\(\int \frac {(a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sec ^{\frac {13}{2}}(c+d x)} \, dx\) [1011]

Optimal result

Integrand size = 43, antiderivative size = 516 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \left (364 a^3 b B+468 a b^3 B+39 b^4 (3 A+5 C)+78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 a \left (192 A b^3+1053 a^3 B+2171 a b^2 B+a^2 (2518 A b+3146 b C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 A b^4+4004 a^3 b B+3458 a b^3 B+77 a^4 (11 A+13 C)+11 a^2 b^2 (491 A+637 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (45 a^4 B+330 a^2 b^2 B+77 b^4 B+44 a b^3 (5 A+7 C)+20 a^3 b (9 A+11 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 A b^2+221 a b B+11 a^2 (11 A+13 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 (8 A b+13 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {11}{2}}(c+d x)} \] Output:

2/195*(364*B*a^3*b+468*B*a*b^3+39*b^4*(3*A+5*C)+78*a^2*b^2*(7*A+9*C)+a^4*( 
77*A+91*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+2/231*(45*B*a^4+330*B*a^2*b^2+77*B*b^4+44*a*b^3*(5*A+7*C)+20*a 
^3*b*(9*A+11*C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*s 
ec(d*x+c)^(1/2)/d+2/9009*a*(192*A*b^3+1053*B*a^3+2171*B*a*b^2+a^2*(2518*A* 
b+3146*C*b))*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/6435*(192*A*b^4+4004*B*a^3*b+ 
3458*B*a*b^3+77*a^4*(11*A+13*C)+11*a^2*b^2*(491*A+637*C))*sin(d*x+c)/d/sec 
(d*x+c)^(3/2)+2/231*(45*B*a^4+330*B*a^2*b^2+77*B*b^4+44*a*b^3*(5*A+7*C)+20 
*a^3*b*(9*A+11*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/1287*(48*A*b^2+221*B*a* 
b+11*a^2*(11*A+13*C))*(a+b*sec(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/1 
43*(8*A*b+13*B*a)*(a+b*sec(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(9/2)+2/13*A* 
(a+b*sec(d*x+c))^4*sin(d*x+c)/d/sec(d*x+c)^(11/2)
                                                                                    
                                                                                    
 

Mathematica [A] (warning: unable to verify)

Time = 15.81 (sec) , antiderivative size = 658, normalized size of antiderivative = 1.28 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\frac {2 \cos ^6(c+d x) \left (\frac {2 \left (5929 a^4 A+42042 a^2 A b^2+9009 A b^4+28028 a^3 b B+36036 a b^3 B+7007 a^4 C+54054 a^2 b^2 C+15015 b^4 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (11700 a^3 A b+14300 a A b^3+2925 a^4 B+21450 a^2 b^2 B+5005 b^4 B+14300 a^3 b C+20020 a b^3 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{15015 d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\left (1897 a^4 A+11856 a^2 A b^2+1872 A b^4+7904 a^3 b B+7488 a b^3 B+1976 a^4 C+11232 a^2 b^2 C\right ) \sin (c+d x)}{9360}+\frac {\left (4164 a^3 A b+4576 a A b^3+1041 a^4 B+6864 a^2 b^2 B+1232 b^4 B+4576 a^3 b C+4928 a b^3 C\right ) \sin (2 (c+d x))}{1848}+\frac {\left (2297 a^4 A+13416 a^2 A b^2+1872 A b^4+8944 a^3 b B+7488 a b^3 B+2236 a^4 C+11232 a^2 b^2 C\right ) \sin (3 (c+d x))}{9360}+\frac {1}{77} a \left (32 a^2 A b+22 A b^3+8 a^3 B+33 a b^2 B+22 a^2 b C\right ) \sin (4 (c+d x))+\frac {a^2 \left (89 a^2 A+312 A b^2+208 a b B+52 a^2 C\right ) \sin (5 (c+d x))}{1872}+\frac {1}{88} a^3 (4 A b+a B) \sin (6 (c+d x))+\frac {1}{208} a^4 A \sin (7 (c+d x))\right )}{d (b+a \cos (c+d x))^4 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {11}{2}}(c+d x)} \] 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]^(13/2),x]
 

Output:

