\(\int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx\) [1500]

Optimal result

Integrand size = 43, antiderivative size = 596 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\frac {\left (15 a^4 b B-29 a^2 b^3 B+8 b^5 B-a^3 b^2 (3 A-65 C)+3 a b^4 (3 A-8 C)-35 a^5 C\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}-\frac {\left (45 a^5 b B-99 a^3 b^3 B+72 a b^5 B-a^4 b^2 (9 A-223 C)+a^2 b^4 (15 A-128 C)-105 a^6 C-8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 b^5 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 A b^6-15 a^5 b B+38 a^3 b^3 B-35 a b^5 B+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^5 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (5 A b^4+3 a^3 b B-9 a b^3 B-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}-\frac {\left (15 a^3 b B-33 a b^3 B-a^2 b^2 (3 A-61 C)+b^4 (21 A-8 C)-35 a^4 C\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \] Output:

1/4*(15*B*a^4*b-29*B*a^2*b^3+8*B*b^5-a^3*b^2*(3*A-65*C)+3*a*b^4*(3*A-8*C)- 
35*C*a^5)*cos(d*x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c 
)^(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-a^4*b^2 
*(9*A-223*C)+a^2*b^4*(15*A-128*C)-105*a^6*C-8*b^6*(3*A+C))*cos(d*x+c)^(1/2 
)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/b^5/(a^2-b^2)^2/ 
d-1/4*a*(15*A*b^6-15*B*a^5*b+38*B*a^3*b^3-35*B*a*b^5+a^4*b^2*(3*A-86*C)-3* 
a^2*b^4*(2*A-21*C)+35*a^6*C)*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c 
),2*b/(a+b),2^(1/2))*sec(d*x+c)^(1/2)/(a-b)^2/b^5/(a+b)^3/d-1/2*(A*b^2-a*( 
B*b-C*a))*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2/sec(d*x+c)^(5/2)+1/4 
*(5*A*b^4+3*B*a^3*b-9*B*a*b^3-7*a^4*C+a^2*b^2*(A+13*C))*sin(d*x+c)/b^2/(a^ 
2-b^2)^2/d/(a+b*cos(d*x+c))/sec(d*x+c)^(3/2)-1/12*(15*B*a^3*b-33*B*a*b^3-a 
^2*b^2*(3*A-61*C)+b^4*(21*A-8*C)-35*a^4*C)*sin(d*x+c)/b^3/(a^2-b^2)^2/d/se 
c(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 7.64 (sec) , antiderivative size = 972, normalized size of antiderivative = 1.63 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3* 
Sec[c + d*x]^(5/2)),x]
 

Output:

-1/48*((2*(3*a^3*A*b^2 + 15*a*A*b^4 - 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)*Cos[c + d*x]^2*(EllipticF[ArcSin[ 
Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], 
-1])*(b + a*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(a + b 
*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a^2*A*b^3 - 48*A*b^5 - 24*a 
^3*b^2*B + 96*a*b^4*B + 56*a^4*b*C - 112*a^2*b^3*C - 16*b^5*C)*Cos[c + d*x 
]^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(b + a*Sec[c + d*x] 
)*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(a + b*Cos[c + d*x])*(1 - Cos[ 
c + d*x]^2)) + ((9*a^3*A*b^2 - 27*a*A*b^4 - 45*a^4*b*B + 87*a^2*b^3*B - 24 
*b^5*B + 105*a^5*C - 195*a^3*b^2*C + 72*a*b^4*C)*Cos[2*(c + d*x)]*(b + a*S 
ec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[ 
Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*(2*a - 
 b)*b*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 
- Sec[c + d*x]^2] - 4*a^2*EllipticPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], - 
1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*b^2*EllipticPi[-(a/b), 
ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2 
])*Sin[c + d*x])/(a*b^2*(a + b*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec 
[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[ 
c + d*x]]*((a*(3*a^2*A*b^2 - 9*A*b^4 - 7*a^3*b*B + 13*a*b^3*B + 11*a^4*C - 
 17*a^2*b^2*C)*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2) - (a^3*A*b^2*Sin[c +...
 

Rubi [A] (verified)

Time = 3.89 (sec) , antiderivative size = 560, normalized size of antiderivative = 0.94, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4709, 3042, 3526, 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.

\(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \cos (c+d x)+C \cos (c+d x)^2}{\sec (c+d x)^{5/2} (a+b \cos (c+d x))^3}dx\)

