Integrand size = 41, antiderivative size = 213 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {7}{16} a^4 (7 A+8 B+10 C) x+\frac {4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d} \] Output:
7/16*a^4*(7*A+8*B+10*C)*x+4/5*a^4*(7*A+8*B+10*C)*sin(d*x+c)/d+27/80*a^4*(7 *A+8*B+10*C)*cos(d*x+c)*sin(d*x+c)/d+1/40*a^4*(7*A+8*B+10*C)*cos(d*x+c)^3* sin(d*x+c)/d+1/15*(2*A+3*B)*cos(d*x+c)^4*(a+a*sec(d*x+c))^4*sin(d*x+c)/d+1 /6*A*cos(d*x+c)^5*(a+a*sec(d*x+c))^4*sin(d*x+c)/d-2/15*a^4*(7*A+8*B+10*C)* sin(d*x+c)^3/d
Time = 0.78 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (1+\cos (c+d x))^4 \sin (c+d x) \left (-A+6 B+5 A (1+\cos (c+d x))+\frac {(7 A+8 B+10 C) \left (210 \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (160+81 \cos (c+d x)+32 \cos ^2(c+d x)+6 \cos ^3(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{8 \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{9/2}}\right )}{30 d} \] Input:
Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(a^4*(1 + Cos[c + d*x])^4*Sin[c + d*x]*(-A + 6*B + 5*A*(1 + Cos[c + d*x]) + ((7*A + 8*B + 10*C)*(210*ArcSin[Sqrt[Sin[(c + d*x)/2]^2]] + (160 + 81*Co s[c + d*x] + 32*Cos[c + d*x]^2 + 6*Cos[c + d*x]^3)*Sqrt[Sin[c + d*x]^2]))/ (8*Sqrt[1 - Cos[c + d*x]]*(1 + Cos[c + d*x])^(9/2))))/(30*d)
Time = 0.79 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4574, 3042, 4501, 3042, 4278, 2009}
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.
| \(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos ^5(c+d x) (\sec (c+d x) a+a)^4 (2 a (2 A+3 B)+a (A+6 C) \sec (c+d x))dx}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (2 a (2 A+3 B)+a (A+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 4501 |
| \(\displaystyle \frac {\frac {3}{5} a (7 A+8 B+10 C) \int \cos ^4(c+d x) (\sec (c+d x) a+a)^4dx+\frac {2 a (2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{5} a (7 A+8 B+10 C) \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {2 a (2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 4278 |
| \(\displaystyle \frac {\frac {3}{5} a (7 A+8 B+10 C) \int \left (\cos ^4(c+d x) a^4+4 \cos ^3(c+d x) a^4+6 \cos ^2(c+d x) a^4+4 \cos (c+d x) a^4+a^4\right )dx+\frac {2 a (2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{5} a (7 A+8 B+10 C) \left (-\frac {4 a^4 \sin ^3(c+d x)}{3 d}+\frac {8 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {27 a^4 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {35 a^4 x}{8}\right )+\frac {2 a (2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}}{6 a}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}\) |
Input:
Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(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])^4*Sin[c + d*x])/(6*d) + ((2*a*(2*A + 3*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) + (3*a*(7 *A + 8*B + 10*C)*((35*a^4*x)/8 + (8*a^4*Sin[c + d*x])/d + (27*a^4*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^4*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (4*a^ 4*Sin[c + d*x]^3)/(3*d)))/5)/(6*a)
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[ExpandTrig[(a + b*csc[e + f*x])^m*(d*csc[e + f *x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] && I GtQ[m, 0] && RationalQ[n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), 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[(a*A*m - b*B*n)/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1), x] , x] /; FreeQ[{a, b, d, e, f, A, B, m, n}, x] && NeQ[A*b - a*B, 0] && EqQ[a ^2 - b^2, 0] && EqQ[m + n + 1, 0] && !LeQ[m, -1]