(2*Cos[c + d*x]^6*((2*(5929*a^4*A + 42042*a^2*A*b^2 + 9009*A*b^4 + 28028*a 
^3*b*B + 36036*a*b^3*B + 7007*a^4*C + 54054*a^2*b^2*C + 15015*b^4*C)*Ellip 
ticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(11700*a 
^3*A*b + 14300*a*A*b^3 + 2925*a^4*B + 21450*a^2*b^2*B + 5005*b^4*B + 14300 
*a^3*b*C + 20020*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqr 
t[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d* 
x]^2))/(15015*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)*(((1897*a^4*A + 11856*a^2*A*b^2 + 1872*A*b^4 + 7904*a^3*b*B + 7488 
*a*b^3*B + 1976*a^4*C + 11232*a^2*b^2*C)*Sin[c + d*x])/9360 + ((4164*a^3*A 
*b + 4576*a*A*b^3 + 1041*a^4*B + 6864*a^2*b^2*B + 1232*b^4*B + 4576*a^3*b* 
C + 4928*a*b^3*C)*Sin[2*(c + d*x)])/1848 + ((2297*a^4*A + 13416*a^2*A*b^2 
+ 1872*A*b^4 + 8944*a^3*b*B + 7488*a*b^3*B + 2236*a^4*C + 11232*a^2*b^2*C) 
*Sin[3*(c + d*x)])/9360 + (a*(32*a^2*A*b + 22*A*b^3 + 8*a^3*B + 33*a*b^2*B 
 + 22*a^2*b*C)*Sin[4*(c + d*x)])/77 + (a^2*(89*a^2*A + 312*A*b^2 + 208*a*b 
*B + 52*a^2*C)*Sin[5*(c + d*x)])/1872 + (a^3*(4*A*b + a*B)*Sin[6*(c + d*x) 
])/88 + (a^4*A*Sin[7*(c + d*x)])/208))/(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 = 3.70 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.95, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4562, 27, 3042, 4535, 3042, 4256, 3042, 4258, 3042, 3120, 4533, 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.

\(\displaystyle \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {2}{13} \int \frac {(a+b \sec (c+d x))^3 \left (b (3 A+13 C) \sec ^2(c+d x)+(11 a A+13 b B+13 a C) \sec (c+d x)+8 A b+13 a B\right )}{2 \sec ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{13} \int \frac {(a+b \sec (c+d x))^3 \left (b (3 A+13 C) \sec ^2(c+d x)+(11 a A+13 b B+13 a C) \sec (c+d x)+8 A b+13 a B\right )}{\sec ^{\frac {11}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (3 A+13 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(11 a A+13 b B+13 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+8 A b+13 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{13} \left (\frac {2}{11} \int \frac {(a+b \sec (c+d x))^2 \left (11 (11 A+13 C) a^2+221 b B a+48 A b^2+b (57 A b+143 C b+39 a B) \sec ^2(c+d x)+\left (117 B a^2+226 A b a+286 b C a+143 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \frac {(a+b \sec (c+d x))^2 \left (11 (11 A+13 C) a^2+221 b B a+48 A b^2+b (57 A b+143 C b+39 a B) \sec ^2(c+d x)+\left (117 B a^2+226 A b a+286 b C a+143 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (11 (11 A+13 C) a^2+221 b B a+48 A b^2+b (57 A b+143 C b+39 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (117 B a^2+226 A b a+286 b C a+143 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \sec (c+d x)) \left (1053 B a^3+2 b (1259 A+1573 C) a^2+2171 b^2 B a+192 A b^3+3 b \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)+\left (77 (11 A+13 C) a^3+2951 b B a^2+3 b^2 (961 A+1287 C) a+1287 b^3 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \sec (c+d x)) \left (1053 B a^3+(2518 A b+3146 C b) a^2+2171 b^2 B a+192 A b^3+3 b \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)+\left (77 (11 A+13 C) a^3+2951 b B a^2+3 b^2 (961 A+1287 C) a+1287 b^3 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (1053 B a^3+(2518 A b+3146 C b) a^2+2171 b^2 B a+192 A b^3+3 b \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (77 (11 A+13 C) a^3+2951 b B a^2+3 b^2 (961 A+1287 C) a+1287 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4562

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}-\frac {2}{7} \int -\frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)+117 \left (45 B a^4+20 b (9 A+11 C) a^3+330 b^2 B a^2+44 b^3 (5 A+7 C) a+77 b^4 B\right ) \sec (c+d x)+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)+117 \left (45 B a^4+20 b (9 A+11 C) a^3+330 b^2 B a^2+44 b^3 (5 A+7 C) a+77 b^4 B\right ) \sec (c+d x)+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+117 \left (45 B a^4+20 b (9 A+11 C) a^3+330 b^2 B a^2+44 b^3 (5 A+7 C) a+77 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x)}dx+\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \sec ^2(c+d x)+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \int \frac {1}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4256

