\(\int \frac {\sqrt {\cos (c+d x)} (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^3} \, dx\) [1112]

Optimal result

Integrand size = 43, antiderivative size = 418 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (a^3 b B-7 a b^3 B+a^2 b^2 (3 A-5 C)+3 a^4 C+b^4 (3 A+8 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a (a-b)^2 b^3 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 a b \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \] Output:

1/4*(A*b^4-B*a^3*b-5*B*a*b^3-3*a^4*C+a^2*b^2*(5*A+9*C))*EllipticE(sin(1/2* 
d*x+1/2*c),2^(1/2))/a/b^2/(a^2-b^2)^2/d+1/4*(B*a^3*b-7*B*a*b^3+a^2*b^2*(3* 
A-5*C)+3*a^4*C+b^4*(3*A+8*C))*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/b^3/( 
a^2-b^2)^2/d+1/4*(A*b^6-B*a^5*b+10*B*a^3*b^3+3*B*a*b^5-3*a^4*b^2*(A-2*C)-3 
*a^6*C-5*a^2*b^4*(2*A+3*C))*EllipticPi(sin(1/2*d*x+1/2*c),2*b/(a+b),2^(1/2 
))/a/(a-b)^2/b^3/(a+b)^3/d-1/2*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^(1/2)*sin(d* 
x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2-1/4*(A*b^4-B*a^3*b-5*B*a*b^3-3*a^4*C 
+a^2*b^2*(5*A+9*C))*cos(d*x+c)^(1/2)*sin(d*x+c)/a/b/(a^2-b^2)^2/d/(a+b*cos 
(d*x+c))
 

Mathematica [A] (warning: unable to verify)

Time = 5.58 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \sqrt {\cos (c+d x)} \left (a \left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right )+b \left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {\frac {\left (-3 A b^4-5 a^3 b B-a b^3 B+a^4 C+a^2 b^2 (9 A+5 C)\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}-\frac {8 a \left (-3 a b B+a^2 (2 A+C)+b^2 (A+2 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-a \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {\left (-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-a^2 b^2 (5 A+9 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 a (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (-2 a^2+b^2\right ) \operatorname {EllipticPi}\left (-\frac {b}{a},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b^2 \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a b d} \] Input:

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

Output:

((2*Sqrt[Cos[c + d*x]]*(a*(A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b 
^2*(A + C)) + b*(-(A*b^4) + a^3*b*B + 5*a*b^3*B + 3*a^4*C - a^2*b^2*(5*A + 
 9*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(a + b*Cos[c + d*x])^2) 
- (((-3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + a^2*b^2*(9*A + 5*C))*Ellipti 
cPi[(2*b)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*a*(-3*a*b*B + a^2*(2*A + 
C) + b^2*(A + 2*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - a*EllipticPi[(2*b 
)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-(A*b^4) + a^3*b*B + 5*a*b^3*B + 
3*a^4*C - a^2*b^2*(5*A + 9*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]))/((a - b)^2*(a + b)^2))/(8*a*b*d)
 

Rubi [A] (verified)

Time = 2.64 (sec) , antiderivative size = 415, normalized size of antiderivative = 0.99, 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, 3534, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\frac {a \left (3 a^4 C+a^3 b B+a^2 b^2 (3 A-5 C)-7 a b^3 B+b^4 (3 A+8 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}+\frac {\left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{b}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\frac {a \left (3 a^4 C+a^3 b B+a^2 b^2 (3 A-5 C)-7 a b^3 B+b^4 (3 A+8 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {\left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{b}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle -\frac {\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {\frac {\left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (3 a^4 C+a^3 b B+a^2 b^2 (3 A-5 C)-7 a b^3 B+b^4 (3 A+8 C)\right )}{b d}}{b}+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d}+\frac {\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (3 a^4 C+a^3 b B+a^2 b^2 (3 A-5 C)-7 a b^3 B+b^4 (3 A+8 C)\right )}{b d}+\frac {2 \left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}}{b}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\)

Input:

Int[(Sqrt[Cos[c + d*x]]*(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))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(a + b*Cos[c + d*x])^2) - (-1/2*((2*(A*b^4 - a^3*b*B - 5*a*b^3*B - 3 
*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(b*d) + ((2*a*(a^ 
3*b*B - 7*a*b^3*B + a^2*b^2*(3*A - 5*C) + 3*a^4*C + b^4*(3*A + 8*C))*Ellip 
ticF[(c + d*x)/2, 2])/(b*d) + (2*(A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5 
*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2* 
b)/(a + b), (c + d*x)/2, 2])/(b*(a + b)*d))/b)/(a*(a^2 - b^2)) + ((A*b^4 - 
 a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*S 
in[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Cos[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 3534

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[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x 
]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b* 
c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2))   Int 
[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b*c - a* 
d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - 
a*b*B + a^2*C) + (m + 1)*(b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A 
*b^2 - a*b*B + a^2*C)*(m + n + 3)*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] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ 
[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) | 
| EqQ[a, 0])))
 

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. \(1949\) vs. \(2(409)=818\).

Time = 7.90 (sec) , antiderivative size = 1950, normalized size of antiderivative = 4.67

method	result	size
default	\(\text {Expression too large to display}\)	\(1950\)

Input:

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

Fricas [F(-1)]

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

integrate(cos(d*x+c)^(1/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c) 
)^3,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

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

integrate(cos(d*x+c)**(1/2)*(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]

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

integrate(cos(d*x+c)^(1/2)*(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)*sqrt(cos(d*x + c))/(b*co 
s(d*x + c) + a)^3, x)
 

Giac [F]

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

integrate(cos(d*x+c)^(1/2)*(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)*sqrt(cos(d*x + c))/(b*co 
s(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

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

int((cos(c + d*x)^(1/2)*(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)^(1/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*co 
s(c + d*x))^3, x)
 

Reduce [F]

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

int(cos(d*x+c)^(1/2)*(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)**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)) 
/(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 + 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)*c