Integrand size = 31, antiderivative size = 160 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {1}{8} a^2 (6 A+7 B) x+\frac {a^2 (9 A+10 B) \sin (c+d x)}{5 d}+\frac {a^2 (6 A+7 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (6 A+5 B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{5 d}-\frac {a^2 (9 A+10 B) \sin ^3(c+d x)}{15 d} \] Output:
1/8*a^2*(6*A+7*B)*x+1/5*a^2*(9*A+10*B)*sin(d*x+c)/d+1/8*a^2*(6*A+7*B)*cos( d*x+c)*sin(d*x+c)/d+1/20*a^2*(6*A+5*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/5*A*cos (d*x+c)^4*(a^2+a^2*sec(d*x+c))*sin(d*x+c)/d-1/15*a^2*(9*A+10*B)*sin(d*x+c) ^3/d
Time = 0.30 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^2 (360 A c+360 A d x+420 B d x+60 (11 A+12 B) \sin (c+d x)+240 (A+B) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+80 B \sin (3 (c+d x))+30 A \sin (4 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x)))}{480 d} \] Input:
Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]
Output:
(a^2*(360*A*c + 360*A*d*x + 420*B*d*x + 60*(11*A + 12*B)*Sin[c + d*x] + 24 0*(A + B)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 80*B*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/ (480*d)
Time = 0.78 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {3042, 4505, 3042, 4484, 25, 3042, 4274, 3042, 3113, 2009, 3115, 24}
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.
| \(\displaystyle \int \cos ^5(c+d x) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a) (a (6 A+5 B)+a (3 A+5 B) \sec (c+d x))dx+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a (6 A+5 B)+a (3 A+5 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {1}{5} \left (\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (4 (9 A+10 B) a^2+5 (6 A+7 B) \sec (c+d x) a^2\right )dx\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (4 (9 A+10 B) a^2+5 (6 A+7 B) \sec (c+d x) a^2\right )dx+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {4 (9 A+10 B) a^2+5 (6 A+7 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (4 a^2 (9 A+10 B) \int \cos ^3(c+d x)dx+5 a^2 (6 A+7 B) \int \cos ^2(c+d x)dx\right )+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 a^2 (6 A+7 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+4 a^2 (9 A+10 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 a^2 (6 A+7 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 a^2 (9 A+10 B) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 a^2 (6 A+7 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {4 a^2 (9 A+10 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (5 a^2 (6 A+7 B) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {4 a^2 (9 A+10 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} \left (\frac {a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (5 a^2 (6 A+7 B) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {4 a^2 (9 A+10 B) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {A \sin (c+d x) \cos ^4(c+d x) \left (a^2 \sec (c+d x)+a^2\right )}{5 d}\) |
Input:
Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]
Output:
(A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d) + ((a^2*(6* A + 5*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (5*a^2*(6*A + 7*B)*(x/2 + (C os[c + d*x]*Sin[c + d*x])/(2*d)) - (4*a^2*(9*A + 10*B)*(-Sin[c + d*x] + Si n[c + d*x]^3/3))/d)/4)/5
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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]
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]
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]
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]
Time = 0.88 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59
| method | result | size |
| parallelrisch | \(\frac {a^{2} \left (\left (8 A +8 B \right ) \sin \left (2 d x +2 c \right )+\left (3 A +\frac {8 B}{3}\right ) \sin \left (3 d x +3 c \right )+\left (A +\frac {B}{2}\right ) \sin \left (4 d x +4 c \right )+\frac {A \sin \left (5 d x +5 c \right )}{5}+\left (22 A +24 B \right ) \sin \left (d x +c \right )+12 \left (A +\frac {7 B}{6}\right ) d x \right )}{16 d}\) | \(94\) |
| risch | \(\frac {3 a^{2} A x}{4}+\frac {7 a^{2} x B}{8}+\frac {11 a^{2} A \sin \left (d x +c \right )}{8 d}+\frac {3 \sin \left (d x +c \right ) B \,a^{2}}{2 d}+\frac {A \,a^{2} \sin \left (5 d x +5 c \right )}{80 d}+\frac {A \,a^{2} \sin \left (4 d x +4 c \right )}{16 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{2}}{32 d}+\frac {3 A \,a^{2} \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2}}{6 d}+\frac {A \,a^{2} \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{2 d}\) | \(172\) |
| derivativedivides | \(\frac {\frac {A \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 B \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {A \,a^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(186\) |
| default | \(\frac {\frac {A \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 B \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {A \,a^{2} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(186\) |