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])
Time = 3.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {3 a^{4} \left (\frac {\left (\frac {127 A}{16}+8 B +7 C \right ) \sin \left (2 d x +2 c \right )}{3}+\left (A +\frac {29 B}{36}+\frac {4 C}{9}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (\frac {5 A}{8}+\frac {B}{3}+\frac {C}{12}\right ) \sin \left (4 d x +4 c \right )}{2}+\frac {\left (A +\frac {B}{4}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {A \sin \left (6 d x +6 c \right )}{144}+\frac {\left (22 A +\frac {49 B}{2}+28 C \right ) \sin \left (d x +c \right )}{3}+\frac {49 \left (\frac {8 B}{7}+\frac {10 C}{7}+A \right ) x d}{12}\right )}{4 d}\) | \(128\) |
| risch | \(\frac {49 a^{4} A x}{16}+\frac {7 a^{4} x B}{2}+\frac {35 a^{4} x C}{8}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {49 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {7 \sin \left (d x +c \right ) a^{4} C}{d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{20 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {15 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{32 d}+\frac {3 a^{4} A \sin \left (3 d x +3 c \right )}{4 d}+\frac {29 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{3 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {2 \sin \left (2 d x +2 c \right ) B \,a^{4}}{d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} C}{4 d}\) | \(284\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{4} \sin \left (d x +c \right )+a^{4} C \left (d x +c \right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \sin \left (d x +c \right )+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(416\) |
| default | \(\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{4} \sin \left (d x +c \right )+a^{4} C \left (d x +c \right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \sin \left (d x +c \right )+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(416\) |
Input:
int(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,meth od=_RETURNVERBOSE)
Output:
3/4*a^4*(1/3*(127/16*A+8*B+7*C)*sin(2*d*x+2*c)+(A+29/36*B+4/9*C)*sin(3*d*x +3*c)+1/2*(5/8*A+1/3*B+1/12*C)*sin(4*d*x+4*c)+1/15*(A+1/4*B)*sin(5*d*x+5*c )+1/144*A*sin(6*d*x+6*c)+1/3*(22*A+49/2*B+28*C)*sin(d*x+c)+49/12*(8/7*B+10 /7*C+A)*x*d)/d
Time = 0.08 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (41 \, A + 24 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \, {\left (18 \, A + 17 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (49 \, A + 56 \, B + 54 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (72 \, A + 83 \, B + 100 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
Output:
1/240*(105*(7*A + 8*B + 10*C)*a^4*d*x + (40*A*a^4*cos(d*x + c)^5 + 48*(4*A + B)*a^4*cos(d*x + c)^4 + 10*(41*A + 24*B + 6*C)*a^4*cos(d*x + c)^3 + 32* (18*A + 17*B + 10*C)*a^4*cos(d*x + c)^2 + 15*(49*A + 56*B + 54*C)*a^4*cos( d*x + c) + 16*(72*A + 83*B + 100*C)*a^4)*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)** 2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (199) = 398\).
