\(\int \frac {\sqrt {\sec (c+d x)} (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [1028]

Optimal result

Integrand size = 43, antiderivative size = 478 \[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (3 A b^4+5 a^3 b B+a b^3 B-a^4 C-a^2 b^2 (9 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^2 b \left (a^2-b^2\right )^2 d}+\frac {\left (3 A b^4-7 a^3 b B+a b^3 B-a^2 b^2 (5 A-3 C)+a^4 (8 A+3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^3 \left (a^2-b^2\right )^2 d}-\frac {\left (3 A b^6-3 a^5 b B-10 a^3 b^3 B+a b^5 B-3 a^2 b^4 (2 A-C)-a^6 C+5 a^4 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^3 (a-b)^2 b (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a b \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \] Output:

-1/4*(3*A*b^4+5*B*a^3*b+B*a*b^3-a^4*C-a^2*b^2*(9*A+5*C))*cos(d*x+c)^(1/2)* 
EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/a^2/b/(a^2-b^2)^2/d 
+1/4*(3*A*b^4-7*B*a^3*b+B*a*b^3-a^2*b^2*(5*A-3*C)+a^4*(8*A+3*C))*cos(d*x+c 
)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/a^3/(a^2-b 
^2)^2/d-1/4*(3*A*b^6-3*a^5*b*B-10*a^3*b^3*B+a*b^5*B-3*a^2*b^4*(2*A-C)-a^6* 
C+5*a^4*b^2*(3*A+2*C))*cos(d*x+c)^(1/2)*EllipticPi(sin(1/2*d*x+1/2*c),2*a/ 
(a+b),2^(1/2))*sec(d*x+c)^(1/2)/a^3/(a-b)^2/b/(a+b)^3/d-1/2*(A*b^2-a*(B*b- 
C*a))*sec(d*x+c)^(1/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(d*x+c))^2+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))*sec(d*x+c)^(1/2)*sin(d*x+c) 
/a/b/(a^2-b^2)^2/d/(a+b*sec(d*x+c))
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1046\) vs. \(2(478)=956\).

Time = 8.97 (sec) , antiderivative size = 1046, normalized size of antiderivative = 2.19 \[ \int \frac {\sqrt {\sec (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 
2)*((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 
)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[- 
(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[ 
c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + 
(2*(16*a^3*A*b + 8*a*A*b^3 - 24*a^2*b^2*B + 8*a^3*b*C + 16*a*b^3*C)*Cos[c 
+ d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + 
 d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - 
 Cos[c + d*x]^2)) + ((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C 
+ 5*a^2*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c 
 + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d 
*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c 
+ d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*Elliptic 
Pi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec 
[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sq 
rt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos 
[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))) 
/(8*a*(a - b)^2*b*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2* 
d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2) 
*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-9*a^2*A*b^2 + 3*A*b^4 + 5*...
 

Rubi [A] (verified)

Time = 3.17 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4586, 27, 3042, 4588, 27, 3042, 4594, 3042, 4274, 3042, 4258, 3042, 3119, 3120, 4336, 3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4588

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4594

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\int \frac {a \left (-C a^4+5 b B a^3-b^2 (9 A+5 C) a^2+b^3 B a+3 A b^4\right )-b \left ((8 A+3 C) a^4-7 b B a^3-b^2 (5 A-3 C) a^2+b^3 B a+3 A b^4\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{a+b \sec (c+d x)}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\int \frac {a \left (-C a^4+5 b B a^3-b^2 (9 A+5 C) a^2+b^3 B a+3 A b^4\right )-b \left ((8 A+3 C) a^4-7 b B a^3-b^2 (5 A-3 C) a^2+b^3 B a+3 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {a \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-b \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right ) \int \sqrt {\sec (c+d x)}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {a \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right ) \int \sqrt {\cos (c+d x)}dx-b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\frac {2 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right )}{d}-b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}+\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}{a+b \csc \left (c+d x+\frac {\pi }{2}\right )}dx}{a^2}+\frac {\frac {2 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right )}{d}-\frac {2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right )}{d}}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 4336

