\(\int \frac {(a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\) [1001]

Optimal result

Integrand size = 43, antiderivative size = 317 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \left (3 a^3 B+15 a b^2 B+5 b^3 (A-C)+3 a^2 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (21 a^2 b B+21 b^3 B+21 a b^2 (A+3 C)+a^3 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (24 A b^2+63 a b B+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 (11 A b+7 a B-35 b C) \sqrt {\sec (c+d x)} \sin (c+d x)}{35 d}+\frac {2 (6 A b+7 a B) (a+b \sec (c+d x))^2 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^3 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \] Output:

2/5*(3*B*a^3+15*B*a*b^2+5*b^3*(A-C)+3*a^2*b*(3*A+5*C))*cos(d*x+c)^(1/2)*El 
lipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/21*(21*B*a^2*b+21 
*B*b^3+21*a*b^2*(A+3*C)+a^3*(5*A+7*C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/ 
2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+2/105*a*(24*A*b^2+63*B*a*b+5*a^2*( 
5*A+7*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)-2/35*b^2*(11*A*b+7*B*a-35*C*b)*sec 
(d*x+c)^(1/2)*sin(d*x+c)/d+2/35*(6*A*b+7*B*a)*(a+b*sec(d*x+c))^2*sin(d*x+c 
)/d/sec(d*x+c)^(3/2)+2/7*A*(a+b*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(5/2 
)
 

Mathematica [A] (verified)

Time = 7.07 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.74 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (168 \left (3 a^3 B+15 a b^2 B+5 b^3 (A-C)+3 a^2 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (21 a^2 b B+21 b^3 B+21 a b^2 (A+3 C)+a^3 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+2 \left (42 \left (3 a^2 A b+a^3 B+10 b^3 C\right )+5 a \left (84 A b^2+84 a b B+a^2 (29 A+28 C)\right ) \cos (c+d x)+42 a^2 (3 A b+a B) \cos (2 (c+d x))+15 a^3 A \cos (3 (c+d x))\right ) \sin (c+d x)\right )}{420 d} \] Input:

Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) 
/Sec[c + d*x]^(7/2),x]
 

Output:

(Sqrt[Sec[c + d*x]]*(168*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*( 
3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(21*a^2*b*B 
+ 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Elli 
pticF[(c + d*x)/2, 2] + 2*(42*(3*a^2*A*b + a^3*B + 10*b^3*C) + 5*a*(84*A*b 
^2 + 84*a*b*B + a^2*(29*A + 28*C))*Cos[c + d*x] + 42*a^2*(3*A*b + a*B)*Cos 
[2*(c + d*x)] + 15*a^3*A*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)
 

Rubi [A] (verified)

Time = 2.28 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.465, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4562, 27, 3042, 4535, 3042, 4258, 3042, 3120, 4534, 3042, 4258, 3042, 3119}

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.

\(\displaystyle \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\)

\(\Big \downarrow \) 4582

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \sec (c+d x)) \left (5 (5 A+7 C) a^2+63 b B a+24 A b^2-b (11 A b-35 C b+7 a B) \sec ^2(c+d x)+\left (21 B a^2+38 A b a+70 b C a+35 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \sec (c+d x)) \left (5 (5 A+7 C) a^2+63 b B a+24 A b^2-b (11 A b-35 C b+7 a B) \sec ^2(c+d x)+\left (21 B a^2+38 A b a+70 b C a+35 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (5 (5 A+7 C) a^2+63 b B a+24 A b^2-b (11 A b-35 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 B a^2+38 A b a+70 b C a+35 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4562

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}-\frac {2}{3} \int -\frac {-3 b^2 (11 A b-35 C b+7 a B) \sec ^2(c+d x)+5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right ) \sec (c+d x)+3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )}{2 \sqrt {\sec (c+d x)}}dx\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {-3 b^2 (11 A b-35 C b+7 a B) \sec ^2(c+d x)+5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right ) \sec (c+d x)+3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {-3 b^2 (11 A b-35 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+5 \left ((5 A+7 C) a^3+21 b B a^2+21 b^2 (A+3 C) a+21 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )-3 b^2 (11 A b-35 C b+7 a B) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}}dx+5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \sqrt {\sec (c+d x)}dx\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (5 \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\int \frac {3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )-3 b^2 (11 A b-35 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )-3 b^2 (11 A b-35 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )-3 b^2 (11 A b-35 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {3 \left (21 B a^3+21 b (3 A+5 C) a^2+98 b^2 B a+24 A b^3\right )-3 b^2 (11 A b-35 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4534

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}-\frac {6 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (7 a B+11 A b-35 b C)}{d}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}-\frac {6 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (7 a B+11 A b-35 b C)}{d}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}-\frac {6 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (7 a B+11 A b-35 b C)}{d}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}-\frac {6 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (7 a B+11 A b-35 b C)}{d}\right )+\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right )}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} \left (\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right )}{d}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right )}{d}-\frac {6 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} (7 a B+11 A b-35 b C)}{d}\right )\right )+\frac {2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

Input:

Int[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c 
 + d*x]^(7/2),x]
 

Output:

(2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*( 
6*A*b + 7*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2 
)) + ((2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(3*d*Sq 
rt[Sec[c + d*x]]) + ((42*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*( 
3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x] 
])/d + (10*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))* 
Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (6*b^ 
2*(11*A*b + 7*a*B - 35*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d)/3)/5)/7
 

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 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 4534

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) 
+ (A_)), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*(m + 1) 
)), x] + Simp[(C*m + A*(m + 1))/(m + 1)   Int[(b*Csc[e + f*x])^m, x], x] /; 
 FreeQ[{b, e, f, A, C, m}, x] && NeQ[C*m + A*(m + 1), 0] &&  !LeQ[m, -1]
 

rule 4535

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* 
(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b   Int[(b*Cs 
c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) 
, x] /; FreeQ[{b, e, f, A, B, C, m}, x]
 

rule 4562

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 
_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si 
mp[1/(d*n)   Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* 
b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ 
{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
 

rule 4582

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*((d*Csc[e 
 + f*x])^n/(f*n)), x] - Simp[1/(d*n)   Int[(a + b*Csc[e + f*x])^(m - 1)*(d* 
Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs 
c[e + f*x] - b*(C*n + A*(m + n + 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] && GtQ[m, 0] && LeQ[n, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1097\) vs. \(2(292)=584\).

