3.14.32 \(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx\) [1332]

3.14.32.1 Optimal result

Integrand size = 43, antiderivative size = 409 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=-\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 a^2 b \left (a^2-b^2\right )^2 d}-\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 C)\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))} \]

output

-1/4*(A*b^4-B*a^3*b-5*B*a*b^3-3*a^4*C+a^2*b^2*(5*A+9*C))*(cos(1/2*d*x+1/2* 
c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a/b^2 
/(a^2-b^2)^2/d+1/4*(A*b^4+3*B*a^3*b+3*B*a*b^3+a^4*C-7*a^2*b^2*(A+C))*(cos( 
1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^ 
(1/2))/a^2/b/(a^2-b^2)^2/d-1/4*(A*b^6-a^5*b*B+10*a^3*b^3*B+3*a*b^5*B-3*a^4 
*b^2*(A-2*C)-3*a^6*C-5*a^2*b^4*(2*A+3*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos 
(1/2*d*x+1/2*c)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/(a+b),2^(1/2))/a^2/(a-b) 
^2/b^2/(a+b)^3/d-1/2*(A*b^2-a*(B*b-C*a))*sin(d*x+c)*cos(d*x+c)^(1/2)/b/(a^ 
2-b^2)/d/(b+a*cos(d*x+c))^2+1/4*(A*b^4-B*a^3*b-5*B*a*b^3-3*a^4*C+a^2*b^2*( 
5*A+9*C))*sin(d*x+c)*cos(d*x+c)^(1/2)/b^2/(a^2-b^2)^2/d/(b+a*cos(d*x+c))
 

3.14.32.2 Mathematica [A] (warning: unable to verify)

Time = 9.62 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.30 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx=\frac {\frac {2 \left (a^2 A b^2+5 A b^4+3 a^3 b B-9 a b^3 B+9 a^4 C-19 a^2 b^2 C+16 b^4 C\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}+\frac {\left (-24 a A b^3+8 a^2 b^2 B+16 b^4 B+8 a^3 b C-32 a b^3 C\right ) \left (2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\frac {2 b \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a+b}\right )}{a}+\frac {2 \left (-5 a^2 A b^2-A b^4+a^3 b B+5 a b^3 B+3 a^4 C-9 a^2 b^2 C\right ) \cos (2 (c+d x)) \left (-2 a b E\left (\left .\arcsin \left (\sqrt {\cos (c+d x)}\right )\right |-1\right )+2 b (a+b) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )+\left (a^2-2 b^2\right ) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\cos (c+d x)}\right ),-1\right )\right ) \sin (c+d x)}{a^2 b \sqrt {1-\cos ^2(c+d x)} \left (-1+2 \cos ^2(c+d x)\right )}}{16 (a-b)^2 b^2 (a+b)^2 d}+\frac {\sqrt {\cos (c+d x)} \left (\frac {A b^2 \sin (c+d x)-a b B \sin (c+d x)+a^2 C \sin (c+d x)}{2 b \left (-a^2+b^2\right ) (b+a \cos (c+d x))^2}+\frac {5 a^2 A b^2 \sin (c+d x)+A b^4 \sin (c+d x)-a^3 b B \sin (c+d x)-5 a b^3 B \sin (c+d x)-3 a^4 C \sin (c+d x)+9 a^2 b^2 C \sin (c+d x)}{4 b^2 \left (-a^2+b^2\right )^2 (b+a \cos (c+d x))}\right )}{d} \]

input

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

output

((2*(a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B + 9*a^4*C - 19*a^2*b^2*C 
+ 16*b^4*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + ((-24*a*A 
*b^3 + 8*a^2*b^2*B + 16*b^4*B + 8*a^3*b*C - 32*a*b^3*C)*(2*EllipticF[(c + 
d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + 
 (2*(-5*a^2*A*b^2 - A*b^4 + a^3*b*B + 5*a*b^3*B + 3*a^4*C - 9*a^2*b^2*C)*C 
os[2*(c + d*x)]*(-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^2*b*Sqrt[1 - Co 
s[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(16*(a - b)^2*b^2*(a + b)^2*d) + ( 
Sqrt[Cos[c + d*x]]*((A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c 
 + d*x])/(2*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (5*a^2*A*b^2*Sin[c + 
d*x] + A*b^4*Sin[c + d*x] - a^3*b*B*Sin[c + d*x] - 5*a*b^3*B*Sin[c + d*x] 
- 3*a^4*C*Sin[c + d*x] + 9*a^2*b^2*C*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2*( 
b + a*Cos[c + d*x]))))/d
 

3.14.32.3 Rubi [A] (verified)

Time = 2.61 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, 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, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3534

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3538

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3481

\(\displaystyle -\frac {-\frac {\frac {\frac {b \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a}-\frac {\left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a}}{a}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {\frac {\frac {b \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}-\frac {\left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle -\frac {-\frac {\frac {\frac {2 b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d}-\frac {\left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a}}{a}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d}}{2 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle -\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}-\frac {-\frac {\sin (c+d x) \sqrt {\cos (c+d x)} \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{b d \left (a^2-b^2\right ) (a \cos (c+d x)+b)}-\frac {\frac {\frac {2 b \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d}-\frac {2 \left (-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a d (a+b)}}{a}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right )}{a d}}{2 b \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}\)

input

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

output

-1/2*((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(b + a*Cos[c + d*x])^2) - (-1/2*((-2*(A*b^4 - a^3*b*B - 5*a*b^3*B - 
3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(a*d) + ((2*b*(A 
*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(c + d 
*x)/2, 2])/(a*d) - (2*(A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4* 
b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*a)/(a + b), 
 (c + d*x)/2, 2])/(a*(a + b)*d))/a)/(b*(a^2 - b^2)) - ((A*b^4 - a^3*b*B - 
5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x] 
)/(b*(a^2 - b^2)*d*(b + a*Cos[c + d*x])))/(4*b*(a^2 - b^2))
 

3.14.32.3.1 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]
 

3.14.32.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1878\) vs. \(2(473)=946\).

Time = 5.43 (sec) , antiderivative size = 1879, normalized size of antiderivative = 4.59

method	result	size
default	\(\text {Expression too large to display}\)	\(1879\)

input

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/cos(d*x+c)^(3/2)/(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/a/(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*(-2*A*b+B*a)/a^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*cos(1/2*d*x+1/2*c 
)^2*a-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^2*(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*cos( 
1/2*d*x+1/2*c)^2*a-a+b)^2+3/4*a^2*(a^2-3*b^2)/b^2/(a^2-b^2)^2*cos(1/2*d...
 

3.14.32.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.14.32.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.14.32.7 Maxima [F(-1)]

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

input

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

output

Timed out
 

3.14.32.8 Giac [F]

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

input

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

3.14.32.9 Mupad [F(-1)]

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

input

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