| norman | \(\frac {\frac {a^{2} \left (6 A +7 B \right ) x}{8}-\frac {2 a^{2} \left (6 A -B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {2 a^{2} \left (6 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}+\frac {a^{2} \left (6 A +7 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{4 d}+\frac {3 a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{8}-\frac {5 a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{8}-\frac {5 a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {3 a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{8}+\frac {a^{2} \left (6 A +7 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{8}-\frac {16 a^{2} \left (12 A +5 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{15 d}+\frac {a^{2} \left (26 A +25 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (38 A -25 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{20 d}+\frac {a^{2} \left (58 A -135 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{2}}\) | \(394\) |
Input:
int(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)),x,method=_RETURNVERBO SE)
Output:
1/16*a^2*((8*A+8*B)*sin(2*d*x+2*c)+(3*A+8/3*B)*sin(3*d*x+3*c)+(A+1/2*B)*si n(4*d*x+4*c)+1/5*A*sin(5*d*x+5*c)+(22*A+24*B)*sin(d*x+c)+12*(A+7/6*B)*d*x) /d
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.69 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (6 \, A + 7 \, B\right )} a^{2} d x + {\left (24 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (9 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (6 \, A + 7 \, B\right )} a^{2} \cos \left (d x + c\right ) + 16 \, {\left (9 \, A + 10 \, B\right )} a^{2}\right )} \sin \left (d x + c\right )}{120 \, d} \] Input:
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="f ricas")
Output:
1/120*(15*(6*A + 7*B)*a^2*d*x + (24*A*a^2*cos(d*x + c)^4 + 30*(2*A + B)*a^ 2*cos(d*x + c)^3 + 8*(9*A + 10*B)*a^2*cos(d*x + c)^2 + 15*(6*A + 7*B)*a^2* cos(d*x + c) + 16*(9*A + 10*B)*a^2)*sin(d*x + c))/d
Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5*(a+a*sec(d*x+c))**2*(A+B*sec(d*x+c)),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.11 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2}}{480 \, d} \] Input:
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="m axima")
Output:
1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^2 - 160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2 + 30*(12*d*x + 12*c + sin(4*d *x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^2 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a ^2 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2)/d
Time = 0.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.31 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (6 \, A a^{2} + 7 \, B a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (90 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 490 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 864 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 800 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 540 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 790 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 390 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 375 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \] Input:
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)),x, algorithm="g iac")
Output:
1/120*(15*(6*A*a^2 + 7*B*a^2)*(d*x + c) + 2*(90*A*a^2*tan(1/2*d*x + 1/2*c) ^9 + 105*B*a^2*tan(1/2*d*x + 1/2*c)^9 + 420*A*a^2*tan(1/2*d*x + 1/2*c)^7 + 490*B*a^2*tan(1/2*d*x + 1/2*c)^7 + 864*A*a^2*tan(1/2*d*x + 1/2*c)^5 + 800 *B*a^2*tan(1/2*d*x + 1/2*c)^5 + 540*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 790*B*a ^2*tan(1/2*d*x + 1/2*c)^3 + 390*A*a^2*tan(1/2*d*x + 1/2*c) + 375*B*a^2*tan (1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d
Time = 14.22 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.54 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {\left (\frac {3\,A\,a^2}{2}+\frac {7\,B\,a^2}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (7\,A\,a^2+\frac {49\,B\,a^2}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {72\,A\,a^2}{5}+\frac {40\,B\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (9\,A\,a^2+\frac {79\,B\,a^2}{6}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a^2}{2}+\frac {25\,B\,a^2}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,A+7\,B\right )}{4\,\left (\frac {3\,A\,a^2}{2}+\frac {7\,B\,a^2}{4}\right )}\right )\,\left (6\,A+7\,B\right )}{4\,d} \] Input:
int(cos(c + d*x)^5*(A + B/cos(c + d*x))*(a + a/cos(c + d*x))^2,x)
Output:
(tan(c/2 + (d*x)/2)*((13*A*a^2)/2 + (25*B*a^2)/4) + tan(c/2 + (d*x)/2)^9*( (3*A*a^2)/2 + (7*B*a^2)/4) + tan(c/2 + (d*x)/2)^7*(7*A*a^2 + (49*B*a^2)/6) + tan(c/2 + (d*x)/2)^3*(9*A*a^2 + (79*B*a^2)/6) + tan(c/2 + (d*x)/2)^5*(( 72*A*a^2)/5 + (40*B*a^2)/3))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d* x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d* x)/2)^10 + 1)) + (a^2*atan((a^2*tan(c/2 + (d*x)/2)*(6*A + 7*B))/(4*((3*A*a ^2)/2 + (7*B*a^2)/4)))*(6*A + 7*B))/(4*d)
Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.84 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx=\frac {a^{2} \left (-60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -30 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b +150 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +135 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +24 \sin \left (d x +c \right )^{5} a -120 \sin \left (d x +c \right )^{3} a -80 \sin \left (d x +c \right )^{3} b +240 \sin \left (d x +c \right ) a +240 \sin \left (d x +c \right ) b +90 a d x +105 b d x \right )}{120 d} \] Input:
int(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)),x)
Output:
(a**2*( - 60*cos(c + d*x)*sin(c + d*x)**3*a - 30*cos(c + d*x)*sin(c + d*x) **3*b + 150*cos(c + d*x)*sin(c + d*x)*a + 135*cos(c + d*x)*sin(c + d*x)*b + 24*sin(c + d*x)**5*a - 120*sin(c + d*x)**3*a - 80*sin(c + d*x)**3*b + 24 0*sin(c + d*x)*a + 240*sin(c + d*x)*b + 90*a*d*x + 105*b*d*x))/(120*d)