\(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx\) [1328]

Optimal result

Integrand size = 43, antiderivative size = 387 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=-\frac {\left (3 a^2 b B-2 b^3 B-a b^2 (A-4 C)-5 a^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 \left (a^2-b^2\right ) d}+\frac {\left (3 A b^2-3 a b B+5 a^2 C-2 b^2 C\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 b^2 \left (a^2-b^2\right ) d}-\frac {\left (3 A b^4+3 a^3 b B-5 a b^3 B-a^2 b^2 (A-7 C)-5 a^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{(a-b) b^3 (a+b)^2 d}+\frac {\left (3 A b^2-3 a b B+5 a^2 C-2 b^2 C\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x)}+\frac {\left (3 a^2 b B-2 b^3 B-a b^2 (A-4 C)-5 a^3 C\right ) \sin (c+d x)}{b^3 \left (a^2-b^2\right ) d \sqrt {\cos (c+d x)}}-\frac {\left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x))} \] Output:

-(3*B*a^2*b-2*B*b^3-a*b^2*(A-4*C)-5*a^3*C)*EllipticE(sin(1/2*d*x+1/2*c),2^ 
(1/2))/b^3/(a^2-b^2)/d+1/3*(3*A*b^2-3*B*a*b+5*C*a^2-2*C*b^2)*InverseJacobi 
AM(1/2*d*x+1/2*c,2^(1/2))/b^2/(a^2-b^2)/d-(3*A*b^4+3*B*a^3*b-5*B*a*b^3-a^2 
*b^2*(A-7*C)-5*a^4*C)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))/(a- 
b)/b^3/(a+b)^2/d+1/3*(3*A*b^2-3*B*a*b+5*C*a^2-2*C*b^2)*sin(d*x+c)/b^2/(a^2 
-b^2)/d/cos(d*x+c)^(3/2)+(3*B*a^2*b-2*B*b^3-a*b^2*(A-4*C)-5*a^3*C)*sin(d*x 
+c)/b^3/(a^2-b^2)/d/cos(d*x+c)^(1/2)-(A*b^2-a*(B*b-C*a))*sin(d*x+c)/b/(a^2 
-b^2)/d/cos(d*x+c)^(3/2)/(b+a*cos(d*x+c))
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 5.38 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.09 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx=\frac {\frac {\frac {2 \left (-27 a^3 b B+30 a b^3 B+a^2 b^2 (9 A-44 C)+45 a^4 C-4 b^4 (3 A+C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {8 b \left (-6 a^2 b B+3 b^3 B+a b^2 (3 A-7 C)+10 a^3 C\right ) \left ((a+b) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )\right )}{a (a+b)}+\frac {6 \left (-3 a^2 b B+2 b^3 B+a b^2 (A-4 C)+5 a^3 C\right ) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a b \sqrt {\sin ^2(c+d x)}}}{(a-b) (a+b)}+\frac {8 b \left (-a^2+b^2\right ) (3 b B-5 a C) \sin (c+d x)+6 a \left (-3 a^2 b B+2 b^3 B+a b^2 (A-4 C)+5 a^3 C\right ) \sin (2 (c+d x))+8 b^2 \left (-a^2+b^2\right ) C \tan (c+d x)}{\left (-a^2+b^2\right ) \sqrt {\cos (c+d x)} (b+a \cos (c+d x))}}{12 b^3 d} \] 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])^2),x]
 

Output:

(((2*(-27*a^3*b*B + 30*a*b^3*B + a^2*b^2*(9*A - 44*C) + 45*a^4*C - 4*b^4*( 
3*A + C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-6*a^ 
2*b*B + 3*b^3*B + a*b^2*(3*A - 7*C) + 10*a^3*C)*((a + b)*EllipticF[(c + d* 
x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + (6* 
(-3*a^2*b*B + 2*b^3*B + a*b^2*(A - 4*C) + 5*a^3*C)*(-2*a*b*EllipticE[ArcSi 
n[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)*(a + b)) + (8*b*(-a^ 
2 + b^2)*(3*b*B - 5*a*C)*Sin[c + d*x] + 6*a*(-3*a^2*b*B + 2*b^3*B + a*b^2* 
(A - 4*C) + 5*a^3*C)*Sin[2*(c + d*x)] + 8*b^2*(-a^2 + b^2)*C*Tan[c + d*x]) 
/((-a^2 + b^2)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])))/(12*b^3*d)
 

