\(\int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [997]

Optimal result

Integrand size = 43, antiderivative size = 397 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{15 d}+\frac {2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{315 d}+\frac {2 (3 b B+2 a C) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{21 d}+\frac {2 C \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{9 d} \] Output:

-2/15*(15*B*a^3+27*B*a*b^2+9*a^2*b*(5*A+3*C)+b^3*(9*A+7*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*(21*B*a^2 
*b+5*B*b^3+7*a^3*(3*A+C)+3*a*b^2*(7*A+5*C))*cos(d*x+c)^(1/2)*InverseJacobi 
AM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+2/15*(15*B*a^3+27*B*a*b^2+9*a 
^2*b*(5*A+3*C)+b^3*(9*A+7*C))*sec(d*x+c)^(1/2)*sin(d*x+c)/d+2/63*(54*B*a^2 
*b+15*B*b^3+8*a^3*C+9*a*b^2*(7*A+5*C))*sec(d*x+c)^(3/2)*sin(d*x+c)/d+2/315 
*b*(63*A*b^2+99*B*a*b+24*C*a^2+49*C*b^2)*sec(d*x+c)^(5/2)*sin(d*x+c)/d+2/2 
1*(3*B*b+2*C*a)*sec(d*x+c)^(3/2)*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+2/9*C*sec 
(d*x+c)^(3/2)*(a+b*sec(d*x+c))^3*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 10.23 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.43 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2 \cos ^5(c+d x) \left (\frac {2 \left (-315 a^2 A b-63 A b^3-105 a^3 B-189 a b^2 B-189 a^2 b C-49 b^3 C\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}+2 \left (105 a^3 A+105 a A b^2+105 a^2 b B+25 b^3 B+35 a^3 C+75 a b^2 C\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}\right ) (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{105 d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}+\frac {(a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {4}{15} \left (45 a^2 A b+9 A b^3+15 a^3 B+27 a b^2 B+27 a^2 b C+7 b^3 C\right ) \sin (c+d x)+\frac {4}{7} \sec ^3(c+d x) \left (b^3 B \sin (c+d x)+3 a b^2 C \sin (c+d x)\right )+\frac {4}{21} \sec (c+d x) \left (21 a A b^2 \sin (c+d x)+21 a^2 b B \sin (c+d x)+5 b^3 B \sin (c+d x)+7 a^3 C \sin (c+d x)+15 a b^2 C \sin (c+d x)\right )+\frac {4}{45} \sec ^2(c+d x) \left (9 A b^3 \sin (c+d x)+27 a b^2 B \sin (c+d x)+27 a^2 b C \sin (c+d x)+7 b^3 C \sin (c+d x)\right )+\frac {4}{9} b^3 C \sec ^3(c+d x) \tan (c+d x)\right )}{d (b+a \cos (c+d x))^3 (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sec ^{\frac {9}{2}}(c+d x)} \] Input:

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

Output:

(2*Cos[c + d*x]^5*((2*(-315*a^2*A*b - 63*A*b^3 - 105*a^3*B - 189*a*b^2*B - 
 189*a^2*b*C - 49*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sq 
rt[Sec[c + d*x]]) + 2*(105*a^3*A + 105*a*A*b^2 + 105*a^2*b*B + 25*b^3*B + 
35*a^3*C + 75*a*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[S 
ec[c + d*x]])*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 
2))/(105*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c 
+ 2*d*x])) + ((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^ 
2)*((4*(45*a^2*A*b + 9*A*b^3 + 15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3* 
C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^3*(b^3*B*Sin[c + d*x] + 3*a*b^2*C*Si 
n[c + d*x]))/7 + (4*Sec[c + d*x]*(21*a*A*b^2*Sin[c + d*x] + 21*a^2*b*B*Sin 
[c + d*x] + 5*b^3*B*Sin[c + d*x] + 7*a^3*C*Sin[c + d*x] + 15*a*b^2*C*Sin[c 
 + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^3*Sin[c + d*x] + 27*a*b^2*B*Sin[c 
+ d*x] + 27*a^2*b*C*Sin[c + d*x] + 7*b^3*C*Sin[c + d*x]))/45 + (4*b^3*C*Se 
c[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Co 
s[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))
 