Time = 19.49 (sec) , antiderivative size = 1098, normalized size of antiderivative = 3.46

method	result	size
parts	\(\text {Expression too large to display}\)	\(1098\)
default	\(\text {Expression too large to display}\)	\(1278\)

Input:

int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2),x, 
method=_RETURNVERBOSE)
 

Output:

-2/5*(3*A*a^2*b+B*a^3)*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^( 
1/2)*(-8*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+8*cos(1/2*d*x+1/2*c)*sin( 
1/2*d*x+1/2*c)^4-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(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*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2 
*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d-2*(B*b^3+3*C*a*b^2)*((2*cos 
(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(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))/sin(1/2*d*x+1/2*c)/ 
(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d+2*(A*b^3+3*B*a*b^2+3*C*a^2*b)*((2*cos(1 
/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2 
)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))/ 
(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2 
*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d-2/3*(3*A*a*b^2+3*B*a^2*b+C*a^3)*((2*cos(1 
/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(4*cos(1/2*d*x+1/2*c)*sin(1 
/2*d*x+1/2*c)^4-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+(sin(1/2*d*x+1/2 
*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c) 
,2^(1/2)))/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d* 
x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d-2*C*b^3*(-2*(-2*sin(1/2*d*x+1/ 
2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=-\frac {5 \, \sqrt {2} {\left (i \, {\left (5 \, A + 7 \, C\right )} a^{3} + 21 i \, B a^{2} b + 21 i \, {\left (A + 3 \, C\right )} a b^{2} + 21 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 7 \, C\right )} a^{3} - 21 i \, B a^{2} b - 21 i \, {\left (A + 3 \, C\right )} a b^{2} - 21 i \, B b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-3 i \, B a^{3} - 3 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b - 15 i \, B a b^{2} - 5 i \, {\left (A - C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, \sqrt {2} {\left (3 i \, B a^{3} + 3 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b + 15 i \, B a b^{2} + 5 i \, {\left (A - C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (15 \, A a^{3} \cos \left (d x + c\right )^{3} + 105 \, C b^{3} + 21 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{3} + 21 \, B a^{2} b + 21 \, A a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d} \] Input:

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7 
/2),x, algorithm="fricas")
 

Output:

-1/105*(5*sqrt(2)*(I*(5*A + 7*C)*a^3 + 21*I*B*a^2*b + 21*I*(A + 3*C)*a*b^2 
 + 21*I*B*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 
 5*sqrt(2)*(-I*(5*A + 7*C)*a^3 - 21*I*B*a^2*b - 21*I*(A + 3*C)*a*b^2 - 21* 
I*B*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sq 
rt(2)*(-3*I*B*a^3 - 3*I*(3*A + 5*C)*a^2*b - 15*I*B*a*b^2 - 5*I*(A - C)*b^3 
)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d 
*x + c))) + 21*sqrt(2)*(3*I*B*a^3 + 3*I*(3*A + 5*C)*a^2*b + 15*I*B*a*b^2 + 
 5*I*(A - C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d* 
x + c) - I*sin(d*x + c))) - 2*(15*A*a^3*cos(d*x + c)^3 + 105*C*b^3 + 21*(B 
*a^3 + 3*A*a^2*b)*cos(d*x + c)^2 + 5*((5*A + 7*C)*a^3 + 21*B*a^2*b + 21*A* 
a*b^2)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
 

Sympy [F]

\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \sec {\left (c + d x \right )}\right )^{3} \left (A + B \sec {\left (c + d x \right )} + C \sec ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {7}{2}}{\left (c + d x \right )}}\, dx \] Input:

integrate((a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)* 
*(7/2),x)
 

Output:

Integral((a + b*sec(c + d*x))**3*(A + B*sec(c + d*x) + C*sec(c + d*x)**2)/ 
sec(c + d*x)**(7/2), x)
                                                                                    
                                                                                    
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7 
/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {7}{2}}} \,d x } \] Input:

integrate((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7 
/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/s 
ec(d*x + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \] Input:

int(((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co 
s(c + d*x))^(7/2),x)
 

Output:

int(((a + b/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co 
s(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a^{3} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} c +6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b^{2}+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{2} b c +4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a \,b^{3}+3 \left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a \,b^{2} c +\left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) b^{4}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) b^{3} c \] Input:

int((a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2),x)
 

Output:

int(sqrt(sec(c + d*x))/sec(c + d*x)**4,x)*a**4 + 4*int(sqrt(sec(c + d*x))/ 
sec(c + d*x)**3,x)*a**3*b + int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**3 
*c + 6*int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**2*b**2 + 3*int(sqrt(se 
c(c + d*x))/sec(c + d*x),x)*a**2*b*c + 4*int(sqrt(sec(c + d*x))/sec(c + d* 
x),x)*a*b**3 + 3*int(sqrt(sec(c + d*x)),x)*a*b**2*c + int(sqrt(sec(c + d*x 
)),x)*b**4 + int(sqrt(sec(c + d*x))*sec(c + d*x),x)*b**3*c