\(\int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1349]

Optimal result

Integrand size = 45, antiderivative size = 452 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \left (a^2-b^2\right ) \left (10 A b^3-75 a^3 B-45 a b^2 B-6 a^2 b (19 A+28 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (10 A b^4-435 a^3 b B-45 a b^3 B-21 a^4 (7 A+9 C)-3 a^2 b^2 (93 A+161 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (5 A b^3+75 a^3 B+135 a b^2 B+a^2 b (163 A+231 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (15 A b^2+90 a b B+7 a^2 (7 A+9 C)\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 (5 A b+9 a B) \cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{63 d}+\frac {2 A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{9 d} \] Output:

-2/315*(a^2-b^2)*(10*A*b^3-75*B*a^3-45*B*a*b^2-6*a^2*b*(19*A+28*C))*((b+a* 
cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2)*(a/(a+b))^( 
1/2))/a^2/d/cos(d*x+c)^(1/2)/(a+b*sec(d*x+c))^(1/2)-2/315*(10*A*b^4-435*B* 
a^3*b-45*B*a*b^3-21*a^4*(7*A+9*C)-3*a^2*b^2*(93*A+161*C))*cos(d*x+c)^(1/2) 
*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(a/(a+b))^(1/2))*(a+b*sec(d*x+c))^(1 
/2)/a^2/d/((b+a*cos(d*x+c))/(a+b))^(1/2)+2/315*(5*A*b^3+75*B*a^3+135*B*a*b 
^2+a^2*b*(163*A+231*C))*cos(d*x+c)^(1/2)*(a+b*sec(d*x+c))^(1/2)*sin(d*x+c) 
/a/d+2/315*(15*A*b^2+90*B*a*b+7*a^2*(7*A+9*C))*cos(d*x+c)^(3/2)*(a+b*sec(d 
*x+c))^(1/2)*sin(d*x+c)/d+2/63*(5*A*b+9*B*a)*cos(d*x+c)^(5/2)*(a+b*sec(d*x 
+c))^(3/2)*sin(d*x+c)/d+2/9*A*cos(d*x+c)^(7/2)*(a+b*sec(d*x+c))^(5/2)*sin( 
d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 26.65 (sec) , antiderivative size = 3785, normalized size of antiderivative = 8.37 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \] Input:

Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x 
] + C*Sec[c + d*x]^2),x]
 

Output:

(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec 
[c + d*x]^2)*(((747*a^2*A*b + 20*A*b^3 + 345*a^3*B + 540*a*b^2*B + 924*a^2 
*b*C)*Sin[c + d*x])/(315*a) + ((133*a^2*A + 150*A*b^2 + 270*a*b*B + 126*a^ 
2*C)*Sin[2*(c + d*x)])/315 + (a*(19*A*b + 9*a*B)*Sin[3*(c + d*x)])/63 + (a 
^2*A*Sin[4*(c + d*x)])/18))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c 
 + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((14*a^3*A*Sqrt[Cos 
[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (62*a*A*b^2 
*Sqrt[Cos[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4 
*A*b^4*Sqrt[Cos[c + d*x]])/(63*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x 
]]) + (58*a^2*b*B*Sqrt[Cos[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Se 
c[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]])/(7*Sqrt[b + a*Cos[c + d*x]]*Sq 
rt[Sec[c + d*x]]) + (6*a^3*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x 
]]*Sqrt[Sec[c + d*x]]) + (46*a*b^2*C*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Co 
s[c + d*x]]*Sqrt[Sec[c + d*x]]) + (58*a^2*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[ 
c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (62*A*b^3*Sqrt[Cos[c + d*x]]*Sq 
rt[Sec[c + d*x]])/(63*Sqrt[b + a*Cos[c + d*x]]) + (10*a^3*B*Sqrt[Cos[c + d 
*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (18*a*b^2*B*Sqrt[ 
Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(7*Sqrt[b + a*Cos[c + d*x]]) + (34*a^2*b 
*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + 
(2*b^3*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]...
 