\(\Big \downarrow \) 4709

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\left (\left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right ) \cos ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)+3 \left (-7 C a^4+3 b B a^3+b^2 (A+13 C) a^2-9 b^3 B a+5 A b^4\right )\right )}{2 (a+b \cos (c+d x))}dx}{b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (-\left (\left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right ) \cos ^2(c+d x)\right )+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \cos (c+d x)+3 \left (-7 C a^4+3 b B a^3+b^2 (A+13 C) a^2-9 b^3 B a+5 A b^4\right )\right )}{a+b \cos (c+d x)}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (35 C a^4-15 b B a^3+b^2 (3 A-61 C) a^2+33 b^3 B a-b^4 (21 A-8 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+4 b \left (C a^3+b B a^2-b^2 (3 A+4 C) a+2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left (-7 C a^4+3 b B a^3+b^2 (A+13 C) a^2-9 b^3 B a+5 A b^4\right )\right )}{a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {\frac {2 \int -\frac {-3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) \cos ^2(c+d x)-4 b \left (-7 C a^4+3 b B a^3+b^2 (3 A+14 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right ) \cos (c+d x)+a \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )}{2 \sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\int \frac {-3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) \cos ^2(c+d x)-4 b \left (-7 C a^4+3 b B a^3+b^2 (3 A+14 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right ) \cos (c+d x)+a \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\int \frac {-3 \left (-35 C a^5+15 b B a^4-b^2 (3 A-65 C) a^3-29 b^3 B a^2+3 b^4 (3 A-8 C) a+8 b^5 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-4 b \left (-7 C a^4+3 b B a^3+b^2 (3 A+14 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3538

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {-\frac {3 \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}-\frac {\int -\frac {a b \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )+\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\frac {\int \frac {a b \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )+\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}-\frac {3 \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \sqrt {\cos (c+d x)}dx}{b}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\frac {\int \frac {a b \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )+\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {3 \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\frac {\int \frac {a b \left (-35 C a^4+15 b B a^3-b^2 (3 A-61 C) a^2-33 b^3 B a+b^4 (21 A-8 C)\right )+\left (-105 C a^6+45 b B a^5-b^2 (9 A-223 C) a^4-99 b^3 B a^3+b^4 (15 A-128 C) a^2+72 b^5 B a-8 b^6 (3 A+C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3481

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\frac {\frac {\left (-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{b}+\frac {3 a \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))}dx}{b}}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\frac {\frac {\left (-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}+\frac {3 a \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {-\frac {-\frac {\frac {\frac {3 a \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}+\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right )}{b d}}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}\right )\)

\(\Big \downarrow \) 3284

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right )}{b d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {-\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)} \left (-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right )}{3 b d}-\frac {\frac {\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (-105 a^6 C+45 a^5 b B-a^4 b^2 (9 A-223 C)-99 a^3 b^3 B+a^2 b^4 (15 A-128 C)+72 a b^5 B-8 b^6 (3 A+C)\right )}{b d}+\frac {6 a \left (35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{b d (a+b)}}{b}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right )}{b d}}{3 b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\right )\)

Input:

Int[(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)/((a + b*Cos[c + d*x])^3*Sec[c 
+ d*x]^(5/2)),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1/2*((A*b^2 - a*(b*B - a*C))*Cos[c 
 + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (-( 
((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*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 - a^3*b^2*(3*A - 65*C) + 3*a*b^4* 
(3*A - 8*C) - 35*a^5*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 - a^4*b^2*(9*A - 223*C) + a^2*b^4*(15*A - 128 
*C) - 105*a^6*C - 8*b^6*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(b*d) + (6*a 
*(15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) 
 - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*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 - a^2*b^2*(3*A - 
61*C) + b^4*(21*A - 8*C) - 35*a^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)))
 

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 3528

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])))
 

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]
 

rule 4709

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a 
+ b*x])^m*(c*Cos[a + b*x])^m   Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u, x]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2266\) vs. \(2(571)=1142\).

Time = 10.53 (sec) , antiderivative size = 2267, normalized size of antiderivative = 3.80

method	result	size
default	\(\text {Expression too large to display}\)	\(2267\)

Input:

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(5/2),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)*(2/3/b^5*(4*C*b 
^2*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^4+3*A*b^2*(sin(1/2*d*x+1/2*c)^2)^ 
(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2 
))-9*B*a*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)*E 
llipticF(cos(1/2*d*x+1/2*c),2^(1/2))-3*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*s 
in(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*b^2-2*C 
*b^2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+18*a^2*C*(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))+C*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)* 
EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+9*C*(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)/( 
-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)+2*a^2/b^5*(3*A*b^2-4*B 
*a*b+5*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)*Ellipti 
cF(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*...
 

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(5 
/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**3/sec(d*x+c)* 
*(5/2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(5 
/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int { \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sec \left (d x + c\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(5 
/2),x, algorithm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)/((b*cos(d*x + c) + a)^3* 
sec(d*x + c)^(5/2)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(5/2)*(a + b 
*cos(c + d*x))^3),x)
 

Output:

int((A + B*cos(c + d*x) + C*cos(c + d*x)^2)/((1/cos(c + d*x))^(5/2)*(a + b 
*cos(c + d*x))^3), x)
 

Reduce [F]

\[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3} a \,b^{2}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3} a^{2} b +\sec \left (d x +c \right )^{3} a^{3}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3} a \,b^{2}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3} a^{2} b +\sec \left (d x +c \right )^{3} a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )^{3} \sec \left (d x +c \right )^{3} b^{3}+3 \cos \left (d x +c \right )^{2} \sec \left (d x +c \right )^{3} a \,b^{2}+3 \cos \left (d x +c \right ) \sec \left (d x +c \right )^{3} a^{2} b +\sec \left (d x +c \right )^{3} a^{3}}d x \right ) c \] Input:

int((A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3/sec(d*x+c)^(5/2),x)
 

Output:

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