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (b+a \cos (c+d x))}dx}{a^2}+\frac {\frac {2 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right )}{d}-\frac {2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right )}{d}}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (b+a \sin \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a^2}+\frac {\frac {2 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right )}{d}-\frac {2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right )}{d}}{a^2}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

\(\Big \downarrow \) 3284

\(\displaystyle \frac {\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right )}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac {\frac {\frac {2 a \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right )}{d}-\frac {2 b \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right )}{d}}{a^2}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right ) \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right )}{a^2 d (a+b)}}{2 a \left (a^2-b^2\right )}}{4 b \left (a^2-b^2\right )}-\frac {\sin (c+d x) \sqrt {\sec (c+d x)} \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \sec (c+d x))^2}\)

Input:

Int[(Sqrt[Sec[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:

-1/2*((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(a + b*Sec[c + d*x])^2) + (-1/2*(((2*a*(3*A*b^4 + 5*a^3*b*B + a*b^3* 
B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 
 2]*Sqrt[Sec[c + d*x]])/d - (2*b*(3*A*b^4 - 7*a^3*b*B + a*b^3*B - a^2*b^2* 
(5*A - 3*C) + a^4*(8*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2 
]*Sqrt[Sec[c + d*x]])/d)/a^2 + (2*(3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a* 
b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + 
d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*( 
a + b)*d))/(a*(a^2 - b^2)) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a 
^2*b^2*(A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*S 
ec[c + d*x])))/(4*b*(a^2 - b^2))
 

Defintions of rubi rules used

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 4258

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] 
)^n*Sin[c + d*x]^n   Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && 
 EqQ[n^2, 1/4]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4336

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(3/2)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[d*Sqrt[d*Sin[e + f*x]]*Sqrt[d*Csc[e + f*x]]   Int[ 
1/(Sqrt[d*Sin[e + f*x]]*(b + a*Sin[e + f*x])), x], x] /; FreeQ[{a, b, d, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4586

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[(-d)*(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + 
b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^(n - 1)/(b*f*(a^2 - b^2)*(m + 1)) 
), x] + Simp[d/(b*(a^2 - b^2)*(m + 1))   Int[(a + b*Csc[e + f*x])^(m + 1)*( 
d*Csc[e + f*x])^(n - 1)*Simp[A*b^2*(n - 1) - a*(b*B - a*C)*(n - 1) + b*(a*A 
 - b*B + a*C)*(m + 1)*Csc[e + f*x] - (b*(A*b - a*B)*(m + n + 1) + C*(a^2*n 
+ b^2*(m + 1)))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C 
}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && GtQ[n, 0]
 

rule 4588

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a + b*Csc 
[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*(m + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f 
*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*(m + n + 
1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m 
+ n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x 
] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] && ILtQ[n, 0])
 

rule 4594

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)/(a^2*d^2)   Int[(d*Csc[e + 
f*x])^(3/2)/(a + b*Csc[e + f*x]), x], x] + Simp[1/a^2   Int[(a*A - (A*b - a 
*B)*Csc[e + f*x])/Sqrt[d*Csc[e + f*x]], x], x] /; FreeQ[{a, b, d, e, f, A, 
B, C}, x] && NeQ[a^2 - b^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1972\) vs. \(2(457)=914\).

Time = 6.96 (sec) , antiderivative size = 1973, normalized size of antiderivative = 4.13

method	result	size
default	\(\text {Expression too large to display}\)	\(1973\)

Input:

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

Fricas [F(-1)]

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

integrate(sec(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(-1)]

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

integrate(sec(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:

Timed out
 

Maxima [F(-1)]

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

integrate(sec(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]

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

integrate(sec(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(sec(d*x + c))/(b*se 
c(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int(sec(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(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)*a + int((sqrt(sec(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(sec(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