Rubi [A] (verified)

Time = 4.28 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.08, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4753, 3042, 4582, 27, 3042, 4582, 27, 3042, 4582, 27, 3042, 4592, 27, 3042, 4523, 3042, 4343, 3042, 3134, 3042, 3132, 4345, 3042, 3142, 3042, 3140}

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.

\(\displaystyle \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^{9/2} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\)

\(\Big \downarrow \) 4753

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \sec (c+d x))^{5/2} \left (C \sec ^2(c+d x)+B \sec (c+d x)+A\right )}{\sec ^{\frac {9}{2}}(c+d x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \csc \left (c+d x+\frac {\pi }{2}\right )^2+B \csc \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\)

\(\Big \downarrow \) 4582

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{9} \int \frac {(a+b \sec (c+d x))^{3/2} \left (b (2 A+9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+5 A b+9 a B\right )}{2 \sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {(a+b \sec (c+d x))^{3/2} \left (b (2 A+9 C) \sec ^2(c+d x)+(7 a A+9 b B+9 a C) \sec (c+d x)+5 A b+9 a B\right )}{\sec ^{\frac {7}{2}}(c+d x)}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (2 A+9 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(7 a A+9 b B+9 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+5 A b+9 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 4582

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {\sqrt {a+b \sec (c+d x)} \left (7 (7 A+9 C) a^2+90 b B a+15 A b^2+3 b (8 A b+21 C b+6 a B) \sec ^2(c+d x)+\left (45 B a^2+88 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)\right )}{2 \sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\sqrt {a+b \sec (c+d x)} \left (7 (7 A+9 C) a^2+90 b B a+15 A b^2+3 b (8 A b+21 C b+6 a B) \sec ^2(c+d x)+\left (45 B a^2+88 A b a+126 b C a+63 b^2 B\right ) \sec (c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )} \left (7 (7 A+9 C) a^2+90 b B a+15 A b^2+3 b (8 A b+21 C b+6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (45 B a^2+88 A b a+126 b C a+63 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 4582

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {b \left (14 (7 A+9 C) a^2+270 b B a+15 b^2 (10 A+21 C)\right ) \sec ^2(c+d x)+\left (21 (7 A+9 C) a^3+585 b B a^2+5 b^2 (121 A+189 C) a+315 b^3 B\right ) \sec (c+d x)+3 \left (75 B a^3+b (163 A+231 C) a^2+135 b^2 B a+5 A b^3\right )}{2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {b \left (14 (7 A+9 C) a^2+270 b B a+15 b^2 (10 A+21 C)\right ) \sec ^2(c+d x)+\left (21 (7 A+9 C) a^3+585 b B a^2+5 b^2 (121 A+189 C) a+315 b^3 B\right ) \sec (c+d x)+3 \left (75 B a^3+b (163 A+231 C) a^2+135 b^2 B a+5 A b^3\right )}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {b \left (14 (7 A+9 C) a^2+270 b B a+15 b^2 (10 A+21 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (21 (7 A+9 C) a^3+585 b B a^2+5 b^2 (121 A+189 C) a+315 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (75 B a^3+b (163 A+231 C) a^2+135 b^2 B a+5 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 4592

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {3 \left (-21 (7 A+9 C) a^4-435 b B a^3-3 b^2 (93 A+161 C) a^2-45 b^3 B a-\left (75 B a^3+3 b (87 A+119 C) a^2+405 b^2 B a+5 b^3 (31 A+63 C)\right ) \sec (c+d x) a+10 A b^4\right )}{2 \sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{3 a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-21 (7 A+9 C) a^4-435 b B a^3-3 b^2 (93 A+161 C) a^2-45 b^3 B a-\left (75 B a^3+3 b (87 A+119 C) a^2+405 b^2 B a+5 b^3 (31 A+63 C)\right ) \sec (c+d x) a+10 A b^4}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}}dx}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\int \frac {-21 (7 A+9 C) a^4-435 b B a^3-3 b^2 (93 A+161 C) a^2-45 b^3 B a-\left (75 B a^3+3 b (87 A+119 C) a^2+405 b^2 B a+5 b^3 (31 A+63 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right ) a+10 A b^4}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 4523

