\(\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [509]

Optimal result

Integrand size = 43, antiderivative size = 311 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{5/2} (1015 A+1132 B+1304 C) \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{512 d}+\frac {a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1015 A+1132 B+1304 C) \cos (c+d x) \sin (c+d x)}{768 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (545 A+628 B+680 C) \cos ^2(c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (115 A+156 B+120 C) \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{480 d}+\frac {a (5 A+12 B) \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{60 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d} \] Output:

1/512*a^(5/2)*(1015*A+1132*B+1304*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d* 
x+c))^(1/2))/d+1/512*a^3*(1015*A+1132*B+1304*C)*sin(d*x+c)/d/(a+a*sec(d*x+ 
c))^(1/2)+1/768*a^3*(1015*A+1132*B+1304*C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*se 
c(d*x+c))^(1/2)+1/960*a^3*(545*A+628*B+680*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a 
+a*sec(d*x+c))^(1/2)+1/480*a^2*(115*A+156*B+120*C)*cos(d*x+c)^3*(a+a*sec(d 
*x+c))^(1/2)*sin(d*x+c)/d+1/60*a*(5*A+12*B)*cos(d*x+c)^4*(a+a*sec(d*x+c))^ 
(3/2)*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.96 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.72 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 a^2 \cos (c+d x) \sqrt {a (1+\sec (c+d x))} \left (693 C \cos ^2(c+d x) \sin (c+d x)+495 B \cos ^3(c+d x) \sin (c+d x)+1584 C \cos ^3(c+d x) \sin (c+d x)+385 A \cos ^4(c+d x) \sin (c+d x)+1210 B \cos ^4(c+d x) \sin (c+d x)+980 A \cos ^5(c+d x) \sin (c+d x)-16137 C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},5,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)-15565 B \operatorname {Hypergeometric2F1}\left (\frac {1}{2},6,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)-15225 A \operatorname {Hypergeometric2F1}\left (\frac {1}{2},7,\frac {3}{2},1-\sec (c+d x)\right ) \tan (c+d x)\right )}{3465 d (1+\cos (c+d x))} \] Input:

Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + 
C*Sec[c + d*x]^2),x]
 

Output:

(-2*a^2*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(693*C*Cos[c + d*x]^2*Sin[ 
c + d*x] + 495*B*Cos[c + d*x]^3*Sin[c + d*x] + 1584*C*Cos[c + d*x]^3*Sin[c 
 + d*x] + 385*A*Cos[c + d*x]^4*Sin[c + d*x] + 1210*B*Cos[c + d*x]^4*Sin[c 
+ d*x] + 980*A*Cos[c + d*x]^5*Sin[c + d*x] - 16137*C*Hypergeometric2F1[1/2 
, 5, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x] - 15565*B*Hypergeometric2F1[1/2, 
6, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x] - 15225*A*Hypergeometric2F1[1/2, 7, 
 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x]))/(3465*d*(1 + Cos[c + d*x]))
 

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4574, 27, 3042, 4505, 27, 3042, 4505, 27, 3042, 4503, 3042, 4292, 3042, 4292, 3042, 4261, 216}

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.

\(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^{5/2} \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\)

\(\Big \downarrow \) 4574

\(\displaystyle \frac {\int \frac {1}{2} \cos ^5(c+d x) (\sec (c+d x) a+a)^{5/2} (a (5 A+12 B)+a (5 A+12 C) \sec (c+d x))dx}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \cos ^5(c+d x) (\sec (c+d x) a+a)^{5/2} (a (5 A+12 B)+a (5 A+12 C) \sec (c+d x))dx}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{5/2} \left (a (5 A+12 B)+a (5 A+12 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{5} \int \frac {1}{2} \cos ^4(c+d x) (\sec (c+d x) a+a)^{3/2} \left ((115 A+156 B+120 C) a^2+15 (5 A+4 B+8 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{10} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^{3/2} \left ((115 A+156 B+120 C) a^2+15 (5 A+4 B+8 C) \sec (c+d x) a^2\right )dx+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{10} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^{3/2} \left ((115 A+156 B+120 C) a^2+15 (5 A+4 B+8 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{4} \int \frac {1}{2} \cos ^3(c+d x) \sqrt {\sec (c+d x) a+a} \left (3 (545 A+628 B+680 C) a^3+5 (235 A+252 B+312 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \int \cos ^3(c+d x) \sqrt {\sec (c+d x) a+a} \left (3 (545 A+628 B+680 C) a^3+5 (235 A+252 B+312 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a} \left (3 (545 A+628 B+680 C) a^3+5 (235 A+252 B+312 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 4503

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \int \cos ^2(c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 4292

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \int \cos (c+d x) \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \int \frac {\sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\csc \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 4292

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sqrt {\sec (c+d x) a+a}dx+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {1}{2} \int \sqrt {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 4261

