\(\int \frac {(a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\) [1009]

Optimal result

Integrand size = 43, antiderivative size = 426 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \left (36 a^3 b B+60 a b^3 B+15 b^4 (A-C)+18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (5 a^4 B+42 a^2 b^2 B+21 b^4 B+28 a b^3 (A+3 C)+4 a^3 b (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 a \left (64 A b^3+75 a^3 B+261 a b^2 B+a^2 (202 A b+294 b C)\right ) \sin (c+d x)}{315 d \sqrt {\sec (c+d x)}}-\frac {2 b^2 \left (162 a b B+3 b^2 (41 A-105 C)+7 a^2 (7 A+9 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 \left (48 A b^2+117 a b B+7 a^2 (7 A+9 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (8 A b+9 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x)} \] Output:

2/15*(36*B*a^3*b+60*B*a*b^3+15*b^4*(A-C)+18*a^2*b^2*(3*A+5*C)+a^4*(7*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) 
/d+2/21*(5*B*a^4+42*B*a^2*b^2+21*B*b^4+28*a*b^3*(A+3*C)+4*a^3*b*(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+2/315*a*(64*A*b^3+75*B*a^3+261*B*a*b^2+a^2*(202*A*b+294*C*b))*sin(d*x+c) 
/d/sec(d*x+c)^(1/2)-2/315*b^2*(162*B*a*b+3*b^2*(41*A-105*C)+7*a^2*(7*A+9*C 
))*sec(d*x+c)^(1/2)*sin(d*x+c)/d+2/315*(48*A*b^2+117*B*a*b+7*a^2*(7*A+9*C) 
)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/63*(8*A*b+9*B*a)*(a+b 
*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/9*A*(a+b*sec(d*x+c))^4*sin( 
d*x+c)/d/sec(d*x+c)^(7/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 11.85 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (49 a^4 A+378 a^2 A b^2+105 A b^4+252 a^3 b B+420 a b^3 B+63 a^4 C+630 a^2 b^2 C-105 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (100 a^3 A b+140 a A b^3+25 a^4 B+210 a^2 b^2 B+105 b^4 B+140 a^3 b C+420 a b^3 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{105 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{90} \left (19 a^4 A+108 a^2 A b^2+72 a^3 b B+18 a^4 C+360 b^4 C\right ) \sin (c+d x)+\frac {1}{21} a \left (52 a^2 A b+56 A b^3+13 a^3 B+84 a b^2 B+56 a^2 b C\right ) \sin (2 (c+d x))+\frac {1}{180} a^2 \left (43 a^2 A+216 A b^2+144 a b B+36 a^2 C\right ) \sin (3 (c+d x))+\frac {1}{14} a^3 (4 A b+a B) \sin (4 (c+d x))+\frac {1}{36} a^4 A \sin (5 (c+d x))\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \] 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]^(9/2),x]
 

Output:

(2*Cos[c + d*x]^6*((2*(49*a^4*A + 378*a^2*A*b^2 + 105*A*b^4 + 252*a^3*b*B 
+ 420*a*b^3*B + 63*a^4*C + 630*a^2*b^2*C - 105*b^4*C)*EllipticE[(c + d*x)/ 
2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(100*a^3*A*b + 140*a*A* 
b^3 + 25*a^4*B + 210*a^2*b^2*B + 105*b^4*B + 140*a^3*b*C + 420*a*b^3*C)*Sq 
rt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[ 
c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos[c + 
 d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[ 
c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((19*a^4*A + 108*a^2* 
A*b^2 + 72*a^3*b*B + 18*a^4*C + 360*b^4*C)*Sin[c + d*x])/90 + (a*(52*a^2*A 
*b + 56*A*b^3 + 13*a^3*B + 84*a*b^2*B + 56*a^2*b*C)*Sin[2*(c + d*x)])/21 + 
 (a^2*(43*a^2*A + 216*A*b^2 + 144*a*b*B + 36*a^2*C)*Sin[3*(c + d*x)])/180 
+ (a^3*(4*A*b + a*B)*Sin[4*(c + d*x)])/14 + (a^4*A*Sin[5*(c + d*x)])/36))/ 
(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x] 
)*Sec[c + d*x]^(11/2))
 

Rubi [A] (verified)

Time = 3.44 (sec) , antiderivative size = 433, 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, 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.

