\(\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [120]

Optimal result

Integrand size = 33, antiderivative size = 254 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{4} a^4 (11 A+14 C) x+\frac {a^4 (454 A+581 C) \sin (c+d x)}{105 d}+\frac {a^4 (11 A+14 C) \cos (c+d x) \sin (c+d x)}{4 d}+\frac {a^4 (247 A+308 C) \cos ^2(c+d x) \sin (c+d x)}{210 d}+\frac {2 a A \cos ^5(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{21 d}+\frac {A \cos ^6(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{7 d}+\frac {(8 A+7 C) \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{35 d}+\frac {(109 A+126 C) \cos ^3(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{210 d} \] Output:

1/4*a^4*(11*A+14*C)*x+1/105*a^4*(454*A+581*C)*sin(d*x+c)/d+1/4*a^4*(11*A+1 
4*C)*cos(d*x+c)*sin(d*x+c)/d+1/210*a^4*(247*A+308*C)*cos(d*x+c)^2*sin(d*x+ 
c)/d+2/21*a*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^3*sin(d*x+c)/d+1/7*A*cos(d*x+c 
)^6*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1/35*(8*A+7*C)*cos(d*x+c)^4*(a^2+a^2*s 
ec(d*x+c))^2*sin(d*x+c)/d+1/210*(109*A+126*C)*cos(d*x+c)^3*(a^4+a^4*sec(d* 
x+c))*sin(d*x+c)/d
 

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.57 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (11760 A c+18480 A d x+23520 C d x+105 (323 A+392 C) \sin (c+d x)+420 (31 A+32 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+4060 C \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+840 C \sin (4 (c+d x))+651 A \sin (5 (c+d x))+84 C \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x)))}{6720 d} \] Input:

Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
 

Output:

(a^4*(11760*A*c + 18480*A*d*x + 23520*C*d*x + 105*(323*A + 392*C)*Sin[c + 
d*x] + 420*(31*A + 32*C)*Sin[2*(c + d*x)] + 5495*A*Sin[3*(c + d*x)] + 4060 
*C*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 840*C*Sin[4*(c + d*x)] + 6 
51*A*Sin[5*(c + d*x)] + 84*C*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 1 
5*A*Sin[7*(c + d*x)]))/(6720*d)
 

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.08, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.576, Rules used = {3042, 4575, 3042, 4505, 27, 3042, 4505, 3042, 4505, 27, 3042, 4484, 25, 3042, 4274, 3042, 3115, 24, 3117}

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.

\(\displaystyle \int \cos ^7(c+d x) (a \sec (c+d x)+a)^4 \left (A+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx\)

\(\Big \downarrow \) 4575

\(\displaystyle \frac {\int \cos ^6(c+d x) (\sec (c+d x) a+a)^4 (4 a A+a (2 A+7 C) \sec (c+d x))dx}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a A+a (2 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{6} \int 2 \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left (3 (8 A+7 C) a^2+(10 A+21 C) \sec (c+d x) a^2\right )dx+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \int \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left (3 (8 A+7 C) a^2+(10 A+21 C) \sec (c+d x) a^2\right )dx+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 (8 A+7 C) a^2+(10 A+21 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left (2 (109 A+126 C) a^3+49 (2 A+3 C) \sec (c+d x) a^3\right )dx+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 (109 A+126 C) a^3+49 (2 A+3 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 4505

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{4} \int 6 \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((247 A+308 C) a^4+2 (69 A+91 C) \sec (c+d x) a^4\right )dx+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \int \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((247 A+308 C) a^4+2 (69 A+91 C) \sec (c+d x) a^4\right )dx+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((247 A+308 C) a^4+2 (69 A+91 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 4484

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (105 (11 A+14 C) a^5+2 (454 A+581 C) \sec (c+d x) a^5\right )dx\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (105 (11 A+14 C) a^5+2 (454 A+581 C) \sec (c+d x) a^5\right )dx+\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \int \frac {105 (11 A+14 C) a^5+2 (454 A+581 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^5}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 4274

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (105 a^5 (11 A+14 C) \int \cos ^2(c+d x)dx+2 a^5 (454 A+581 C) \int \cos (c+d x)dx\right )+\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (2 a^5 (454 A+581 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+105 a^5 (11 A+14 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (2 a^5 (454 A+581 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+105 a^5 (11 A+14 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {1}{3} \left (2 a^5 (454 A+581 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+105 a^5 (11 A+14 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )+\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {\frac {2 a^2 A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{3 d}+\frac {1}{3} \left (\frac {1}{5} \left (\frac {3}{2} \left (\frac {a^5 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {2 a^5 (454 A+581 C) \sin (c+d x)}{d}+105 a^5 (11 A+14 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{2 d}\right )+\frac {3 a^3 (8 A+7 C) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\right )}{7 a}+\frac {A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}\)

Input:

Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]
 

Output:

(A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((2*a^2*A*C 
os[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((3*a^3*(8*A + 
7*C)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (((109*A 
+ 126*C)*Cos[c + d*x]^3*(a^5 + a^5*Sec[c + d*x])*Sin[c + d*x])/(2*d) + (3* 
((a^5*(247*A + 308*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((2*a^5*(454*A 
+ 581*C)*Sin[c + d*x])/d + 105*a^5*(11*A + 14*C)*(x/2 + (Cos[c + d*x]*Sin[ 
c + d*x])/(2*d)))/3))/2)/5)/3)/(7*a)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3115

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

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 4274

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d   In 
t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
 

rule 4484

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

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 4575

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

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.49

method	result	size
parallelrisch	\(\frac {5 a^{4} \left (\frac {\left (31 A +32 C \right ) \sin \left (2 d x +2 c \right )}{5}+\frac {\left (\frac {157 A}{4}+29 C \right ) \sin \left (3 d x +3 c \right )}{15}+\left (A +\frac {2 C}{5}\right ) \sin \left (4 d x +4 c \right )+\frac {\left (\frac {31 A}{4}+C \right ) \sin \left (5 d x +5 c \right )}{25}+\frac {A \sin \left (6 d x +6 c \right )}{15}+\frac {A \sin \left (7 d x +7 c \right )}{140}+\frac {\left (\frac {323 A}{4}+98 C \right ) \sin \left (d x +c \right )}{5}+\frac {44 d \left (A +\frac {14 C}{11}\right ) x}{5}\right )}{16 d}\)	\(125\)
risch	\(\frac {11 a^{4} A x}{4}+\frac {7 a^{4} x C}{2}+\frac {323 \sin \left (d x +c \right ) a^{4} A}{64 d}+\frac {49 \sin \left (d x +c \right ) a^{4} C}{8 d}+\frac {a^{4} A \sin \left (7 d x +7 c \right )}{448 d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{48 d}+\frac {31 a^{4} A \sin \left (5 d x +5 c \right )}{320 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} C}{80 d}+\frac {5 a^{4} A \sin \left (4 d x +4 c \right )}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{8 d}+\frac {157 a^{4} A \sin \left (3 d x +3 c \right )}{192 d}+\frac {29 \sin \left (3 d x +3 c \right ) a^{4} C}{48 d}+\frac {31 \sin \left (2 d x +2 c \right ) a^{4} A}{16 d}+\frac {2 \sin \left (2 d x +2 c \right ) a^{4} C}{d}\)	\(226\)
derivativedivides	\(\frac {\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} C \sin \left (d x +c \right )}{d}\)	\(322\)
default	\(\frac {\frac {a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+\frac {a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+4 a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {6 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+2 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+4 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} C \sin \left (d x +c \right )}{d}\)	\(322\)

Input:

int(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x,method=_RETURNVER 
BOSE)
 

Output:

5/16*a^4*(1/5*(31*A+32*C)*sin(2*d*x+2*c)+1/15*(157/4*A+29*C)*sin(3*d*x+3*c 
)+(A+2/5*C)*sin(4*d*x+4*c)+1/25*(31/4*A+C)*sin(5*d*x+5*c)+1/15*A*sin(6*d*x 
+6*c)+1/140*A*sin(7*d*x+7*c)+1/5*(323/4*A+98*C)*sin(d*x+c)+44/5*d*(A+14/11 
*C)*x)/d
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.57 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (11 \, A + 14 \, C\right )} a^{4} d x + {\left (60 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, A a^{4} \cos \left (d x + c\right )^{5} + 12 \, {\left (48 \, A + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (11 \, A + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 4 \, {\left (227 \, A + 238 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (11 \, A + 14 \, C\right )} a^{4} \cos \left (d x + c\right ) + 4 \, {\left (454 \, A + 581 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{420 \, d} \] Input:

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= 
"fricas")
 

Output:

1/420*(105*(11*A + 14*C)*a^4*d*x + (60*A*a^4*cos(d*x + c)^6 + 280*A*a^4*co 
s(d*x + c)^5 + 12*(48*A + 7*C)*a^4*cos(d*x + c)^4 + 70*(11*A + 6*C)*a^4*co 
s(d*x + c)^3 + 4*(227*A + 238*C)*a^4*cos(d*x + c)^2 + 105*(11*A + 14*C)*a^ 
4*cos(d*x + c) + 4*(454*A + 581*C)*a^4)*sin(d*x + c))/d
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**7*(a+a*sec(d*x+c))**4*(A+C*sec(d*x+c)**2),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.26 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {48 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} A a^{4} - 672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 560 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 112 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} + 3360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 1680 \, C a^{4} \sin \left (d x + c\right )}{1680 \, d} \] Input:

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= 
"maxima")
 

Output:

-1/1680*(48*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35 
*sin(d*x + c))*A*a^4 - 672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin( 
d*x + c))*A*a^4 + 35*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 
 4*c) - 48*sin(2*d*x + 2*c))*A*a^4 + 560*(sin(d*x + c)^3 - 3*sin(d*x + c)) 
*A*a^4 - 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 
 - 112*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 33 
60*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 - 210*(12*d*x + 12*c + sin(4*d* 
x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 - 1680*(2*d*x + 2*c + sin(2*d*x + 2*c 
))*C*a^4 - 1680*C*a^4*sin(d*x + c))/d
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.09 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (11 \, A a^{4} + 14 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (1155 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1470 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 7700 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 9800 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 21791 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 27734 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 33792 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 43008 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 31521 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 39914 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 14700 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21560 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5565 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5250 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{7}}}{420 \, d} \] Input:

integrate(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x, algorithm= 
"giac")
 

Output:

1/420*(105*(11*A*a^4 + 14*C*a^4)*(d*x + c) + 2*(1155*A*a^4*tan(1/2*d*x + 1 
/2*c)^13 + 1470*C*a^4*tan(1/2*d*x + 1/2*c)^13 + 7700*A*a^4*tan(1/2*d*x + 1 
/2*c)^11 + 9800*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 21791*A*a^4*tan(1/2*d*x + 
1/2*c)^9 + 27734*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 33792*A*a^4*tan(1/2*d*x + 
1/2*c)^7 + 43008*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 31521*A*a^4*tan(1/2*d*x + 
1/2*c)^5 + 39914*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 14700*A*a^4*tan(1/2*d*x + 
1/2*c)^3 + 21560*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 5565*A*a^4*tan(1/2*d*x + 1 
/2*c) + 5250*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d
 

Mupad [B] (verification not implemented)

Time = 13.92 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.27 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {11\,A\,a^4}{2}+7\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {110\,A\,a^4}{3}+\frac {140\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {3113\,A\,a^4}{30}+\frac {1981\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {5632\,A\,a^4}{35}+\frac {1024\,C\,a^4}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1501\,A\,a^4}{10}+\frac {2851\,C\,a^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (70\,A\,a^4+\frac {308\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {53\,A\,a^4}{2}+25\,C\,a^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (11\,A+14\,C\right )}{2\,\left (\frac {11\,A\,a^4}{2}+7\,C\,a^4\right )}\right )\,\left (11\,A+14\,C\right )}{2\,d} \] Input:

int(cos(c + d*x)^7*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^4,x)
 

Output:

(tan(c/2 + (d*x)/2)*((53*A*a^4)/2 + 25*C*a^4) + tan(c/2 + (d*x)/2)^13*((11 
*A*a^4)/2 + 7*C*a^4) + tan(c/2 + (d*x)/2)^11*((110*A*a^4)/3 + (140*C*a^4)/ 
3) + tan(c/2 + (d*x)/2)^3*(70*A*a^4 + (308*C*a^4)/3) + tan(c/2 + (d*x)/2)^ 
5*((1501*A*a^4)/10 + (2851*C*a^4)/15) + tan(c/2 + (d*x)/2)^9*((3113*A*a^4) 
/30 + (1981*C*a^4)/15) + tan(c/2 + (d*x)/2)^7*((5632*A*a^4)/35 + (1024*C*a 
^4)/5))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 
+ (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan( 
c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1)) + (a^4*atan((a^4*tan(c/2 + 
 (d*x)/2)*(11*A + 14*C))/(2*((11*A*a^4)/2 + 7*C*a^4)))*(11*A + 14*C))/(2*d 
)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.68 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{4} \left (280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -1330 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -420 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +2205 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +1890 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c -60 \sin \left (d x +c \right )^{7} a +756 \sin \left (d x +c \right )^{5} a +84 \sin \left (d x +c \right )^{5} c -2240 \sin \left (d x +c \right )^{3} a -1120 \sin \left (d x +c \right )^{3} c +3360 \sin \left (d x +c \right ) a +3360 \sin \left (d x +c \right ) c +1155 a d x +1470 c d x \right )}{420 d} \] Input:

int(cos(d*x+c)^7*(a+a*sec(d*x+c))^4*(A+C*sec(d*x+c)^2),x)
 

Output:

(a**4*(280*cos(c + d*x)*sin(c + d*x)**5*a - 1330*cos(c + d*x)*sin(c + d*x) 
**3*a - 420*cos(c + d*x)*sin(c + d*x)**3*c + 2205*cos(c + d*x)*sin(c + d*x 
)*a + 1890*cos(c + d*x)*sin(c + d*x)*c - 60*sin(c + d*x)**7*a + 756*sin(c 
+ d*x)**5*a + 84*sin(c + d*x)**5*c - 2240*sin(c + d*x)**3*a - 1120*sin(c + 
 d*x)**3*c + 3360*sin(c + d*x)*a + 3360*sin(c + d*x)*c + 1155*a*d*x + 1470 
*c*d*x))/(420*d)