\(\int \frac {(a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\) [1008]

Optimal result

Integrand size = 43, antiderivative size = 429 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {2 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b \left (217 a^2 b B-105 b^3 B+12 a b^2 (19 A-35 C)+10 a^3 (5 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}-\frac {2 b^2 \left (98 a b B+b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+77 a b B+5 a^2 (5 A+7 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 (8 A b+7 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{35 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d \sec ^{\frac {5}{2}}(c+d x)} \] Output:

2/5*(3*B*a^4+30*B*a^2*b^2-5*B*b^4+20*a*b^3*(A-C)+4*a^3*b*(3*A+5*C))*cos(d* 
x+c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/21*( 
28*B*a^3*b+84*B*a*b^3+7*b^4*(3*A+C)+42*a^2*b^2*(A+3*C)+a^4*(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-2/1 
05*b*(217*B*a^2*b-105*B*b^3+12*a*b^2*(19*A-35*C)+10*a^3*(5*A+7*C))*sec(d*x 
+c)^(1/2)*sin(d*x+c)/d-2/105*b^2*(98*B*a*b+b^2*(87*A-35*C)+5*a^2*(5*A+7*C) 
)*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/105*(48*A*b^2+77*B*a*b+5*a^2*(5*A+7*C))* 
(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/35*(8*A*b+7*B*a)*(a+b*s 
ec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/7*A*(a+b*sec(d*x+c))^4*sin(d* 
x+c)/d/sec(d*x+c)^(5/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 10.59 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (168 \left (3 a^4 B+30 a^2 b^2 B-5 b^4 B+20 a b^3 (A-C)+4 a^3 b (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+40 \left (28 a^3 b B+84 a b^3 B+7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+168 a^3 A b \sin (c+d x)+42 a^4 B \sin (c+d x)+840 b^4 B \sin (c+d x)+3360 a b^3 C \sin (c+d x)+130 a^4 A \sin (2 (c+d x))+840 a^2 A b^2 \sin (2 (c+d x))+560 a^3 b B \sin (2 (c+d x))+140 a^4 C \sin (2 (c+d x))+168 a^3 A b \sin (3 (c+d x))+42 a^4 B \sin (3 (c+d x))+15 a^4 A \sin (4 (c+d x))+280 b^4 C \tan (c+d x)\right )}{210 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 (c+d x))) \sec ^{\frac {11}{2}}(c+d x)} \] Input:

Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)) 
/Sec[c + d*x]^(7/2),x]
 

Output:

((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(168*(3*a^ 
4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqr 
t[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(28*a^3*b*B + 84*a*b^3*B + 
7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x 
]]*EllipticF[(c + d*x)/2, 2] + 168*a^3*A*b*Sin[c + d*x] + 42*a^4*B*Sin[c + 
 d*x] + 840*b^4*B*Sin[c + d*x] + 3360*a*b^3*C*Sin[c + d*x] + 130*a^4*A*Sin 
[2*(c + d*x)] + 840*a^2*A*b^2*Sin[2*(c + d*x)] + 560*a^3*b*B*Sin[2*(c + d* 
x)] + 140*a^4*C*Sin[2*(c + d*x)] + 168*a^3*A*b*Sin[3*(c + d*x)] + 42*a^4*B 
*Sin[3*(c + d*x)] + 15*a^4*A*Sin[4*(c + d*x)] + 280*b^4*C*Tan[c + d*x]))/( 
210*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d* 
x)])*Sec[c + d*x]^(11/2))
 

Rubi [A] (verified)

Time = 3.27 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.02, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4564, 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.

\(\displaystyle \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx\)

\(\Big \downarrow \) 4582

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \sec (c+d x))^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \sec ^2(c+d x)+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \sec (c+d x))^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \sec ^2(c+d x)+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+7 C) a^2+77 b B a+48 A b^2-b (39 A b-35 C b+21 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 B a^2+34 A b a+70 b C a+35 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {(a+b \sec (c+d x)) \left (63 B a^3+2 b (101 A+175 C) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \sec ^2(c+d x)+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \sec (c+d x)\right )}{2 \sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {(a+b \sec (c+d x)) \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \sec ^2(c+d x)+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3-3 b \left (5 (5 A+7 C) a^2+98 b B a+b^2 (87 A-35 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 (5 A+7 C) a^3+77 b B a^2+3 b^2 (11 A+105 C) a+105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4564

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2}{3} \int \frac {3 \left (-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \sec ^2(c+d x)+5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right ) \sec (c+d x)+a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )\right )}{2 \sqrt {\sec (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \sec ^2(c+d x)+5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right ) \sec (c+d x)+a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )}{\sqrt {\sec (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+5 \left ((5 A+7 C) a^4+28 b B a^3+42 b^2 (A+3 C) a^2+84 b^3 B a+7 b^4 (3 A+C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}}dx+5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \sqrt {\sec (c+d x)}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+5 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\int \frac {a \left (63 B a^3+(202 A b+350 C b) a^2+413 b^2 B a+192 A b^3\right )-b \left (10 (5 A+7 C) a^3+217 b B a^2+12 b^2 (19 A-35 C) a-105 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4534

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}\right )+\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{3 d \sqrt {\sec (c+d x)}}+\frac {1}{3} \left (-\frac {2 b^2 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right )}{d}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)} \left (10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right )}{d}+\frac {10 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right )}{d}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right )}{d}\right )\right )+\frac {2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac {5}{2}}(c+d x)}\)

Input:

Int[((a + b*Sec[c + d*x])^4*(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])^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*( 
8*A*b + 7*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2 
)) + ((2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Sec[c + d*x])^2* 
Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + ((42*(3*a^4*B + 30*a^2*b^2*B - 5* 
b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*Ellipti 
cE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (10*(28*a^3*b*B + 84*a*b^3*B + 
7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x 
]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - (2*b*(217*a^2*b*B - 1 
05*b^3*B + 12*a*b^2*(19*A - 35*C) + 10*a^3*(5*A + 7*C))*Sqrt[Sec[c + d*x]] 
*Sin[c + d*x])/d - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C 
))*Sec[c + d*x]^(3/2)*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 4564

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[(-b)*C*Csc[e + f*x]*Cot[e + f*x]*((d*Csc[e + f*x])^ 
n/(f*(n + 2))), x] + Simp[1/(n + 2)   Int[(d*Csc[e + f*x])^n*Simp[A*a*(n + 
2) + (B*a*(n + 2) + b*(C*(n + 1) + A*(n + 2)))*Csc[e + f*x] + (a*C + B*b)*( 
n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, 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. \(1337\) vs. \(2(400)=800\).

Time = 33.16 (sec) , antiderivative size = 1338, normalized size of antiderivative = 3.12

method	result	size
parts	\(\text {Expression too large to display}\)	\(1338\)
default	\(\text {Expression too large to display}\)	\(2507\)

Input:

int((a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/sec(d*x+c)^(7/2),x, 
method=_RETURNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b \sec (c+d x))^4 \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^{4} + 28 i \, B a^{3} b + 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} + 84 i \, B a b^{3} + 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\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^{4} - 28 i \, B a^{3} b - 42 i \, {\left (A + 3 \, C\right )} a^{2} b^{2} - 84 i \, B a b^{3} - 7 i \, {\left (3 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\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^{4} - 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b - 30 i \, B a^{2} b^{2} - 20 i \, {\left (A - C\right )} a b^{3} + 5 i \, B b^{4}\right )} \cos \left (d x + c\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^{4} + 4 i \, {\left (3 \, A + 5 \, C\right )} a^{3} b + 30 i \, B a^{2} b^{2} + 20 i \, {\left (A - C\right )} a b^{3} - 5 i \, B b^{4}\right )} \cos \left (d x + c\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^{4} \cos \left (d x + c\right )^{4} + 35 \, C b^{4} + 21 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 5 \, {\left ({\left (5 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )} \] Input:

integrate((a+b*sec(d*x+c))^4*(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^4 + 28*I*B*a^3*b + 42*I*(A + 3*C)*a^2*b 
^2 + 84*I*B*a*b^3 + 7*I*(3*A + C)*b^4)*cos(d*x + c)*weierstrassPInverse(-4 
, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-I*(5*A + 7*C)*a^4 - 28*I 
*B*a^3*b - 42*I*(A + 3*C)*a^2*b^2 - 84*I*B*a*b^3 - 7*I*(3*A + C)*b^4)*cos( 
d*x + c)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 21*sq 
rt(2)*(-3*I*B*a^4 - 4*I*(3*A + 5*C)*a^3*b - 30*I*B*a^2*b^2 - 20*I*(A - C)* 
a*b^3 + 5*I*B*b^4)*cos(d*x + c)*weierstrassZeta(-4, 0, weierstrassPInverse 
(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(3*I*B*a^4 + 4*I*(3*A 
 + 5*C)*a^3*b + 30*I*B*a^2*b^2 + 20*I*(A - C)*a*b^3 - 5*I*B*b^4)*cos(d*x + 
 c)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin 
(d*x + c))) - 2*(15*A*a^4*cos(d*x + c)^4 + 35*C*b^4 + 21*(B*a^4 + 4*A*a^3* 
b)*cos(d*x + c)^3 + 5*((5*A + 7*C)*a^4 + 28*B*a^3*b + 42*A*a^2*b^2)*cos(d* 
x + c)^2 + 105*(4*C*a*b^3 + B*b^4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x 
 + c)))/(d*cos(d*x + c))
 

Sympy [F(-1)]

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

integrate((a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/sec(d*x+c)* 
*(7/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-1)]

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

integrate((a+b*sec(d*x+c))^4*(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]

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

integrate((a+b*sec(d*x+c))^4*(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)^4/s 
ec(d*x + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}} \,d x \] Input:

int(((a + b/cos(c + d*x))^4*(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))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2))/(1/co 
s(c + d*x))^(7/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{5}+5 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{4} c +10 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{3} b c +10 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{2} b^{3}+6 \left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a^{2} b^{2} c +5 \left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a \,b^{4}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{2}d x \right ) b^{4} c +4 \left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) a \,b^{3} c +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) b^{5} \] Input:

int((a+b*sec(d*x+c))^4*(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**5 + 5*int(sqrt(sec(c + d*x))/ 
sec(c + d*x)**3,x)*a**4*b + int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**4 
*c + 10*int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**3*b**2 + 4*int(sqrt(s 
ec(c + d*x))/sec(c + d*x),x)*a**3*b*c + 10*int(sqrt(sec(c + d*x))/sec(c + 
d*x),x)*a**2*b**3 + 6*int(sqrt(sec(c + d*x)),x)*a**2*b**2*c + 5*int(sqrt(s 
ec(c + d*x)),x)*a*b**4 + int(sqrt(sec(c + d*x))*sec(c + d*x)**2,x)*b**4*c 
+ 4*int(sqrt(sec(c + d*x))*sec(c + d*x),x)*a*b**3*c + int(sqrt(sec(c + d*x 
))*sec(c + d*x),x)*b**5