\(\int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx\) [1409]

Optimal result

Integrand size = 35, antiderivative size = 493 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {a \left (a^2 b^2 (3 A-65 C)-3 b^4 (3 A-8 C)+35 a^4 C\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^4 \left (a^2-b^2\right )^2 d}+\frac {\left (a^4 b^2 (9 A-223 C)-a^2 b^4 (15 A-128 C)+105 a^6 C+8 b^6 (3 A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 b^5 \left (a^2-b^2\right )^2 d}-\frac {a \left (15 A b^6+a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+35 a^6 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 (a-b)^2 b^5 (a+b)^3 d}-\frac {\left (A b^2+a^2 C\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2 \sec ^{\frac {5}{2}}(c+d x)}+\frac {\left (5 A b^4-7 a^4 C+a^2 b^2 (A+13 C)\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (a^2 b^2 (3 A-61 C)-b^4 (21 A-8 C)+35 a^4 C\right ) \sin (c+d x)}{12 b^3 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \] Output:

-1/4*a*(a^2*b^2*(3*A-65*C)-3*b^4*(3*A-8*C)+35*a^4*C)*cos(d*x+c)^(1/2)*Elli 
pticE(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* 
(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+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+C*a^2)*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^4-7*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*(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/sec 
(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 7.40 (sec) , antiderivative size = 857, normalized size of antiderivative = 1.74 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \, dx=-\frac {\frac {2 \left (3 a^3 A b^2+15 a A b^4+35 a^5 C-73 a^3 b^2 C+56 a b^4 C\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (-24 a^2 A b^3-48 A b^5+56 a^4 b C-112 a^2 b^3 C-16 b^5 C\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (9 a^3 A b^2-27 a A b^4+105 a^5 C-195 a^3 b^2 C+72 a b^4 C\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 (a-b)^2 b^3 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (\frac {a \left (3 a^2 A b^2-9 A b^4+11 a^4 C-17 a^2 b^2 C\right ) \sin (c+d x)}{4 b^4 \left (a^2-b^2\right )^2}-\frac {a^3 A b^2 \sin (c+d x)+a^5 C \sin (c+d x)}{2 b^4 \left (-a^2+b^2\right ) (a+b \cos (c+d x))^2}+\frac {-5 a^4 A b^2 \sin (c+d x)+11 a^2 A b^4 \sin (c+d x)-13 a^6 C \sin (c+d x)+19 a^4 b^2 C \sin (c+d x)}{4 b^4 \left (-a^2+b^2\right )^2 (a+b \cos (c+d x))}+\frac {C \sin (2 (c+d x))}{3 b^3}\right )}{d} \] Input:

Integrate[(A + 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 + 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 + 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 + 105*a^5*C - 195*a^3*b^2 
*C + 72*a*b^4*C)*Cos[2*(c + d*x)]*(b + a*Sec[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*Ellipt 
icPi[-(a/b), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - S 
ec[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*C 
os[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 + 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 + d*x] + a^5*C*Sin[c + d*x])/(2*b^4*(-a^2 + b^2)*(a + b*Cos[c 
 + d*x])^2) + (-5*a^4*A*b^2*Sin[c + d*x] + 11*a^2*A*b^4*Sin[c + d*x] - ...
 

Rubi [A] (verified)

Time = 3.27 (sec) , antiderivative size = 457, normalized size of antiderivative = 0.93, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 4709, 3042, 3527, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4709

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3527

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3284

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

Input:

Int[(A + 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^2*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 - 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]))) - (((-6*a*(a^2*b^2*(3*A - 65*C) - 3*b^4*( 
3*A - 8*C) + 35*a^4*C)*EllipticE[(c + d*x)/2, 2])/(b*d) + ((2*(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))*Ellipti 
cF[(c + d*x)/2, 2])/(b*d) - (6*a*(15*A*b^6 + 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)/(3*b) + (2*(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 3527

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> 
Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b 
*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( 
A*d^2*(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, 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. \(2239\) vs. \(2(468)=936\).

Time = 7.69 (sec) , antiderivative size = 2240, normalized size of antiderivative = 4.54

method	result	size
default	\(\text {Expression too large to display}\)	\(2240\)

Input:

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

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*(A*b^2+6*C*a 
^2+3*C*a*b+C*b^2)/b^5*(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))+2/b^5*a^2*(3*A*b^2+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)*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*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))+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)*E 
llipticPi(cos(1/2*d*x+1/2*c),-2*b/(a-b),2^(1/2)))+4/3*C/b^3*(2*cos(1/2*d*x 
+1/2*c)*sin(1/2*d*x+1/2*c)^4-sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+2*(si 
n(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(co...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((A+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)**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