\(\displaystyle \frac {\frac {1}{10} \left (\frac {1}{8} \left (\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}-\frac {a \int \frac {1}{\frac {a^2 \tan ^2(c+d x)}{\sec (c+d x) a+a}+a}d\left (-\frac {a \tan (c+d x)}{\sqrt {\sec (c+d x) a+a}}\right )}{d}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}\right )+\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {a^2 (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac {1}{10} \left (\frac {a^3 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{4 d}+\frac {1}{8} \left (\frac {a^4 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{d \sqrt {a \sec (c+d x)+a}}+\frac {5}{2} a^3 (1015 A+1132 B+1304 C) \left (\frac {3}{4} \left (\frac {\sqrt {a} \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}\right )+\frac {a \sin (c+d x) \cos (c+d x)}{2 d \sqrt {a \sec (c+d x)+a}}\right )\right )\right )}{12 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}\)

Input:

Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[ 
c + d*x]^2),x]
 

Output:

(A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d) + ((a^2*( 
5*A + 12*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) 
+ ((a^3*(115*A + 156*B + 120*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Si 
n[c + d*x])/(4*d) + ((a^4*(545*A + 628*B + 680*C)*Cos[c + d*x]^2*Sin[c + d 
*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (5*a^3*(1015*A + 1132*B + 1304*C)*((a* 
Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]]) + (3*((Sqrt[a]*A 
rcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*Sin[c + d*x 
])/(d*Sqrt[a + a*Sec[c + d*x]])))/4))/2)/8)/10)/(12*a)
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4261

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) 
  Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], 
 x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
 

rule 4292

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
 + (a_)], x_Symbol] :> Simp[a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n*Sqrt[a 
+ b*Csc[e + f*x]])), x] + Simp[a*((2*n + 1)/(2*b*d*n))   Int[Sqrt[a + b*Csc 
[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && 
 EqQ[a^2 - b^2, 0] && LtQ[n, -2^(-1)] && IntegerQ[2*n]
 

rule 4503

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) 
 + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*b^2*Co 
t[e + f*x]*((d*Csc[e + f*x])^n/(a*f*n*Sqrt[a + b*Csc[e + f*x]])), x] + Simp 
[(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n)   Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[ 
e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a 
*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] && LtQ[n, 0]
 

rule 4505

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot 
[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim 
p[b/(a*d*n)   Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim 
p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] 
/; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 
] && GtQ[m, 1/2] && LtQ[n, -1]
 

rule 4574

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

Maple [A] (warning: unable to verify)

Time = 1.02 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.45

\[\frac {a^{2} \left (\left (15225 \cos \left (d x +c \right )+15225\right ) A \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (16980 \cos \left (d x +c \right )+16980\right ) B \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\left (19560 \cos \left (d x +c \right )+19560\right ) C \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sqrt {2}\, \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{\sqrt {\csc \left (d x +c \right )^{2}-2 \csc \left (d x +c \right ) \cot \left (d x +c \right )+\cot \left (d x +c \right )^{2}-1}}\right )+\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (1280 \cos \left (d x +c \right )^{5}+4480 \cos \left (d x +c \right )^{4}+6960 \cos \left (d x +c \right )^{3}+8120 \cos \left (d x +c \right )^{2}+10150 \cos \left (d x +c \right )+15225\right ) A +\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (1536 \cos \left (d x +c \right )^{4}+5568 \cos \left (d x +c \right )^{3}+9056 \cos \left (d x +c \right )^{2}+11320 \cos \left (d x +c \right )+16980\right ) B +\sin \left (d x +c \right ) \cos \left (d x +c \right ) \left (1920 \cos \left (d x +c \right )^{3}+7360 \cos \left (d x +c \right )^{2}+13040 \cos \left (d x +c \right )+19560\right ) C \right ) \sqrt {a \left (1+\sec \left (d x +c \right )\right )}}{7680 d \left (\cos \left (d x +c \right )+1\right )}\]

Input:

int(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
 

Output:

1/7680/d*a^2*((15225*cos(d*x+c)+15225)*A*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2 
)*arctanh(2^(1/2)/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+cot(d*x+c)^2-1)^(1 
/2)*(csc(d*x+c)-cot(d*x+c)))+(16980*cos(d*x+c)+16980)*B*(-cos(d*x+c)/(cos( 
d*x+c)+1))^(1/2)*arctanh(2^(1/2)/(csc(d*x+c)^2-2*csc(d*x+c)*cot(d*x+c)+cot 
(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c)))+(19560*cos(d*x+c)+19560)*C*(-c 
os(d*x+c)/(cos(d*x+c)+1))^(1/2)*arctanh(2^(1/2)/(csc(d*x+c)^2-2*csc(d*x+c) 
*cot(d*x+c)+cot(d*x+c)^2-1)^(1/2)*(csc(d*x+c)-cot(d*x+c)))+sin(d*x+c)*cos( 
d*x+c)*(1280*cos(d*x+c)^5+4480*cos(d*x+c)^4+6960*cos(d*x+c)^3+8120*cos(d*x 
+c)^2+10150*cos(d*x+c)+15225)*A+sin(d*x+c)*cos(d*x+c)*(1536*cos(d*x+c)^4+5 
568*cos(d*x+c)^3+9056*cos(d*x+c)^2+11320*cos(d*x+c)+16980)*B+sin(d*x+c)*co 
s(d*x+c)*(1920*cos(d*x+c)^3+7360*cos(d*x+c)^2+13040*cos(d*x+c)+19560)*C)*( 
a*(1+sec(d*x+c)))^(1/2)/(cos(d*x+c)+1)
 

Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.72 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="fricas")
 

Output:

[1/15360*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c) + (1015*A + 1132 
*B + 1304*C)*a^2)*sqrt(-a)*log((2*a*cos(d*x + c)^2 - 2*sqrt(-a)*sqrt((a*co 
s(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c) - 
 a)/(cos(d*x + c) + 1)) + 2*(1280*A*a^2*cos(d*x + c)^6 + 128*(35*A + 12*B) 
*a^2*cos(d*x + c)^5 + 48*(145*A + 116*B + 40*C)*a^2*cos(d*x + c)^4 + 8*(10 
15*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 10*(1015*A + 1132*B + 1304*C)* 
a^2*cos(d*x + c)^2 + 15*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c))*sqrt( 
(a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d), -1/ 
7680*(15*((1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c) + (1015*A + 1132*B + 
 1304*C)*a^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d 
*x + c)/(sqrt(a)*sin(d*x + c))) - (1280*A*a^2*cos(d*x + c)^6 + 128*(35*A + 
 12*B)*a^2*cos(d*x + c)^5 + 48*(145*A + 116*B + 40*C)*a^2*cos(d*x + c)^4 + 
 8*(1015*A + 1132*B + 920*C)*a^2*cos(d*x + c)^3 + 10*(1015*A + 1132*B + 13 
04*C)*a^2*cos(d*x + c)^2 + 15*(1015*A + 1132*B + 1304*C)*a^2*cos(d*x + c)) 
*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d 
)]
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+ 
c)**2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

Timed out
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2319 vs. \(2 (279) = 558\).

Time = 1.83 (sec) , antiderivative size = 2319, normalized size of antiderivative = 7.46 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c) 
^2),x, algorithm="giac")
 

Output:

-1/15360*(15*(1015*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 1132*B*sqrt(-a)*a^2* 
sgn(cos(d*x + c)) + 1304*C*sqrt(-a)*a^2*sgn(cos(d*x + c)))*log(abs((sqrt(- 
a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 - a*(2*sq 
rt(2) + 3))) - 15*(1015*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 1132*B*sqrt(-a) 
*a^2*sgn(cos(d*x + c)) + 1304*C*sqrt(-a)*a^2*sgn(cos(d*x + c)))*log(abs((s 
qrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2 + a* 
(2*sqrt(2) - 3))) + 4*(15225*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt 
(-a*tan(1/2*d*x + 1/2*c)^2 + a))^22*A*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 169 
80*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 
 + a))^22*B*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 19560*sqrt(2)*(sqrt(-a)*tan(1 
/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^22*C*sqrt(-a)*a^3*s 
gn(cos(d*x + c)) - 502425*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a 
*tan(1/2*d*x + 1/2*c)^2 + a))^20*A*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 560340 
*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + 
 a))^20*B*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 645480*sqrt(2)*(sqrt(-a)*tan(1/ 
2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^20*C*sqrt(-a)*a^4*sg 
n(cos(d*x + c)) + 6518495*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a 
*tan(1/2*d*x + 1/2*c)^2 + a))^18*A*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 796302 
0*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 
+ a))^18*B*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 8467800*sqrt(2)*(sqrt(-a)*t...
 

Mupad [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\cos \left (c+d\,x\right )}^6\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \] Input:

int(cos(c + d*x)^6*(a + a/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)^6*(a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos( 
c + d*x)^2), x)
 

Reduce [F]

\[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\sqrt {a}\, a^{2} \left (\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{3}d x \right ) b +2 \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{3}d x \right ) c +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{2}d x \right ) a +2 \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{2}d x \right ) c +2 \left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )+1}\, \cos \left (d x +c \right )^{6}d x \right ) a \right ) \] Input:

int(cos(d*x+c)^6*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
 

Output:

sqrt(a)*a**2*(int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**6*sec(c + d*x)**4,x 
)*c + int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**6*sec(c + d*x)**3,x)*b + 2* 
int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**6*sec(c + d*x)**3,x)*c + int(sqrt 
(sec(c + d*x) + 1)*cos(c + d*x)**6*sec(c + d*x)**2,x)*a + 2*int(sqrt(sec(c 
 + d*x) + 1)*cos(c + d*x)**6*sec(c + d*x)**2,x)*b + int(sqrt(sec(c + d*x) 
+ 1)*cos(c + d*x)**6*sec(c + d*x)**2,x)*c + 2*int(sqrt(sec(c + d*x) + 1)*c 
os(c + d*x)**6*sec(c + d*x),x)*a + int(sqrt(sec(c + d*x) + 1)*cos(c + d*x) 
**6*sec(c + d*x),x)*b + int(sqrt(sec(c + d*x) + 1)*cos(c + d*x)**6,x)*a)