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}}dx}{a}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \int \frac {\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 4343

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \cos (c+d x)}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{a \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+b}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3134

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {\left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} \int \sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{a \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3132

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a}+\frac {2 \left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 4345

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \cos (c+d x)}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \sqrt {a \cos (c+d x)+b} \int \frac {1}{\sqrt {b+a \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3142

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {\left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )+\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

\(\Big \downarrow \) 3140

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2 \sin (c+d x) \left (7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right ) \sqrt {a+b \sec (c+d x)}}{5 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\sec (c+d x)}}-\frac {\frac {2 \left (a^2-b^2\right ) \sqrt {\sec (c+d x)} \left (-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}}{a}\right )\right )+\frac {2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac {7}{2}}(c+d x)}\right )\)

Input:

Int[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C* 
Sec[c + d*x]^2),x]
 

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*A*(a + b*Sec[c + d*x])^(5/2)*Sin 
[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + ((2*(5*A*b + 9*a*B)*(a + b*Sec[c + d 
*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + ((2*(15*A*b^2 + 90*a*b 
*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c 
+ d*x]^(3/2)) + (-(((2*(a^2 - b^2)*(10*A*b^3 - 75*a^3*B - 45*a*b^2*B - 6*a 
^2*b*(19*A + 28*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x) 
/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2 
*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93 
*A + 161*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x] 
])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]))/a) + (2*(5 
*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Sec[c 
+ d*x]]*Sin[c + d*x])/(a*d*Sqrt[Sec[c + d*x]]))/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 3132

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a 
 + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, 
b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3134

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + 
b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)]   Int[Sqrt[a/(a + b) + ( 
b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 
, 0] &&  !GtQ[a + b, 0]
 

rule 3140

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S 
qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ 
{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
 

rule 3142

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a 
 + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]]   Int[1/Sqrt[a/(a + b) 
 + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - 
 b^2, 0] &&  !GtQ[a + b, 0]
 

rule 4343

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] 
*(d_.)], x_Symbol] :> Simp[Sqrt[a + b*Csc[e + f*x]]/(Sqrt[d*Csc[e + f*x]]*S 
qrt[b + a*Sin[e + f*x]])   Int[Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[{a 
, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4345

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
+ (a_)], x_Symbol] :> Simp[Sqrt[d*Csc[e + f*x]]*(Sqrt[b + a*Sin[e + f*x]]/S 
qrt[a + b*Csc[e + f*x]])   Int[1/Sqrt[b + a*Sin[e + f*x]], x], x] /; FreeQ[ 
{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4523

Int[(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_))/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(d 
_.)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]), x_Symbol] :> Simp[A/a   I 
nt[Sqrt[a + b*Csc[e + f*x]]/Sqrt[d*Csc[e + f*x]], x], x] - Simp[(A*b - a*B) 
/(a*d)   Int[Sqrt[d*Csc[e + f*x]]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ 
[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0]
 

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]
 

rule 4592

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 + 1)*((d 
*Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m 
*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* 
Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d 
, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

rule 4753

Int[(cos[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cos[a 
+ b*x])^m*(c*Sec[a + b*x])^m   Int[ActivateTrig[u]/(c*Sec[a + b*x])^m, x], 
x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSecantIntegrandQ[u, x 
]
 

Maple [F(-1)]

Timed out.

hanged

Input:

int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 
),x)
 

Output:

int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 
),x)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 717, normalized size of antiderivative = 1.59 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d* 
x+c)^2),x, algorithm="fricas")
 

Output:

2/945*(3*(35*A*a^5*cos(d*x + c)^3 + 75*B*a^5 + (163*A + 231*C)*a^4*b + 135 
*B*a^3*b^2 + 5*A*a^2*b^3 + 5*(9*B*a^5 + 19*A*a^4*b)*cos(d*x + c)^2 + (7*(7 
*A + 9*C)*a^5 + 135*B*a^4*b + 75*A*a^3*b^2)*cos(d*x + c))*sqrt((a*cos(d*x 
+ c) + b)/cos(d*x + c))*sqrt(cos(d*x + c))*sin(d*x + c) - sqrt(1/2)*(225*I 
*B*a^5 + 3*I*(163*A + 231*C)*a^4*b + 345*I*B*a^3*b^2 - 3*I*(31*A + 7*C)*a^ 
2*b^3 - 90*I*B*a*b^4 + 20*I*A*b^5)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 
 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*s 
in(d*x + c) + 2*b)/a) - sqrt(1/2)*(-225*I*B*a^5 - 3*I*(163*A + 231*C)*a^4* 
b - 345*I*B*a^3*b^2 + 3*I*(31*A + 7*C)*a^2*b^3 + 90*I*B*a*b^4 - 20*I*A*b^5 
)*sqrt(a)*weierstrassPInverse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8* 
b^3)/a^3, 1/3*(3*a*cos(d*x + c) - 3*I*a*sin(d*x + c) + 2*b)/a) - 3*sqrt(1/ 
2)*(-21*I*(7*A + 9*C)*a^5 - 435*I*B*a^4*b - 3*I*(93*A + 161*C)*a^3*b^2 - 4 
5*I*B*a^2*b^3 + 10*I*A*a*b^4)*sqrt(a)*weierstrassZeta(-4/3*(3*a^2 - 4*b^2) 
/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInverse(-4/3*(3*a^2 - 4*b^2) 
/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x + c) + 3*I*a*sin(d*x + 
c) + 2*b)/a)) - 3*sqrt(1/2)*(21*I*(7*A + 9*C)*a^5 + 435*I*B*a^4*b + 3*I*(9 
3*A + 161*C)*a^3*b^2 + 45*I*B*a^2*b^3 - 10*I*A*a*b^4)*sqrt(a)*weierstrassZ 
eta(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, weierstrassPInve 
rse(-4/3*(3*a^2 - 4*b^2)/a^2, 8/27*(9*a^2*b - 8*b^3)/a^3, 1/3*(3*a*cos(d*x 
 + c) - 3*I*a*sin(d*x + c) + 2*b)/a)))/(a^3*d)
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(9/2)*(a+b*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec( 
d*x+c)**2),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d* 
x+c)^2),x, algorithm="maxima")
 

Output:

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(b*sec(d*x + c) + a)^(5/ 
2)*cos(d*x + c)^(9/2), x)
 

Giac [F]

\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \] Input:

integrate(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(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)^(5/ 
2)*cos(d*x + c)^(9/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^{9/2}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:

int(cos(c + d*x)^(9/2)*(a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/ 
cos(c + d*x)^2),x)
 

Output:

int(cos(c + d*x)^(9/2)*(a + b/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/ 
cos(c + d*x)^2), x)
 

Reduce [F]

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

int(cos(d*x+c)^(9/2)*(a+b*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2 
),x)
 

Output:

int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d* 
x)**4,x)*b**2*c + 2*int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c 
+ d*x)**4*sec(c + d*x)**3,x)*a*b*c + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos 
(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**3,x)*b**3 + int(sqrt(sec(c + d*x) 
*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x)**2,x)*a**2*c + 3*i 
nt(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**4*sec(c + d*x 
)**2,x)*a*b**2 + 3*int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + 
 d*x)**4*sec(c + d*x),x)*a**2*b + int(sqrt(sec(c + d*x)*b + a)*sqrt(cos(c 
+ d*x))*cos(c + d*x)**4,x)*a**3