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {1}{3} \int \sqrt {\sec (c+d x)}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {1}{3} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {1}{3} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\int \frac {21 b^2 \left (11 (11 A+13 C) a^2+338 b B a+3 b^2 (73 A+143 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+7 \left (77 (11 A+13 C) a^4+4004 b B a^3+11 b^2 (491 A+637 C) a^2+3458 b^3 B a+192 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4533

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \left (a^4 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}}dx+\frac {14 \sin (c+d x) \left (77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \left (a^4 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right ) \int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 \sin (c+d x) \left (77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right ) \int \sqrt {\cos (c+d x)}dx+\frac {14 \sin (c+d x) \left (77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {231}{5} \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (a^4 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {14 \sin (c+d x) \left (77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )+\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \sin (c+d x) \left (11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right ) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {1}{9} \left (\frac {2 a \sin (c+d x) \left (1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right )}{7 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{7} \left (\frac {14 \sin (c+d x) \left (77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right )}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {462 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (a^4 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right )}{5 d}+117 \left (45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right ) \left (\frac {2 \sin (c+d x)}{3 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}\right )\right )\right )\right )+\frac {2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac {9}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac {11}{2}}(c+d x)}\)

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]^(13/2),x]
 

Output:

(2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + ((2 
*(8*A*b + 13*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^ 
(9/2)) + ((2*(48*A*b^2 + 221*a*b*B + 11*a^2*(11*A + 13*C))*(a + b*Sec[c + 
d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*a*(192*A*b^3 + 1053*a 
^3*B + 2171*a*b^2*B + a^2*(2518*A*b + 3146*b*C))*Sin[c + d*x])/(7*d*Sec[c 
+ d*x]^(5/2)) + ((462*(364*a^3*b*B + 468*a*b^3*B + 39*b^4*(3*A + 5*C) + 78 
*a^2*b^2*(7*A + 9*C) + a^4*(77*A + 91*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c 
+ d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (14*(192*A*b^4 + 4004*a^3*b*B + 3 
458*a*b^3*B + 77*a^4*(11*A + 13*C) + 11*a^2*b^2*(491*A + 637*C))*Sin[c + d 
*x])/(5*d*Sec[c + d*x]^(3/2)) + 117*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 
 44*a*b^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*((2*Sqrt[Cos[c + d*x]]*Elli 
pticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*Sin[c + d*x])/(3*d*Sq 
rt[Sec[c + d*x]])))/7)/9)/11)/13
 

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 4256

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( 
b*Csc[c + d*x])^(n + 1)/(b*d*n)), x] + Simp[(n + 1)/(b^2*n)   Int[(b*Csc[c 
+ d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2* 
n]
 

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 4533

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) 
+ (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + 
Simp[(C*m + A*(m + 1))/(b^2*m)   Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr 
eeQ[{b, e, f, A, C}, 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. \(1406\) vs. \(2(483)=966\).

Time = 83.34 (sec) , antiderivative size = 1407, normalized size of antiderivative = 2.73

method	result	size
default	\(\text {Expression too large to display}\)	\(1407\)
parts	\(\text {Expression too large to display}\)	\(1485\)

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)^(13/2),x 
,method=_RETURNVERBOSE)
 

Output:

-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520* 
a^4*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(1330560*A*a^4+1048320*A*a^ 
3*b+262080*B*a^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-1798720*A*a^4 
-2620800*A*a^3*b-960960*A*a^2*b^2-655200*B*a^4-640640*B*a^3*b-160160*C*a^4 
)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(1379840*A*a^4+2957760*A*a^3*b+ 
1921920*A*a^2*b^2+411840*A*a*b^3+739440*B*a^4+1281280*B*a^3*b+617760*B*a^2 
*b^2+320320*C*a^4+411840*C*a^3*b)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+ 
(-666512*A*a^4-1815840*A*a^3*b-1777776*A*a^2*b^2-617760*A*a*b^3-72072*A*b^ 
4-453960*B*a^4-1185184*B*a^3*b-926640*B*a^2*b^2-288288*B*a*b^3-296296*C*a^ 
4-617760*C*a^3*b-432432*C*a^2*b^2)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c) 
+(198352*A*a^4+720720*A*a^3*b+816816*A*a^2*b^2+480480*A*a*b^3+72072*A*b^4+ 
180180*B*a^4+544544*B*a^3*b+720720*B*a^2*b^2+288288*B*a*b^3+60060*B*b^4+13 
6136*C*a^4+480480*C*a^3*b+432432*C*a^2*b^2+240240*C*a*b^3)*sin(1/2*d*x+1/2 
*c)^4*cos(1/2*d*x+1/2*c)+(-27258*A*a^4-145080*A*a^3*b-144144*A*a^2*b^2-137 
280*A*a*b^3-18018*A*b^4-36270*B*a^4-96096*B*a^3*b-205920*B*a^2*b^2-72072*B 
*a*b^3-30030*B*b^4-24024*C*a^4-137280*C*a^3*b-108108*C*a^2*b^2-120120*C*a* 
b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+35100*A*a^3*b*(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))+42900*a*A*b^3*(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))-17787*A*(sin(1/2*d*x...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.09 \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx =\text {Too large to display} \] 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)^(1 
3/2),x, algorithm="fricas")
 

