\(\int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\) [1062]

Optimal result

Integrand size = 45, antiderivative size = 350 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\frac {2 \left (8 A b^2-6 a b B+a^2 (A+3 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 a^3 d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (8 A b^3+3 a^3 B-6 a b^2 B-a^2 (5 A b-3 b C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^3 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (4 A b^2-3 a b B-a^2 (A-3 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d \sqrt {\sec (c+d x)}} \] Output:

2/3*(8*A*b^2-6*B*a*b+a^2*(A+3*C))*((b+a*cos(d*x+c))/(a+b))^(1/2)*InverseJa 
cobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^(1/2))*sec(d*x+c)^(1/2)/a^3/d/(a+b* 
sec(d*x+c))^(1/2)+2/3*(8*A*b^3+3*B*a^3-6*B*a*b^2-a^2*(5*A*b-3*C*b))*Ellipt 
icE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1/2)/a^3 
/(a^2-b^2)/d/((b+a*cos(d*x+c))/(a+b))^(1/2)/sec(d*x+c)^(1/2)+2*(A*b^2-a*(B 
*b-C*a))*sin(d*x+c)/a/(a^2-b^2)/d/sec(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)- 
2/3*(4*A*b^2-3*B*a*b-a^2*(A-3*C))*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c)/a^2/(a 
^2-b^2)/d/sec(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 8.26 (sec) , antiderivative size = 4557, normalized size of antiderivative = 13.02 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(-5*a 
^2*A*b + 11*A*b^3 + 3*a^3*B - 9*a*b^2*B + 6*a^2*b*C - 5*a^2*A*b*Cos[2*c] + 
 5*A*b^3*Cos[2*c] + 3*a^3*B*Cos[2*c] - 3*a*b^2*B*Cos[2*c])*Csc[c]*Sec[c])/ 
(3*a^3*(a^2 - b^2)*d) + (4*A*Cos[d*x]*Sin[c])/(3*a^2*d) + (4*A*Cos[c]*Sin[ 
d*x])/(3*a^2*d) + (4*Sec[c]*(A*b^4*Sin[c] - a*b^3*B*Sin[c] + a^2*b^2*C*Sin 
[c] - a*A*b^3*Sin[d*x] + a^2*b^2*B*Sin[d*x] - a^3*b*C*Sin[d*x]))/(a^3*(a^2 
 - b^2)*d*(b + a*Cos[c + d*x]))))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c 
 + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (4*A*AppellF1[ 
1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - Arc 
Tan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2] 
))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/ 
(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Co 
s[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - 
 ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x 
 - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Co 
t[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[ 
c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[C 
ot[c]]]])/(3*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x 
])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (8* 
A*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Si...
 

Rubi [A] (verified)

Time = 2.66 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.422, Rules used = {3042, 4588, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4588

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4592

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4523

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4343

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

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

\(\Big \downarrow \) 4345

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3142

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3140

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

Input:

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

Output:

(2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x 
]]*Sqrt[a + b*Sec[c + d*x]]) - (-1/3*((2*(a^2 - b^2)*(8*A*b^2 - 6*a*b*B + 
a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, ( 
2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A 
*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*EllipticE[(c + d*x)/2, ( 
2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a 
+ b)]*Sqrt[Sec[c + d*x]]))/a + (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt 
[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]))/(a*(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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 4343

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

rule 4345

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

rule 4523

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d 
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a   I 
nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) 
/(a*d)   Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ 
[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 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 4592

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*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d 
*Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m 
*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* 
Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d 
, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1509\) vs. \(2(333)=666\).

Time = 32.27 (sec) , antiderivative size = 1510, normalized size of antiderivative = 4.31

method	result	size
default	\(\text {Expression too large to display}\)	\(1510\)
parts	\(\text {Expression too large to display}\)	\(1648\)

Input:

int((A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(3/2)/(a+b*sec(d*x+c))^(3/2 
),x,method=_RETURNVERBOSE)
 

Output:

2/3/d/a^3/(a+b)/((a-b)/(a+b))^(1/2)*(a+b*sec(d*x+c))^(1/2)/(cos(d*x+c)^2*a 
+a*cos(d*x+c)+b*cos(d*x+c)+b)/sec(d*x+c)^(3/2)*(A*(1/(cos(d*x+c)+1))^(1/2) 
*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*EllipticE(((a-b)/(a 
+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(5*cos(d*x+c)+10 
+5*sec(d*x+c))+A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d 
*x+c)+1))^(1/2)*b^3*EllipticE(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)) 
,(-(a+b)/(a-b))^(1/2))*(-8*cos(d*x+c)-16-8*sec(d*x+c))+B*(1/(cos(d*x+c)+1) 
)^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*EllipticE(((a- 
b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-3*cos(d*x 
+c)-6-3*sec(d*x+c))+B*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/( 
cos(d*x+c)+1))^(1/2)*a*b^2*EllipticE(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot( 
d*x+c)),(-(a+b)/(a-b))^(1/2))*(6*cos(d*x+c)+12+6*sec(d*x+c))+C*(1/(cos(d*x 
+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*Ellipt 
icE(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-3 
*cos(d*x+c)-6-3*sec(d*x+c))+A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d 
*x+c))/(cos(d*x+c)+1))^(1/2)*a^3*EllipticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c 
)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*(-cos(d*x+c)-2-sec(d*x+c))+A*(1/(cos(d 
*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*a^2*b*Elli 
pticF(((a-b)/(a+b))^(1/2)*(-csc(d*x+c)+cot(d*x+c)),(-(a+b)/(a-b))^(1/2))*( 
-6*cos(d*x+c)-12-6*sec(d*x+c))+A*(1/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 860, normalized size of antiderivative = 2.46 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9*(sqrt(2)*(3*I*(A + 3*C)*a^4*b - 15*I*B*a^3*b^2 + 2*I*(8*A - 3*C)*a^2* 
b^3 + 12*I*B*a*b^4 - 16*I*A*b^5 + (3*I*(A + 3*C)*a^5 - 15*I*B*a^4*b + 2*I* 
(8*A - 3*C)*a^3*b^2 + 12*I*B*a^2*b^3 - 16*I*A*a*b^4)*cos(d*x + c))*sqrt(a) 
*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 
 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + c) + 2*b)/a) + sqrt(2)*(-3*I*(A + 
 3*C)*a^4*b + 15*I*B*a^3*b^2 - 2*I*(8*A - 3*C)*a^2*b^3 - 12*I*B*a*b^4 + 16 
*I*A*b^5 + (-3*I*(A + 3*C)*a^5 + 15*I*B*a^4*b - 2*I*(8*A - 3*C)*a^3*b^2 - 
12*I*B*a^2*b^3 + 16*I*A*a*b^4)*cos(d*x + c))*sqrt(a)*weierstrassPInverse(- 
4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) 
 - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(2)*(3*I*B*a^4*b - I*(5*A - 3*C)*a 
^3*b^2 - 6*I*B*a^2*b^3 + 8*I*A*a*b^4 + (3*I*B*a^5 - I*(5*A - 3*C)*a^4*b - 
6*I*B*a^3*b^2 + 8*I*A*a^2*b^3)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3* 
(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3* 
(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3 
*I*a*sin(d*x + c) + 2*b)/a)) - 3*sqrt(2)*(-3*I*B*a^4*b + I*(5*A - 3*C)*a^3 
*b^2 + 6*I*B*a^2*b^3 - 8*I*A*a*b^4 + (-3*I*B*a^5 + I*(5*A - 3*C)*a^4*b + 6 
*I*B*a^3*b^2 - 8*I*A*a^2*b^3)*cos(d*x + c))*sqrt(a)*weierstrassZeta(-4/3*( 
3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*( 
3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3* 
I*a*sin(d*x + c) + 2*b)/a)) - 6*((A*a^5 - A*a^3*b^2)*cos(d*x + c)^2 + (...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{4} b^{2}+2 \sec \left (d x +c \right )^{3} a b +\sec \left (d x +c \right )^{2} a^{2}}d x \right ) a +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{3} b^{2}+2 \sec \left (d x +c \right )^{2} a b +\sec \left (d x +c \right ) a^{2}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sqrt {\sec \left (d x +c \right ) b +a}}{\sec \left (d x +c \right )^{2} b^{2}+2 \sec \left (d x +c \right ) a b +a^{2}}d x \right ) c \] Input:

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

Output:

int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + a))/(sec(c + d*x)**4*b**2 + 
2*sec(c + d*x)**3*a*b + sec(c + d*x)**2*a**2),x)*a + int((sqrt(sec(c + d*x 
))*sqrt(sec(c + d*x)*b + a))/(sec(c + d*x)**3*b**2 + 2*sec(c + d*x)**2*a*b 
 + sec(c + d*x)*a**2),x)*b + int((sqrt(sec(c + d*x))*sqrt(sec(c + d*x)*b + 
 a))/(sec(c + d*x)**2*b**2 + 2*sec(c + d*x)*a*b + a**2),x)*c