Rubi [A] (verified)

Time = 3.06 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.95, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

\(\displaystyle \frac {\frac {\frac {2 \int -\frac {-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2+3 \left (-5 C a^3+3 b B a^2-b^2 (A-4 C) a-2 b^3 B\right ) \cos ^2(c+d x) a-12 b^3 B a+2 b^4 (3 A+C)+2 b \left (-10 C a^3+6 b B a^2-b^2 (3 A-7 C) a-3 b^3 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}+\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2+3 \left (-5 C a^3+3 b B a^2-b^2 (A-4 C) a-2 b^3 B\right ) \cos ^2(c+d x) a-12 b^3 B a+2 b^4 (3 A+C)+2 b \left (-10 C a^3+6 b B a^2-b^2 (3 A-7 C) a-3 b^3 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{b}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2+3 \left (-5 C a^3+3 b B a^2-b^2 (A-4 C) a-2 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a-12 b^3 B a+2 b^4 (3 A+C)+2 b \left (-10 C a^3+6 b B a^2-b^2 (3 A-7 C) a-3 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{b}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 3538

\(\displaystyle \frac {\frac {\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {\int -\frac {a \left (-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right )-a^2 b \left (5 C a^2-3 b B a+3 A b^2-2 b^2 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{b}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right ) \int \sqrt {\cos (c+d x)}dx+\frac {\int \frac {a \left (-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right )-a^2 b \left (5 C a^2-3 b B a+3 A b^2-2 b^2 C\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{b}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}-\frac {3 \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {\int \frac {a \left (-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right )-a^2 b \left (5 C a^2-3 b B a+3 A b^2-2 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{b}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\frac {6 \sin (c+d x) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {\int \frac {a \left (-15 C a^4+9 b B a^3-b^2 (3 A-16 C) a^2-12 b^3 B a+2 b^4 (3 A+C)\right )-a^2 b \left (5 C a^2-3 b B a+3 A b^2-2 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}+\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right )}{d}}{b}}{3 b}+\frac {2 \sin (c+d x) \left (5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right )}{3 b d \cos ^{\frac {3}{2}}(c+d x)}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b d \left (a^2-b^2\right ) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+b)}\)

\(\Big \downarrow \) 3481

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 3284

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

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])^2),x]
 

Output:

-(((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/ 
2)*(b + a*Cos[c + d*x]))) + ((2*(3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Si 
n[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) + (-(((6*(3*a^2*b*B - 2*b^3*B - a*b 
^2*(A - 4*C) - 5*a^3*C)*EllipticE[(c + d*x)/2, 2])/d + ((-2*a*b*(3*A*b^2 - 
 3*a*b*B + 5*a^2*C - 2*b^2*C)*EllipticF[(c + d*x)/2, 2])/d + (6*a*(3*A*b^4 
 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*a)/( 
a + b), (c + d*x)/2, 2])/((a + b)*d))/a)/b) + (6*(3*a^2*b*B - 2*b^3*B - a* 
b^2*(A - 4*C) - 5*a^3*C)*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]))/(3*b))/(2 
*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] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1003\) vs. \(2(382)=764\).

Time = 7.51 (sec) , antiderivative size = 1004, normalized size of antiderivative = 2.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1004\)

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))^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*C/b^2*(-1/6* 
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)/(c 
os(1/2*d*x+1/2*c)^2-1/2)^2+1/3*(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 
llipticF(cos(1/2*d*x+1/2*c),2^(1/2)))+2*(B*b-2*C*a)/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-B*a*b+C*a^2)/b^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*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*co 
s(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))-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)*Ellipti 
cPi(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*(...
 

Fricas [F(-1)]

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

Output:

Timed out
 

Sympy [F(-1)]

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

Output:

Timed out
 

Maxima [F(-1)]

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

Output:

Timed out
 

Giac [F]

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

Output:

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

Mupad [F(-1)]

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

Reduce [F]

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

Output:

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