3.894 \(\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)

Optimal. Leaf size=314 \[ \frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{60 d}+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{120 d}+\frac {\sin (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{30 d}+\frac {1}{8} x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac {b^4 C \tanh ^{-1}(\sin (c+d x))}{d} \]

[Out]

1/8*(3*a^4*B+24*a^2*b^2*B+8*b^4*B+16*a*b^3*(A+2*C)+4*a^3*b*(3*A+4*C))*x+b^4*C*arctanh(sin(d*x+c))/d+1/30*(12*A
*b^4+80*a^3*b*B+95*a*b^3*B+4*a^4*(4*A+5*C)+2*a^2*b^2*(56*A+85*C))*sin(d*x+c)/d+1/120*a*(24*A*b^3+45*a^3*B+130*
a*b^2*B+4*a^2*b*(29*A+40*C))*cos(d*x+c)*sin(d*x+c)/d+1/60*(12*A*b^2+35*a*b*B+4*a^2*(4*A+5*C))*cos(d*x+c)^2*(a+
b*sec(d*x+c))^2*sin(d*x+c)/d+1/20*(4*A*b+5*B*a)*cos(d*x+c)^3*(a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/5*A*cos(d*x+c)^
4*(a+b*sec(d*x+c))^4*sin(d*x+c)/d

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Rubi [A]  time = 1.05, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4094, 4074, 4047, 8, 4045, 3770} \[ \frac {\sin (c+d x) \left (2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right )}{30 d}+\frac {a \sin (c+d x) \cos (c+d x) \left (4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right )}{120 d}+\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{60 d}+\frac {1}{8} x \left (4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 B+16 a b^3 (A+2 C)+8 b^4 B\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac {b^4 C \tanh ^{-1}(\sin (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*x)/8 + (b^4*C*ArcTanh[Sin[c + d
*x]])/d + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(3
0*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((1
2*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((4*A*b +
5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*
Sin[c + d*x])/(5*d)

Rule 8

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4045

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e
+ f*x]*(b*Csc[e + f*x])^m)/(f*m), x] + Dist[(C*m + A*(m + 1))/(b^2*m), Int[(b*Csc[e + f*x])^(m + 2), x], x] /
; FreeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]

Rule 4047

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(
C_.)), x_Symbol] :> Dist[B/b, Int[(b*Csc[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x
]^2), x] /; FreeQ[{b, e, f, A, B, C, m}, x]

Rule 4074

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_)), 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) + (n*(a*C + B*b) + A*a*(n + 1))*Csc[e + f*x]
+ b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]

Rule 4094

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

Rubi steps

\begin {align*} \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+5 a B+(4 a A+5 b B+5 a C) \sec (c+d x)+5 b C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)+\left (28 a A b+15 a^2 B+20 b^2 B+40 a b C\right ) \sec (c+d x)+20 b^2 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{60} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)+\left (115 a^2 b B+60 b^3 B+36 a b^2 (3 A+5 C)+8 a^3 (4 A+5 C)\right ) \sec (c+d x)+60 b^3 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}-\frac {1}{120} \int \cos (c+d x) \left (-4 \left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right )-15 \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) \sec (c+d x)-120 b^4 C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}-\frac {1}{120} \int \cos (c+d x) \left (-4 \left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right )-120 b^4 C \sec ^2(c+d x)\right ) \, dx-\frac {1}{8} \left (-3 a^4 B-24 a^2 b^2 B-8 b^4 B-16 a b^3 (A+2 C)-4 a^3 b (3 A+4 C)\right ) \int 1 \, dx\\ &=\frac {1}{8} \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) x+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\left (b^4 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) x+\frac {b^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}\\ \end {align*}

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Mathematica [A]  time = 1.27, size = 382, normalized size = 1.22 \[ \frac {50 a^4 A \sin (3 (c+d x))+6 a^4 A \sin (5 (c+d x))+15 a^4 B \sin (4 (c+d x))+180 a^4 B c+180 a^4 B d x+40 a^4 C \sin (3 (c+d x))+60 a^3 A b \sin (4 (c+d x))+720 a^3 A b c+720 a^3 A b d x+160 a^3 b B \sin (3 (c+d x))+960 a^3 b c C+960 a^3 b C d x+240 a^2 A b^2 \sin (3 (c+d x))+1440 a^2 b^2 B c+1440 a^2 b^2 B d x+120 a \sin (2 (c+d x)) \left (a^3 B+4 a^2 b (A+C)+6 a b^2 B+4 A b^3\right )+60 \sin (c+d x) \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 A b^4\right )+960 a A b^3 c+960 a A b^3 d x+1920 a b^3 c C+1920 a b^3 C d x+480 b^4 B c+480 b^4 B d x-480 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 b^4 C \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{480 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(720*a^3*A*b*c + 960*a*A*b^3*c + 180*a^4*B*c + 1440*a^2*b^2*B*c + 480*b^4*B*c + 960*a^3*b*c*C + 1920*a*b^3*c*C
+ 720*a^3*A*b*d*x + 960*a*A*b^3*d*x + 180*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 480*b^4*B*d*x + 960*a^3*b*C*d*x +
1920*a*b^3*C*d*x - 480*b^4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*b^4*C*Log[Cos[(c + d*x)/2] + Sin[(
c + d*x)/2]] + 60*(8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*Sin[c + d*x]
+ 120*a*(4*A*b^3 + a^3*B + 6*a*b^2*B + 4*a^2*b*(A + C))*Sin[2*(c + d*x)] + 50*a^4*A*Sin[3*(c + d*x)] + 240*a^2
*A*b^2*Sin[3*(c + d*x)] + 160*a^3*b*B*Sin[3*(c + d*x)] + 40*a^4*C*Sin[3*(c + d*x)] + 60*a^3*A*b*Sin[4*(c + d*x
)] + 15*a^4*B*Sin[4*(c + d*x)] + 6*a^4*A*Sin[5*(c + d*x)])/(480*d)