Time = 0.05 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.88 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 960 \, {\left (d x + c\right )} C a^{4} + 960 \, B a^{4} \sin \left (d x + c\right ) + 3840 \, C a^{4} \sin \left (d x + c\right )}{960 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
Output:
1/960*(256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 - 5*(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 - 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 + 180*(12 *d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 + 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 - 1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^4 + 1 20*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 + 960*(2* d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 1280*(sin(d*x + c)^3 - 3*sin(d*x + c ))*C*a^4 + 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^ 4 + 1440*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4 + 960*(d*x + c)*C*a^4 + 96 0*B*a^4*sin(d*x + c) + 3840*C*a^4*sin(d*x + c))/d
Time = 0.30 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.64 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5950 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 13860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10690 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2790 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
Output:
1/240*(105*(7*A*a^4 + 8*B*a^4 + 10*C*a^4)*(d*x + c) + 2*(735*A*a^4*tan(1/2 *d*x + 1/2*c)^11 + 840*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 1050*C*a^4*tan(1/2* d*x + 1/2*c)^11 + 4165*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 4760*B*a^4*tan(1/2*d *x + 1/2*c)^9 + 5950*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 9702*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 11088*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 13860*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 11802*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 13488*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 16860*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 7355*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 9320*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 10690*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 3105*A*a^4*tan(1/2*d*x + 1/2*c) + 3000*B*a^4*tan(1/2*d*x + 1/2 *c) + 2790*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 14.73 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.57 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4+\frac {35\,C\,a^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {119\,B\,a^4}{3}+\frac {595\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1617\,A\,a^4}{20}+\frac {462\,B\,a^4}{5}+\frac {231\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {562\,B\,a^4}{5}+\frac {281\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1471\,A\,a^4}{24}+\frac {233\,B\,a^4}{3}+\frac {1069\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+25\,B\,a^4+\frac {93\,C\,a^4}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+8\,B+10\,C\right )}{8\,\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4+\frac {35\,C\,a^4}{4}\right )}\right )\,\left (7\,A+8\,B+10\,C\right )}{8\,d} \] Input:
int(cos(c + d*x)^6*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
(tan(c/2 + (d*x)/2)^11*((49*A*a^4)/8 + 7*B*a^4 + (35*C*a^4)/4) + tan(c/2 + (d*x)/2)^9*((833*A*a^4)/24 + (119*B*a^4)/3 + (595*C*a^4)/12) + tan(c/2 + (d*x)/2)^7*((1617*A*a^4)/20 + (462*B*a^4)/5 + (231*C*a^4)/2) + tan(c/2 + ( d*x)/2)^3*((1471*A*a^4)/24 + (233*B*a^4)/3 + (1069*C*a^4)/12) + tan(c/2 + (d*x)/2)^5*((1967*A*a^4)/20 + (562*B*a^4)/5 + (281*C*a^4)/2) + tan(c/2 + ( d*x)/2)*((207*A*a^4)/8 + 25*B*a^4 + (93*C*a^4)/4))/(d*(6*tan(c/2 + (d*x)/2 )^2 + 15*tan(c/2 + (d*x)/2)^4 + 20*tan(c/2 + (d*x)/2)^6 + 15*tan(c/2 + (d* x)/2)^8 + 6*tan(c/2 + (d*x)/2)^10 + tan(c/2 + (d*x)/2)^12 + 1)) + (7*a^4*a tan((7*a^4*tan(c/2 + (d*x)/2)*(7*A + 8*B + 10*C))/(8*((49*A*a^4)/8 + 7*B*a ^4 + (35*C*a^4)/4)))*(7*A + 8*B + 10*C))/(8*d)
Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.03 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{4} \left (40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -490 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +1185 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +1080 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +870 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +192 \sin \left (d x +c \right )^{5} a +48 \sin \left (d x +c \right )^{5} b -960 \sin \left (d x +c \right )^{3} a -640 \sin \left (d x +c \right )^{3} b -320 \sin \left (d x +c \right )^{3} c +1920 \sin \left (d x +c \right ) a +1920 \sin \left (d x +c \right ) b +1920 \sin \left (d x +c \right ) c +735 a d x +840 b d x +1050 c d x \right )}{240 d} \] Input:
int(cos(d*x+c)^6*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
(a**4*(40*cos(c + d*x)*sin(c + d*x)**5*a - 490*cos(c + d*x)*sin(c + d*x)** 3*a - 240*cos(c + d*x)*sin(c + d*x)**3*b - 60*cos(c + d*x)*sin(c + d*x)**3 *c + 1185*cos(c + d*x)*sin(c + d*x)*a + 1080*cos(c + d*x)*sin(c + d*x)*b + 870*cos(c + d*x)*sin(c + d*x)*c + 192*sin(c + d*x)**5*a + 48*sin(c + d*x) **5*b - 960*sin(c + d*x)**3*a - 640*sin(c + d*x)**3*b - 320*sin(c + d*x)** 3*c + 1920*sin(c + d*x)*a + 1920*sin(c + d*x)*b + 1920*sin(c + d*x)*c + 73 5*a*d*x + 840*b*d*x + 1050*c*d*x))/(240*d)