3.81 \(\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx\)

Optimal. Leaf size=241 \[ -\frac{a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac{a^4 (276 A+301 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{a^4 (44 A+49 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(10 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 d}+\frac{7 (A+B) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{15 d}+\frac{1}{16} a^4 x (44 A+49 B)+\frac{a A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]

[Out]

(a^4*(44*A + 49*B)*x)/16 + (a^4*(227*A + 252*B)*Sin[c + d*x])/(35*d) + (a^4*(44*A + 49*B)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a^4*(276*A + 301*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*A*Cos[c + d*x]^6*(a + a*Sec[c +
d*x])^3*Sin[c + d*x])/(7*d) + ((10*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*d) +
(7*(A + B)*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(227*A + 252*B)*Sin[c + d*x]^3)
/(105*d)

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Rubi [A]  time = 0.566977, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4017, 3996, 3787, 2633, 2635, 8} \[ -\frac{a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac{a^4 (276 A+301 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{a^4 (44 A+49 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(10 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{42 d}+\frac{7 (A+B) \sin (c+d x) \cos ^4(c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{15 d}+\frac{1}{16} a^4 x (44 A+49 B)+\frac{a A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(44*A + 49*B)*x)/16 + (a^4*(227*A + 252*B)*Sin[c + d*x])/(35*d) + (a^4*(44*A + 49*B)*Cos[c + d*x]*Sin[c +
d*x])/(16*d) + (a^4*(276*A + 301*B)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (a*A*Cos[c + d*x]^6*(a + a*Sec[c +
d*x])^3*Sin[c + d*x])/(7*d) + ((10*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*d) +
(7*(A + B)*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^4*(227*A + 252*B)*Sin[c + d*x]^3)
/(105*d)

Rule 4017

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]
- Dist[b/(a*d*n), Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Simp[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 3996

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] + Dist[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 3787

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

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] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx &=\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{1}{7} \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (a (10 A+7 B)+a (3 A+7 B) \sec (c+d x)) \, dx\\ &=\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{1}{42} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (98 a^2 (A+B)+3 a^2 (16 A+21 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{210} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^3 (276 A+301 B)+3 a^3 (178 A+203 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{1}{840} \int \cos ^3(c+d x) \left (-24 a^4 (227 A+252 B)-105 a^4 (44 A+49 B) \sec (c+d x)\right ) \, dx\\ &=\frac{a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{8} \left (a^4 (44 A+49 B)\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{35} \left (a^4 (227 A+252 B)\right ) \int \cos ^3(c+d x) \, dx\\ &=\frac{a^4 (44 A+49 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac{1}{16} \left (a^4 (44 A+49 B)\right ) \int 1 \, dx-\frac{\left (a^4 (227 A+252 B)\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{1}{16} a^4 (44 A+49 B) x+\frac{a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac{a^4 (44 A+49 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac{a^4 (276 A+301 B) \cos ^3(c+d x) \sin (c+d x)}{280 d}+\frac{a A \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{7 d}+\frac{(10 A+7 B) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{42 d}+\frac{7 (A+B) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac{a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d}\\ \end{align*}

Mathematica [A]  time = 0.696459, size = 156, normalized size = 0.65 \[ \frac{a^4 (105 (323 A+352 B) \sin (c+d x)+105 (124 A+127 B) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+651 A \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+18480 A c+18480 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(18480*A*c + 18480*A*d*x + 20580*B*d*x + 105*(323*A + 352*B)*Sin[c + d*x] + 105*(124*A + 127*B)*Sin[2*(c
+ d*x)] + 5495*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)
] + 651*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A*Si
n[7*(c + d*x)]))/(6720*d)

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Maple [A]  time = 0.116, size = 358, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({\frac{A{a}^{4}\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+B{a}^{4} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +4\,A{a}^{4} \left ( 1/6\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) \sin \left ( dx+c \right ) +{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{4\,B{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{\frac{6\,A{a}^{4}\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+6\,B{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +4\,A{a}^{4} \left ( 1/4\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{\frac{4\,B{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{A{a}^{4} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+B{a}^{4} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(1/7*A*a^4*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c)+B*a^4*(1/6*(cos(d*x+c)^5+5/4*c
os(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+4*A*a^4*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(
d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+4/5*B*a^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+6/5*A*a^4*(8/3+co
s(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+6*B*a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4
*A*a^4*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/3*B*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+1/3*
A*a^4*(2+cos(d*x+c)^2)*sin(d*x+c)+B*a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c))

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Maxima [A]  time = 1.03232, size = 481, normalized size = 2. \begin{align*} -\frac{192 \,{\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} - 2688 \,{\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} + 140 \,{\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} + 2240 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 840 \,{\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} - 1792 \,{\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} + 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 )} B a^{4} + 8960 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} - 1260 \,{\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} - 1680 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4}}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6720*(192*(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 - 2688*(3*sin(
d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 140*(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 + 2240*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 - 840*(12*d*x + 12*c +
sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 - 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B
*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))*B*a^4 + 8960*(sin(
d*x + c)^3 - 3*sin(d*x + c))*B*a^4 - 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^4 - 1680
*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4)/d

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Fricas [A]  time = 0.504256, size = 396, normalized size = 1.64 \begin{align*} \frac{105 \,{\left (44 \, A + 49 \, B\right )} a^{4} d x +{\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \,{\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{5} + 192 \,{\left (12 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \,{\left (44 \, A + 41 \, B\right )} a^{4} \cos \left (d x + c\right )^{3} + 16 \,{\left (227 \, A + 252 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \,{\left (44 \, A + 49 \, B\right )} a^{4} \cos \left (d x + c\right ) + 32 \,{\left (227 \, A + 252 \, B\right )} a^{4}\right )} \sin \left (d x + c\right )}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(105*(44*A + 49*B)*a^4*d*x + (240*A*a^4*cos(d*x + c)^6 + 280*(4*A + B)*a^4*cos(d*x + c)^5 + 192*(12*A +
7*B)*a^4*cos(d*x + c)^4 + 70*(44*A + 41*B)*a^4*cos(d*x + c)^3 + 16*(227*A + 252*B)*a^4*cos(d*x + c)^2 + 105*(
44*A + 49*B)*a^4*cos(d*x + c) + 32*(227*A + 252*B)*a^4)*sin(d*x + c))/d

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.44387, size = 375, normalized size = 1.56 \begin{align*} \frac{105 \,{\left (44 \, A a^{4} + 49 \, B a^{4}\right )}{\left (d x + c\right )} + \frac{2 \,{\left (4620 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 5145 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 30800 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 34300 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 87164 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 97069 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 135168 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 150528 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 126084 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 134099 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 58800 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 73220 \, B a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 22260 \, A a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21735 \, B 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}}}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1680*(105*(44*A*a^4 + 49*B*a^4)*(d*x + c) + 2*(4620*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 5145*B*a^4*tan(1/2*d*x +
1/2*c)^13 + 30800*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 34300*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 87164*A*a^4*tan(1/2*d
*x + 1/2*c)^9 + 97069*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 135168*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 150528*B*a^4*tan(1/
2*d*x + 1/2*c)^7 + 126084*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 134099*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 58800*A*a^4*tan
(1/2*d*x + 1/2*c)^3 + 73220*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 22260*A*a^4*tan(1/2*d*x + 1/2*c) + 21735*B*a^4*tan(
1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^7)/d