Output:

-1/45045*(195*sqrt(2)*(45*I*B*a^4 + 20*I*(9*A + 11*C)*a^3*b + 330*I*B*a^2* 
b^2 + 44*I*(5*A + 7*C)*a*b^3 + 77*I*B*b^4)*weierstrassPInverse(-4, 0, cos( 
d*x + c) + I*sin(d*x + c)) + 195*sqrt(2)*(-45*I*B*a^4 - 20*I*(9*A + 11*C)* 
a^3*b - 330*I*B*a^2*b^2 - 44*I*(5*A + 7*C)*a*b^3 - 77*I*B*b^4)*weierstrass 
PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-7*I*(11*A + 
 13*C)*a^4 - 364*I*B*a^3*b - 78*I*(7*A + 9*C)*a^2*b^2 - 468*I*B*a*b^3 - 39 
*I*(3*A + 5*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( 
d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(7*I*(11*A + 13*C)*a^4 + 364*I*B 
*a^3*b + 78*I*(7*A + 9*C)*a^2*b^2 + 468*I*B*a*b^3 + 39*I*(3*A + 5*C)*b^4)* 
weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x 
 + c))) - 2*(3465*A*a^4*cos(d*x + c)^6 + 4095*(B*a^4 + 4*A*a^3*b)*cos(d*x 
+ c)^5 + 385*((11*A + 13*C)*a^4 + 52*B*a^3*b + 78*A*a^2*b^2)*cos(d*x + c)^ 
4 + 585*(9*B*a^4 + 4*(9*A + 11*C)*a^3*b + 66*B*a^2*b^2 + 44*A*a*b^3)*cos(d 
*x + c)^3 + 77*(7*(11*A + 13*C)*a^4 + 364*B*a^3*b + 78*(7*A + 9*C)*a^2*b^2 
 + 468*B*a*b^3 + 117*A*b^4)*cos(d*x + c)^2 + 195*(45*B*a^4 + 20*(9*A + 11* 
C)*a^3*b + 330*B*a^2*b^2 + 44*(5*A + 7*C)*a*b^3 + 77*B*b^4)*cos(d*x + c))* 
sin(d*x + c)/sqrt(cos(d*x + c)))/d
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)* 
*(13/2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)^(1 
3/2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {13}{2}}} \,d x } \] 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)^(1 
3/2),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)^(13/2), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{13/2}} \,d x \] 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))^(13/2),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))^(13/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {13}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{7}}d x \right ) a^{5}+5 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{6}}d x \right ) a^{4} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a^{4} c +10 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{5}}d x \right ) a^{3} b^{2}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{3} b c +10 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{4}}d x \right ) a^{2} b^{3}+6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a^{2} b^{2} c +5 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{3}}d x \right ) a \,b^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{3} c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{5}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )}d x \right ) b^{4} c \] 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)^(13/2),x 
)
 

Output:

int(sqrt(sec(c + d*x))/sec(c + d*x)**7,x)*a**5 + 5*int(sqrt(sec(c + d*x))/ 
sec(c + d*x)**6,x)*a**4*b + int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x)*a**4 
*c + 10*int(sqrt(sec(c + d*x))/sec(c + d*x)**5,x)*a**3*b**2 + 4*int(sqrt(s 
ec(c + d*x))/sec(c + d*x)**4,x)*a**3*b*c + 10*int(sqrt(sec(c + d*x))/sec(c 
 + d*x)**4,x)*a**2*b**3 + 6*int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x)*a**2 
*b**2*c + 5*int(sqrt(sec(c + d*x))/sec(c + d*x)**3,x)*a*b**4 + 4*int(sqrt( 
sec(c + d*x))/sec(c + d*x)**2,x)*a*b**3*c + int(sqrt(sec(c + d*x))/sec(c + 
 d*x)**2,x)*b**5 + int(sqrt(sec(c + d*x))/sec(c + d*x),x)*b**4*c