Rubi [A] (verified)

Time = 2.53 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.93, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4584, 27, 3042, 4584, 27, 3042, 4564, 27, 3042, 4535, 3042, 4255, 3042, 4258, 3042, 3119, 4534, 3042, 4258, 3042, 3120}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4584

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4584

\(\displaystyle \frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\left (24 C a^2+99 b B a+63 A b^2+49 b^2 C\right ) \sec ^2(c+d x)+\left (63 B a^2+126 A b a+86 b C a+45 b^2 B\right ) \sec (c+d x)+a (63 a A+9 b B+13 a C)\right )dx+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x)) \left (\left (24 C a^2+99 b B a+63 A b^2+49 b^2 C\right ) \sec ^2(c+d x)+\left (63 B a^2+126 A b a+86 b C a+45 b^2 B\right ) \sec (c+d x)+a (63 a A+9 b B+13 a C)\right )dx+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (24 C a^2+99 b B a+63 A b^2+49 b^2 C\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (63 B a^2+126 A b a+86 b C a+45 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a (63 a A+9 b B+13 a C)\right )dx+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 4564

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\sec (c+d x)} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \sec ^2(c+d x)+21 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \sec (c+d x)\right )dx+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \sqrt {\sec (c+d x)} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \sec ^2(c+d x)+21 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \sec (c+d x)\right )dx+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+21 \left (15 B a^3+9 b (5 A+3 C) a^2+27 b^2 B a+b^3 (9 A+7 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \int \sec ^{\frac {3}{2}}(c+d x)dx+\int \sqrt {\sec (c+d x)} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \sec ^2(c+d x)\right )dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 4255

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\sec (c+d x)}}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\int \frac {1}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \left (5 (63 a A+9 b B+13 a C) a^2+15 \left (8 C a^3+54 b B a^2+9 b^2 (7 A+5 C) a+15 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 4534

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \sqrt {\sec (c+d x)}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 4258

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (15 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {1}{9} \left (\frac {1}{7} \left (\frac {2 b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{5 d}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{d}+\frac {30 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{d}+21 \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right ) \left (\frac {2 \sin (c+d x) \sqrt {\sec (c+d x)}}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {6 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}\right )+\frac {2 C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}\)

Input:

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

Output:

(2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*( 
3*b*B + 2*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7* 
d) + ((2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sec[c + d*x]^(5/2)* 
Sin[c + d*x])/(5*d) + ((30*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b 
^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + 
d*x]])/d + (10*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sec 
[c + d*x]^(3/2)*Sin[c + d*x])/d + 21*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A 
 + 3*C) + b^3*(9*A + 7*C))*((-2*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 
2]*Sqrt[Sec[c + d*x]])/d + (2*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 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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 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 4584

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[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Cs 
c[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(m + n + 1)   Int[(a + b*Csc[e 
+ f*x])^(m - 1)*(d*Csc[e + f*x])^n*Simp[a*A*(m + n + 1) + a*C*n + ((A*b + a 
*B)*(m + n + 1) + b*C*(m + n))*Csc[e + f*x] + (b*B*(m + n + 1) + a*C*m)*Csc 
[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, 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. \(1264\) vs. \(2(368)=736\).

Time = 18.97 (sec) , antiderivative size = 1265, normalized size of antiderivative = 3.19

method	result	size
default	\(\text {Expression too large to display}\)	\(1265\)
parts	\(\text {Expression too large to display}\)	\(1582\)

Input:

int(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, 
method=_RETURNVERBOSE)
 

Output:

-(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*a^3*A*(sin(1 
/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2)/(-2*sin(1/2*d*x+1 
/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+ 
2*C*b^3*(-1/144*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)/(cos(1/2*d*x+1/2*c)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-2*si 
n(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)^2-1/2)^ 
3-14/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2 
+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(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))-7/15*(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))-EllipticE(cos(1/2*d 
*x+1/2*c),2^(1/2))))+2*a*(3*A*b^2+3*B*a*b+C*a^2)*(-1/6*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)/(cos(1/2*d*x+1/2*c)^2 
-1/2)^2+1/3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1)^(1/2) 
/(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d* 
x+1/2*c),2^(1/2)))+2*a^2*(3*A*b+B*a)/sin(1/2*d*x+1/2*c)^2/(2*sin(1/2*d*x+1 
/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2 
*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*Elliptic 
E(cos(1/2*d*x+1/2*c),2^(1/2))*(sin(1/2*d*x+1/2*c)^2)^(1/2))+2/5*b*(A*b^...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.12 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.18 \[ \int \sqrt {\sec (c+d x)} (a+b \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {15 \, \sqrt {2} {\left (7 i \, {\left (3 \, A + C\right )} a^{3} + 21 i \, B a^{2} b + 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2} + 5 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-7 i \, {\left (3 \, A + C\right )} a^{3} - 21 i \, B a^{2} b - 3 i \, {\left (7 \, A + 5 \, C\right )} a b^{2} - 5 i \, B b^{3}\right )} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, \sqrt {2} {\left (15 i \, B a^{3} + 9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b + 27 i \, B a b^{2} + i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} {\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 (-15 i \, B a^{3} - 9 i \, {\left (5 \, A + 3 \, C\right )} a^{2} b - 27 i \, B a b^{2} - i \, {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} {\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 (21 \, {\left (15 \, B a^{3} + 9 \, {\left (5 \, A + 3 \, C\right )} a^{2} b + 27 \, B a b^{2} + {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 35 \, C b^{3} + 15 \, {\left (7 \, C a^{3} + 21 \, B a^{2} b + 3 \, {\left (7 \, A + 5 \, C\right )} a b^{2} + 5 \, B b^{3}\right )} \cos \left (d x + c\right )^{3} + 7 \, {\left (27 \, C a^{2} b + 27 \, B a b^{2} + {\left (9 \, A + 7 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, C a b^{2} + B 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 \cos \left (d x + c\right )^{4}} \] Input:

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="fricas")
 

Output:

-1/315*(15*sqrt(2)*(7*I*(3*A + C)*a^3 + 21*I*B*a^2*b + 3*I*(7*A + 5*C)*a*b 
^2 + 5*I*B*b^3)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) + I 
*sin(d*x + c)) + 15*sqrt(2)*(-7*I*(3*A + C)*a^3 - 21*I*B*a^2*b - 3*I*(7*A 
+ 5*C)*a*b^2 - 5*I*B*b^3)*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d* 
x + c) - I*sin(d*x + c)) + 21*sqrt(2)*(15*I*B*a^3 + 9*I*(5*A + 3*C)*a^2*b 
+ 27*I*B*a*b^2 + I*(9*A + 7*C)*b^3)*cos(d*x + c)^4*weierstrassZeta(-4, 0, 
weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 21*sqrt(2)*(- 
15*I*B*a^3 - 9*I*(5*A + 3*C)*a^2*b - 27*I*B*a*b^2 - I*(9*A + 7*C)*b^3)*cos 
(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) 
 - I*sin(d*x + c))) - 2*(21*(15*B*a^3 + 9*(5*A + 3*C)*a^2*b + 27*B*a*b^2 + 
 (9*A + 7*C)*b^3)*cos(d*x + c)^4 + 35*C*b^3 + 15*(7*C*a^3 + 21*B*a^2*b + 3 
*(7*A + 5*C)*a*b^2 + 5*B*b^3)*cos(d*x + c)^3 + 7*(27*C*a^2*b + 27*B*a*b^2 
+ (9*A + 7*C)*b^3)*cos(d*x + c)^2 + 45*(3*C*a*b^2 + B*b^3)*cos(d*x + c))*s 
in(d*x + c)/sqrt(cos(d*x + c)))/(d*cos(d*x + c)^4)
 

Sympy [F(-1)]

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

integrate(sec(d*x+c)**(1/2)*(a+b*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x+ 
c)**2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

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

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

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

integrate(sec(d*x+c)^(1/2)*(a+b*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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