\(\int \frac {\sqrt {\cos (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [1330]

Optimal result

Integrand size = 43, antiderivative size = 426 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^3 \left (a^2-b^2\right )^2 d}-\frac {\left (15 A b^5-8 a^5 B+5 a^3 b^2 B-3 a b^4 B-a^2 b^3 (33 A+C)+a^4 b (24 A+7 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (15 A b^6-15 a^5 b B+6 a^3 b^3 B-3 a b^5 B+3 a^6 C-a^2 b^4 (38 A+C)+5 a^4 b^2 (7 A+2 C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^4 (a-b)^2 (a+b)^3 d}+\frac {\left (A b^2-a (b B-a C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}-\frac {\left (5 A b^4+7 a^3 b B-a b^3 B-3 a^4 C-a^2 b^2 (11 A+3 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 5.11 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\frac {2 \sqrt {\cos (c+d x)} \left (b \left (-5 A b^4-7 a^3 b B+a b^3 B+3 a^4 C+a^2 b^2 (11 A+3 C)\right )+a \left (-7 A b^4-9 a^3 b B+3 a b^3 B+5 a^4 C+a^2 b^2 (13 A+C)\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}+\frac {\frac {\left (5 A b^4-5 a^3 b B-a b^3 B+a^4 (8 A+C)+a^2 b^2 (-7 A+5 C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 \left (A b^3+2 a^3 B+a b^2 B-a^2 b (4 A+3 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a+b}+\frac {\left (15 A b^4+9 a^3 b B-3 a b^3 B+a^4 (8 A-5 C)-a^2 b^2 (29 A+C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a^2 b \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a^2 d} \] Input:

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

Output:

((2*Sqrt[Cos[c + d*x]]*(b*(-5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + a^2* 
b^2*(11*A + 3*C)) + a*(-7*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + 5*a^4*C + a^2*b^ 
2*(13*A + C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d* 
x])^2) + (((5*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*(8*A + C) + a^2*b^2*(-7*A 
+ 5*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(A*b^3 + 2 
*a^3*B + a*b^2*B - a^2*b*(4*A + 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - 
 b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((15*A*b^4 + 9*a^ 
3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*(-2*a*b*Elliptic 
E[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[ 
c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x 
]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2 
))/(8*a^2*d)
 

Rubi [A] (verified)

Time = 2.87 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.395, Rules used = {3042, 4600, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3284

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

Input:

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

Output:

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

rule 4600

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x 
_)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) 
*(x_)]^2), x_Symbol] :> Simp[d^(m + 2)   Int[(b + a*Cos[e + f*x])^m*(d*Cos[ 
e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr 
eeQ[{a, b, d, e, f, A, B, C, n}, x] &&  !IntegerQ[n] && IntegerQ[m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2021\) vs. \(2(417)=834\).

Time = 8.72 (sec) , antiderivative size = 2022, normalized size of antiderivative = 4.75

method	result	size
default	\(\text {Expression too large to display}\)	\(2022\)

Input:

int(cos(d*x+c)^(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+b*sec(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/a^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)*(3*A*b*EllipticF(cos(1/2*d*x+1/2*c),2^(1 
/2))+a*A*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-B*EllipticF(cos(1/2*d*x+1/2 
*c),2^(1/2))*a)-2/a^3*(6*A*b^2-3*B*a*b+C*a^2)/(a^2-a*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*a/(a-b),2^(1/2))+2 
*b^2*(A*b^2-B*a*b+C*a^2)/a^4*(1/2*a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c)*(-2*s 
in(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*a*cos(1/2*d*x+1/2*c)^2- 
a+b)^2+3/4*a^2*(a^2-3*b^2)/b^2/(a^2-b^2)^2*cos(1/2*d*x+1/2*c)*(-2*sin(1/2* 
d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*a*cos(1/2*d*x+1/2*c)^2-a+b)-3/ 
8/(a+b)/(a^2-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)*EllipticF( 
cos(1/2*d*x+1/2*c),2^(1/2))*a^2-1/4/(a+b)/(a^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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*a+7/8/(a+b)/(a 
^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2 
*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/ 
2*c),2^(1/2))+3/8*a^3/b^2/(a^2-b^2)^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos 
(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

integrate(cos(d*x+c)**(1/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+ 
c))**3,x)
 

Output:

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

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

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