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fricas [A]  time = 0.74, size = 262, normalized size = 0.83 \[ \frac {60 \, C b^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C b^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, {\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 16 \, {\left (4 \, A + 5 \, C\right )} a^{4} + 320 \, B a^{3} b + 240 \, {\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 480 \, B a b^{3} + 120 \, A b^{4} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 20 \, B a^{3} b + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

1/120*(60*C*b^4*log(sin(d*x + c) + 1) - 60*C*b^4*log(-sin(d*x + c) + 1) + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b +
24*B*a^2*b^2 + 16*(A + 2*C)*a*b^3 + 8*B*b^4)*d*x + (24*A*a^4*cos(d*x + c)^4 + 16*(4*A + 5*C)*a^4 + 320*B*a^3*b
+ 240*(2*A + 3*C)*a^2*b^2 + 480*B*a*b^3 + 120*A*b^4 + 30*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^3 + 8*((4*A + 5*C)*
a^4 + 20*B*a^3*b + 30*A*a^2*b^2)*cos(d*x + c)^2 + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^
3)*cos(d*x + c))*sin(d*x + c))/d

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giac [B]  time = 0.38, size = 1094, normalized size = 3.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

1/120*(120*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 120*C*b^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 15*(3*B*a
^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*(d*x + c) + 2*(120*A*a^4*tan(
1/2*d*x + 1/2*c)^9 - 75*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 120*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 300*A*a^3*b*tan(1/2*
d*x + 1/2*c)^9 + 480*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan(1
/2*d*x + 1/2*c)^9 - 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 240*A*a*b^3*
tan(1/2*d*x + 1/2*c)^9 + 480*B*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 160*A*a^4*tan
(1/2*d*x + 1/2*c)^7 - 30*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 320*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 120*A*a^3*b*tan(1/2
*d*x + 1/2*c)^7 + 1280*B*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 1920*A*a^2*b^2*ta
n(1/2*d*x + 1/2*c)^7 - 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 2880*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 480*A*a*
b^3*tan(1/2*d*x + 1/2*c)^7 + 1920*B*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 480*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^
4*tan(1/2*d*x + 1/2*c)^5 + 400*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 2400*A*a^2
*b^2*tan(1/2*d*x + 1/2*c)^5 + 4320*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 2880*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 72
0*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 160*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 30*B*a^4*tan(1/2*d*x + 1/2*c)^3 + 320*C*a^
4*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 1280*B*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 480*C*a^
3*b*tan(1/2*d*x + 1/2*c)^3 + 1920*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 28
80*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 1920*B*a*b^3*tan(1/2*d*x + 1/2*c)^3
+ 480*A*b^4*tan(1/2*d*x + 1/2*c)^3 + 120*A*a^4*tan(1/2*d*x + 1/2*c) + 75*B*a^4*tan(1/2*d*x + 1/2*c) + 120*C*a
^4*tan(1/2*d*x + 1/2*c) + 300*A*a^3*b*tan(1/2*d*x + 1/2*c) + 480*B*a^3*b*tan(1/2*d*x + 1/2*c) + 240*C*a^3*b*ta
n(1/2*d*x + 1/2*c) + 720*A*a^2*b^2*tan(1/2*d*x + 1/2*c) + 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c) + 720*C*a^2*b^2*t
an(1/2*d*x + 1/2*c) + 240*A*a*b^3*tan(1/2*d*x + 1/2*c) + 480*B*a*b^3*tan(1/2*d*x + 1/2*c) + 120*A*b^4*tan(1/2*
d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d

