2 b^3 B-a b^4 (7 A+24 C)+8 b^5 B\right )}{b d}+\frac {2 \left (15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}}{b}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\)

Input:

Int[(Cos[c + d*x]^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Co 
s[c + d*x])^3,x]
 

Output:

-1/2*((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(a + b*Cos[c + d*x])^2) - (-1/2*((-2*(3*a^3*b*B - 9*a*b^3*B + b^4*(5 
*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(c + d*x)/2, 2])/(b*d 
) + ((2*(3*a^4*b*B - 5*a^2*b^3*B + 8*b^5*B - 15*a^5*C - a*b^4*(7*A + 24*C) 
 + a^3*b^2*(A + 33*C))*EllipticF[(c + d*x)/2, 2])/(b*d) + (2*(3*A*b^6 - 3* 
a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^ 
4*b^2*(A + 38*C))*EllipticPi[(2*b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d) 
)/b)/(b*(a^2 - b^2)) - ((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2 
*(3*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Co 
s[c + d*x])))/(4*b*(a^2 - b^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3119

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]
 

rule 3120

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]
 

rule 3284

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]
 

rule 3481

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]
 

rule 3526

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]
 

rule 3538

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1999\) vs. \(2(414)=828\).

Time = 10.61 (sec) , antiderivative size = 2000, normalized size of antiderivative = 4.73

method	result	size
default	\(\text {Expression too large to display}\)	\(2000\)

Input:

int(cos(d*x+c)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, 
method=_RETURNVERBOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2/b^4/(-2*sin( 
1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1 
/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(B*b*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2) 
)-3*C*a*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-C*b*EllipticE(cos(1/2*d*x+1/ 
2*c),2^(1/2)))-4/b^3*(A*b^2-3*B*a*b+6*C*a^2)/(-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))+2*a^2*(A*b^2-B*a*b+C*a^2)/b^4*(-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+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c 
)^2*b+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*si 
n(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*cos(1/2*d*x+1/2*c)^2*b+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)*Elliptic 
F(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)*EllipticF(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)^...
 

Fricas [F]

\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="fricas")
 

Output:

integral((C*cos(d*x + c)^3 + B*cos(d*x + c)^2 + A*cos(d*x + c))*sqrt(cos(d 
*x + c))/(b^3*cos(d*x + c)^3 + 3*a*b^2*cos(d*x + c)^2 + 3*a^2*b*cos(d*x + 
c) + a^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+ 
c))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(3/2)/(b*co 
s(d*x + c) + a)^3, x)
 

Giac [F]

\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*cos(d*x + c)^(3/2)/(b*co 
s(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:

int((cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*co 
s(c + d*x))^3,x)
 

Output:

int((cos(c + d*x)^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*co 
s(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} a \,b^{2}+3 \cos \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b \] Input:

int(cos(d*x+c)^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x)
 

Output:

int((sqrt(cos(c + d*x))*cos(c + d*x))/(cos(c + d*x)**3*b**3 + 3*cos(c + d* 
x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + a**3),x)*a + int((sqrt(cos(c + d*x) 
)*cos(c + d*x)**3)/(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3*co 
s(c + d*x)*a**2*b + a**3),x)*c + int((sqrt(cos(c + d*x))*cos(c + d*x)**2)/ 
(cos(c + d*x)**3*b**3 + 3*cos(c + d*x)**2*a*b**2 + 3*cos(c + d*x)*a**2*b + 
 a**3),x)*b