\(\int \frac {\sec ^{\frac {5}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [1026]

Optimal result

Integrand size = 43, antiderivative size = 556 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\frac {\left (3 a^3 b B-9 a b^3 B+b^4 (5 A-8 C)-15 a^4 C+a^2 b^2 (A+29 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 b^3 \left (a^2-b^2\right )^2 d}+\frac {\left (3 A b^4+a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (3 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a b^2 \left (a^2-b^2\right )^2 d}-\frac {\left (3 A b^6-3 a^5 b B+6 a^3 b^3 B-15 a b^5 B+15 a^6 C+5 a^2 b^4 (2 A+7 C)-a^4 b^2 (A+38 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a (a-b)^2 b^3 (a+b)^3 d}-\frac {\left (3 a^3 b B-9 a b^3 B+b^4 (5 A-8 C)-15 a^4 C+a^2 b^2 (A+29 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^3 \left (a^2-b^2\right )^2 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (3 A b^4+a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (3 A+11 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 9.65 (sec) , antiderivative size = 1087, normalized size of antiderivative = 1.96 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + 
d*x]^2)*((2*(-3*a^3*A*b^2 + 9*a*A*b^4 - 9*a^4*b*B + 19*a^2*b^3*B - 16*b^5* 
B + 45*a^5*C - 95*a^3*b^2*C + 56*a*b^4*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*(-8*a^2*A*b^3 - 16*A*b^5 - 8*a^ 
3*b^2*B + 32*a*b^4*B + 40*a^4*b*C - 80*a^2*b^3*C + 16*b^5*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)) + ((-(a^3*A*b^2) - 5*a*A*b^4 - 3*a^4*b*B + 9*a^2*b^3*B + 15*a^5 
*C - 29*a^3*b^2*C + 8*a*b^4*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[Ar 
cSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] 
+ 2*a^2*EllipticPi[-(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]*Sqrt[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 - Se 
c[c + d*x]^2))))/((a - b)^2*b^3*(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*S...
 

Rubi [A] (verified)

Time = 4.00 (sec) , antiderivative size = 547, normalized size of antiderivative = 0.98, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4586, 27, 3042, 4586, 27, 3042, 4590, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4590

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4594

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4274

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 4336

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

Int[(Sec[c + d*x]^(5/2)*(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))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(a + b*Sec[c + d*x])^2) - (-(((3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4 
*C + a^2*b^2*(3*A + 11*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2) 
*d*(a + b*Sec[c + d*x]))) - ((((2*a^2*(3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 
8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d* 
x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*b*(3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5 
*a^4*C + a^2*b^2*(3*A + 11*C))*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 + 6*a^3*b^3*B - 15* 
a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[ 
c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a 
*(a + b)*d))/b - (2*(3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + 
a^2*b^2*(A + 29*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d))/(2*b*(a^2 - b^ 
2)))/(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 4590

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[(-C)*d*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1 
)*((d*Csc[e + f*x])^(n - 1)/(b*f*(m + n + 1))), x] + Simp[d/(b*(m + n + 1)) 
   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n - 1)*Simp[a*C*(n - 1) + ( 
A*b*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) - a*C*n)*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] && GtQ[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] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(2020\) vs. \(2(531)=1062\).

Time = 154.19 (sec) , antiderivative size = 2021, normalized size of antiderivative = 3.63

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

Input:

int(sec(d*x+c)^(5/2)*(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*C/b^3/sin(1/ 
2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2 
*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(2*sin(1/2 
*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+ 
1/2*c)^2)^(1/2))+2*(A*b^2-C*a^2)/b^2/a*(a^2/b/(a^2-b^2)*cos(1/2*d*x+1/2*c) 
*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*a*cos(1/2*d*x+1/2 
*c)^2-a+b)-1/2/(a+b)/b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c) 
^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF 
(cos(1/2*d*x+1/2*c),2^(1/2))+1/2*a/b/(a^2-b^2)*(sin(1/2*d*x+1/2*c)^2)^(1/2 
)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1 
/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-1/2*a/b/(a^2-b^2)*(si 
n(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d* 
x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2 
))-1/2/b/(a^2-b^2)/(a^2-a*b)*a^3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* 
d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2) 
*EllipticPi(cos(1/2*d*x+1/2*c),2*a/(a-b),2^(1/2))+3/2*b/(a^2-b^2)/(a^2-a*b 
)*a*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin 
(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticPi(cos(1/2*d*x+1/2*c 
),2*a/(a-b),2^(1/2)))-2*(A*b^2-B*a*b+C*a^2)/a/b*(1/2*a^2/b/(a^2-b^2)*cos(1 
/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(2*a...
 

Fricas [F(-1)]

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

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

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

integrate(sec(d*x+c)^(5/2)*(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]

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

integrate(sec(d*x+c)^(5/2)*(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)*sec(d*x + c)^(5/2)/(b*se 
c(d*x + c) + a)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3} \,d x \] Input:

int(((1/cos(c + d*x))^(5/2)*(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))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(a + 
b/cos(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{3} b^{3}+3 \sec \left (d x +c \right )^{2} a \,b^{2}+3 \sec \left (d x +c \right ) a^{2} b +a^{3}}d x \right ) a \] Input:

int(sec(d*x+c)^(5/2)*(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)**4)/(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)**3)/(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 + 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)*a