\(\displaystyle \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\)

\(\Big \downarrow \) 4582

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \int \frac {(a+b \sec (c+d x))^3 \left (-b (A-9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+8 A b+9 a B\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \sec (c+d x))^2 \left (7 (7 A+9 C) a^2+117 b B a+48 A b^2-3 b (5 A b-21 C b+3 a B) \sec ^2(c+d x)+\left (45 B a^2+82 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \sec (c+d x))^2 \left (7 (7 A+9 C) a^2+117 b B a+48 A b^2-3 b (5 A b-21 C b+3 a B) \sec ^2(c+d x)+\left (45 B a^2+82 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (7 (7 A+9 C) a^2+117 b B a+48 A b^2-3 b (5 A b-21 C b+3 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+82 A b a+126 b C a+63 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \sec (c+d x)) \left (-b \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sec ^2(c+d x)+\left (21 (7 A+9 C) a^3+531 b B a^2+b^2 (479 A+945 C) a+315 b^3 B\right ) \sec (c+d x)+3 \left (75 B a^3+(202 A b+294 C b) a^2+261 b^2 B a+64 A b^3\right )\right )}{2 \sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(a+b \sec (c+d x)) \left (-b \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sec ^2(c+d x)+\left (21 (7 A+9 C) a^3+531 b B a^2+b^2 (479 A+945 C) a+315 b^3 B\right ) \sec (c+d x)+3 \left (75 B a^3+(202 A b+294 C b) a^2+261 b^2 B a+64 A b^3\right )\right )}{\sec ^{\frac {3}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (-b \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 (7 A+9 C) a^3+531 b B a^2+b^2 (479 A+945 C) a+315 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (75 B a^3+(202 A b+294 C b) a^2+261 b^2 B a+64 A b^3\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4562

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}-\frac {2}{3} \int -\frac {3 \left (21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sec ^2(c+d x)+15 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right ) \sec (c+d x)\right )}{2 \sqrt {\sec (c+d x)}}dx\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sec ^2(c+d x)+15 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (5 B a^4+4 b (5 A+7 C) a^3+42 b^2 B a^2+28 b^3 (A+3 C) a+21 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)}}dx+15 \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \sqrt {\sec (c+d x)}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {21 (7 A+9 C) a^4+756 b B a^3+7 b^2 (155 A+261 C) a^2+1098 b^3 B a+192 A b^4-b^2 \left (7 (7 A+9 C) a^2+162 b B a+3 b^2 (41 A-105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4534

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right ) \int \sqrt {\cos (c+d x)}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (-\frac {2 b^2 \sin (c+d x) \sqrt {\sec (c+d x)} \left (7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right )}{d}+\frac {2 a \sin (c+d x) \left (75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right )}{d \sqrt {\sec (c+d x)}}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right )}{d}+\frac {42 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right )}{d}\right )\right )+\frac {2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac {7}{2}}(c+d x)}\)

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]^(9/2),x]
 

Output:

(2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*( 
8*A*b + 9*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2 
)) + ((2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Sec[c + d*x])^2 
*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + ((42*(36*a^3*b*B + 60*a*b^3*B + 
15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d* 
x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (30*(5*a^4*B + 42*a^ 
2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*Sqrt[Cos[c 
+ d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(64*A*b^3 + 
 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(d*Sqrt[S 
ec[c + d*x]]) - (2*b^2*(162*a*b*B + 3*b^2*(41*A - 105*C) + 7*a^2*(7*A + 9* 
C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d)/5)/7)/9
 

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. \(1336\) vs. \(2(397)=794\).

