\(\int \frac {\sec ^{\frac {3}{2}}(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^3} \, dx\) [1027]

Optimal result

Integrand size = 43, antiderivative size = 469 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx=-\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a b^2 \left (a^2-b^2\right )^2 d}+\frac {\left (A b^4+3 a^3 b B+3 a b^3 B+a^4 C-7 a^2 b^2 (A+C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^2 b \left (a^2-b^2\right )^2 d}-\frac {\left (A b^6-a^5 b B+10 a^3 b^3 B+3 a b^5 B-3 a^4 b^2 (A-2 C)-3 a^6 C-5 a^2 b^4 (2 A+3 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^2 (a-b)^2 b^2 (a+b)^3 d}-\frac {\left (A b^2-a (b B-a C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2}+\frac {\left (A b^4-a^3 b B-5 a b^3 B-3 a^4 C+a^2 b^2 (5 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 b^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))} \] Output:

-1/4*(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(d*x+c)^(1/2)* 
EllipticE(sin(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*(A*b^4+3*B*a^3*b+3*B*a*b^3+a^4*C-7*a^2*b^2*(A+C))*cos(d*x+c)^(1/2)*In 
verseJacobiAM(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*(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(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^2/(a-b)^2/b^2/(a+b)^3/d-1/2*(A*b^2-a*(B*b-C*a))* 
sec(d*x+c)^(3/2)*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*sec(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))*sec(d*x+c)^(1/2)*sin(d*x+c)/b^2 
/(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. \(2(469)=938\).

Time = 8.97 (sec) , antiderivative size = 1046, normalized size of antiderivative = 2.23 \[ \int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx =\text {Too large to display} \] Input:

Integrate[(Sec[c + d*x]^(3/2)*(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*(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)*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*(-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)*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)) + ((-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[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]*S 
qrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSi 
n[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 - Sec[c 
 + d*x]^2))))/(8*(a - b)^2*b^2*(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)*(((5*a^2*A*b^2 + ...
 

Rubi [A] (verified)

Time = 3.08 (sec) , antiderivative size = 462, normalized size of antiderivative = 0.99, 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, 4586, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4586

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4594

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4274

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4258

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3119

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

\(\Big \downarrow \) 3120

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

\(\Big \downarrow \) 4336

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3284

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

Input:

Int[(Sec[c + d*x]^(3/2)*(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]^(3/2)*Sin[c + d*x])/(b*(a^2 - b 
^2)*d*(a + b*Sec[c + d*x])^2) - ((((2*a*(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]]*EllipticE[(c + d*x)/2, 2]*S 
qrt[Sec[c + d*x]])/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))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d* 
x]])/d)/a^2 + (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))*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))/(2*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[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[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 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. \(1879\) vs. \(2(448)=896\).

Time = 7.78 (sec) , antiderivative size = 1880, normalized size of antiderivative = 4.01

method	result	size
default	\(\text {Expression too large to display}\)	\(1880\)

Input:

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

Fricas [F(-1)]

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

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

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

integrate(sec(d*x+c)^(3/2)*(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]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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