\(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx\) [1333]

Optimal result

Integrand size = 43, antiderivative size = 496 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {\left (3 a^3 b B-9 a b^3 B+b^4 (5 A-8 C)-15 a^4 C+a^2 b^2 (A+29 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (3 A b^4+a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (3 A+11 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 a b^2 \left (a^2-b^2\right )^2 d}-\frac {\left (3 A b^6-3 a^5 b B+6 a^3 b^3 B-15 a b^5 B+15 a^6 C+5 a^2 b^4 (2 A+7 C)-a^4 b^2 (A+38 C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a (a-b)^2 b^3 (a+b)^3 d}-\frac {\left (3 a^3 b B-9 a b^3 B+b^4 (5 A-8 C)-15 a^4 C+a^2 b^2 (A+29 C)\right ) \sin (c+d x)}{4 b^3 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)}}-\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)} (b+a \cos (c+d x))^2}+\frac {\left (3 A b^4+a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (3 A+11 C)\right ) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d \sqrt {\cos (c+d x)} (b+a \cos (c+d x))} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 5.42 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {-\frac {\frac {\left (-9 a^4 b B+19 a^2 b^3 B-16 b^5 B+45 a^5 C+a b^4 (9 A+56 C)-a^3 b^2 (3 A+95 C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}-\frac {8 b \left (a^3 b B-4 a b^3 B+2 b^4 (A-C)-5 a^4 C+a^2 b^2 (A+10 C)\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a (a+b)}+\frac {\left (-3 a^3 b B+9 a b^3 B+15 a^4 C+b^4 (-5 A+8 C)-a^2 b^2 (A+29 C)\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}+\frac {\sqrt {\cos (c+d x)} \left (2 a b \left (-5 a^3 b B+11 a b^3 B+a^2 b^2 (A-47 C)+25 a^4 C+b^4 (-7 A+16 C)\right ) \sin (c+d x)+a^2 \left (-3 a^3 b B+9 a b^3 B+15 a^4 C+b^4 (-5 A+8 C)-a^2 b^2 (A+29 C)\right ) \sin (2 (c+d x))+16 \left (-a^2 b+b^3\right )^2 C \tan (c+d x)\right )}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^2}}{8 b^3 d} \] Input:

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

Output:

(-((((-9*a^4*b*B + 19*a^2*b^3*B - 16*b^5*B + 45*a^5*C + a*b^4*(9*A + 56*C) 
 - a^3*b^2*(3*A + 95*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b 
) - (8*b*(a^3*b*B - 4*a*b^3*B + 2*b^4*(A - C) - 5*a^4*C + a^2*b^2*(A + 10* 
C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + 
d*x)/2, 2]))/(a*(a + b)) + ((-3*a^3*b*B + 9*a*b^3*B + 15*a^4*C + b^4*(-5*A 
 + 8*C) - a^2*b^2*(A + 29*C))*(-2*a*b*EllipticE[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*b 
*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a* 
b*(-5*a^3*b*B + 11*a*b^3*B + a^2*b^2*(A - 47*C) + 25*a^4*C + b^4*(-7*A + 1 
6*C))*Sin[c + d*x] + a^2*(-3*a^3*b*B + 9*a*b^3*B + 15*a^4*C + b^4*(-5*A + 
8*C) - a^2*b^2*(A + 29*C))*Sin[2*(c + d*x)] + 16*(-(a^2*b) + b^3)^2*C*Tan[ 
c + d*x]))/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(8*b^3*d)
 

Rubi [A] (verified)

Time = 3.64 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.97, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4600, 3042, 3534, 27, 3042, 3534, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\sin (c+d x) \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}-\frac {\frac {2 \int -\frac {-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2+\left (-15 C a^4+3 b B a^3+b^2 (A+29 C) a^2-9 b^3 B a+b^4 (5 A-8 C)\right ) \cos ^2(c+d x) a-b^4 (7 A+24 C) a+8 b^5 B+4 b \left (-5 C a^4+b B a^3+b^2 (A+10 C) a^2-4 b^3 B a+2 b^4 (A-C)\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}+\frac {2 \sin (c+d x) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d \sqrt {\cos (c+d x)}}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sin (c+d x) \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}-\frac {\frac {2 \sin (c+d x) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2+\left (-15 C a^4+3 b B a^3+b^2 (A+29 C) a^2-9 b^3 B a+b^4 (5 A-8 C)\right ) \cos ^2(c+d x) a-b^4 (7 A+24 C) a+8 b^5 B+4 b \left (-5 C a^4+b B a^3+b^2 (A+10 C) a^2-4 b^3 B a+2 b^4 (A-C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sin (c+d x) \left (-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right )}{b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)}-\frac {\frac {2 \sin (c+d x) \left (-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right )}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 C a^5+3 b B a^4+b^2 (A+33 C) a^3-5 b^3 B a^2+\left (-15 C a^4+3 b B a^3+b^2 (A+29 C) a^2-9 b^3 B a+b^4 (5 A-8 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a-b^4 (7 A+24 C) a+8 b^5 B+4 b \left (-5 C a^4+b B a^3+b^2 (A+10 C) a^2-4 b^3 B a+2 b^4 (A-C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} (a \cos (c+d x)+b)^2}\)

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3284

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

Input:

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

Output:

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

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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2020\) vs. \(2(483)=966\).

Time = 8.63 (sec) , antiderivative size = 2021, normalized size of antiderivative = 4.07

method	result	size
default	\(\text {Expression too large to display}\)	\(2021\)

Input:

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(5/2)/(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*C/b^3/sin(1/ 
2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-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*cos(1/2*d*x+1/2*c)-(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))*(sin(1/2*d*x+ 
1/2*c)^2)^(1/2))+2*(A*b^2-C*a^2)/b^2/a*(a^2/b/(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*a*cos(1/2*d*x+1/2 
*c)^2-a+b)-1/2/(a+b)/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))+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)*(si 
n(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 
))-1/2/b/(a^2-b^2)/(a^2-a*b)*a^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*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^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)*EllipticPi(cos(1/2*d*x+1/2*c 
),2*a/(a-b),2^(1/2)))-2*(A*b^2-B*a*b+C*a^2)/a/b*(1/2*a^2/b/(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*a...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

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