Time = 48.43 (sec) , antiderivative size = 1337, normalized size of antiderivative = 3.14

method	result	size
parts	\(\text {Expression too large to display}\)	\(1337\)
default	\(\text {Expression too large to display}\)	\(1652\)

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)^(9/2),x, 
method=_RETURNVERBOSE)
 

Output:

-2/21*(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)*(48*cos(1/2*d*x+1/2*c)^9-120*cos(1/2*d*x+1/2*c)^7+128*cos(1/2*d*x+1/ 
2*c)^5-72*cos(1/2*d*x+1/2*c)^3+5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* 
d*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+16*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)/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*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*co 
s(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)*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*co 
s(1/2*d*x+1/2*c)^2-1)^(1/2)/d-2/3*(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)*((2*co 
s(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)*si 
n(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-2/5*(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)^(1/2)*(-8*sin(...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 442, 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 {9}{2}}(c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (5 i \, B a^{4} + 4 i \, {\left (5 \, A + 7 \, C\right )} a^{3} b + 42 i \, B a^{2} b^{2} + 28 i \, {\left (A + 3 \, C\right )} a b^{3} + 21 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-5 i \, B a^{4} - 4 i \, {\left (5 \, A + 7 \, C\right )} a^{3} b - 42 i \, B a^{2} b^{2} - 28 i \, {\left (A + 3 \, C\right )} a b^{3} - 21 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (-i \, {\left (7 \, A + 9 \, C\right )} a^{4} - 36 i \, B a^{3} b - 18 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} - 60 i \, B a b^{3} - 15 i \, {\left (A - C\right )} b^{4}\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 (i \, {\left (7 \, A + 9 \, C\right )} a^{4} + 36 i \, B a^{3} b + 18 i \, {\left (3 \, A + 5 \, C\right )} a^{2} b^{2} + 60 i \, B a b^{3} + 15 i \, {\left (A - C\right )} b^{4}\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 (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 315 \, C b^{4} + 45 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left ({\left (7 \, A + 9 \, C\right )} a^{4} + 36 \, B a^{3} b + 54 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 7 \, C\right )} a^{3} b + 42 \, B a^{2} b^{2} + 28 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{315 \, d} \] 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)^(9 
/2),x, algorithm="fricas")
 

Output:

-1/315*(15*sqrt(2)*(5*I*B*a^4 + 4*I*(5*A + 7*C)*a^3*b + 42*I*B*a^2*b^2 + 2 
8*I*(A + 3*C)*a*b^3 + 21*I*B*b^4)*weierstrassPInverse(-4, 0, cos(d*x + c) 
+ I*sin(d*x + c)) + 15*sqrt(2)*(-5*I*B*a^4 - 4*I*(5*A + 7*C)*a^3*b - 42*I* 
B*a^2*b^2 - 28*I*(A + 3*C)*a*b^3 - 21*I*B*b^4)*weierstrassPInverse(-4, 0, 
cos(d*x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(-I*(7*A + 9*C)*a^4 - 36*I*B*a 
^3*b - 18*I*(3*A + 5*C)*a^2*b^2 - 60*I*B*a*b^3 - 15*I*(A - C)*b^4)*weierst 
rassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) 
 + 21*sqrt(2)*(I*(7*A + 9*C)*a^4 + 36*I*B*a^3*b + 18*I*(3*A + 5*C)*a^2*b^2 
 + 60*I*B*a*b^3 + 15*I*(A - C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInv 
erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(35*A*a^4*cos(d*x + c)^4 + 
 315*C*b^4 + 45*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^3 + 7*((7*A + 9*C)*a^4 + 
36*B*a^3*b + 54*A*a^2*b^2)*cos(d*x + c)^2 + 15*(5*B*a^4 + 4*(5*A + 7*C)*a^ 
3*b + 42*B*a^2*b^2 + 28*A*a*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + 
 c)))/d
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)* 
*(9/2),x)
 

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)^(9 
/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {9}{2}}} \,d x } \] 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)^(9 
/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)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}} \,d x \] 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))^(9/2),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))^(9/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a^{5}+5 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a^{4} c +10 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} b c +10 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b^{3}+6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a^{2} b^{2} c +5 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) a \,b^{4}+4 \left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) a \,b^{3} c +\left (\int \sqrt {\sec \left (d x +c \right )}d x \right ) b^{5}+\left (\int \sqrt {\sec \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) b^{4} c \] 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)^(9/2),x)
 

Output:

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