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maple [A]  time = 1.47, size = 543, normalized size = 1.73 \[ \frac {3 a^{4} B x}{8}+\frac {3 a^{4} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {2 a^{4} C \sin \left (d x +c \right )}{3 d}+2 a^{3} b C x +3 B \,a^{2} b^{2} x +2 A a \,b^{3} x +\frac {3 a^{4} B c}{8 d}+\frac {8 A \,a^{4} \sin \left (d x +c \right )}{15 d}+\frac {2 a^{3} b C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+\frac {2 C \,a^{3} b c}{d}+\frac {A \,a^{4} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {C \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{4}}{3 d}+\frac {3 a^{2} b^{2} B \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+\frac {2 a A \,b^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d}+\frac {3 A \,a^{3} b \sin \left (d x +c \right ) \cos \left (d x +c \right )}{2 d}+4 a \,b^{3} C x +\frac {A \,b^{4} \sin \left (d x +c \right )}{d}+\frac {a^{4} B \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {8 B \,a^{3} b \sin \left (d x +c \right )}{3 d}+\frac {4 A \,a^{2} b^{2} \sin \left (d x +c \right )}{d}+\frac {A \,a^{3} b \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {B \,b^{4} c}{d}+B \,b^{4} x +\frac {2 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{2} b^{2}}{d}+\frac {4 B \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{3} b}{3 d}+\frac {3 B \,a^{2} b^{2} c}{d}+\frac {2 A a \,b^{3} c}{d}+\frac {3 A \,a^{3} b c}{2 d}+\frac {6 C \,a^{2} b^{2} \sin \left (d x +c \right )}{d}+\frac {4 B a \,b^{3} \sin \left (d x +c \right )}{d}+\frac {4 C a \,b^{3} c}{d}+\frac {4 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{4}}{15 d}+\frac {3 a^{3} A b x}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

3/8*a^4*B*x+3/8/d*a^4*B*cos(d*x+c)*sin(d*x+c)+1/d*C*b^4*ln(sec(d*x+c)+tan(d*x+c))+2/3/d*a^4*C*sin(d*x+c)+2*a^3
*b*C*x+3*B*a^2*b^2*x+2*A*a*b^3*x+3/8/d*a^4*B*c+8/15/d*A*a^4*sin(d*x+c)+2/d*a^3*b*C*cos(d*x+c)*sin(d*x+c)+2/d*A
*cos(d*x+c)^2*sin(d*x+c)*a^2*b^2+4/3/d*B*cos(d*x+c)^2*sin(d*x+c)*a^3*b+2/d*C*a^3*b*c+3/d*a^2*b^2*B*cos(d*x+c)*
sin(d*x+c)+2/d*a*A*b^3*cos(d*x+c)*sin(d*x+c)+1/d*A*a^3*b*sin(d*x+c)*cos(d*x+c)^3+3/2/d*A*a^3*b*sin(d*x+c)*cos(
d*x+c)+4*a*b^3*C*x+1/d*A*b^4*sin(d*x+c)+8/3/d*B*a^3*b*sin(d*x+c)+4/d*A*a^2*b^2*sin(d*x+c)+1/d*B*b^4*c+B*b^4*x+
3/d*B*a^2*b^2*c+2/d*A*a*b^3*c+3/2/d*A*a^3*b*c+6/d*C*a^2*b^2*sin(d*x+c)+4/d*B*a*b^3*sin(d*x+c)+1/5/d*A*a^4*sin(
d*x+c)*cos(d*x+c)^4+1/3/d*C*cos(d*x+c)^2*sin(d*x+c)*a^4+1/4/d*a^4*B*sin(d*x+c)*cos(d*x+c)^3+4/d*C*a*b^3*c+4/15
/d*A*cos(d*x+c)^2*sin(d*x+c)*a^4+3/2*a^3*A*b*x

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maxima [A]  time = 0.36, size = 347, normalized size = 1.11 \[ \frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 1920 \, {\left (d x + c\right )} C a b^{3} + 480 \, {\left (d x + c\right )} B b^{4} + 240 \, C b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2880 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 1920 \, B a b^{3} \sin \left (d x + c\right ) + 480 \, A b^{4} \sin \left (d x + c\right )}{480 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c
) + 8*sin(2*d*x + 2*c))*B*a^4 - 160*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^4 + 60*(12*d*x + 12*c + sin(4*d*x +
4*c) + 8*sin(2*d*x + 2*c))*A*a^3*b - 640*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3*b + 480*(2*d*x + 2*c + sin(2*
d*x + 2*c))*C*a^3*b - 960*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2*b^2 + 720*(2*d*x + 2*c + sin(2*d*x + 2*c))*B
*a^2*b^2 + 480*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a*b^3 + 1920*(d*x + c)*C*a*b^3 + 480*(d*x + c)*B*b^4 + 240*C
*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 2880*C*a^2*b^2*sin(d*x + c) + 1920*B*a*b^3*sin(d*x + c)
+ 480*A*b^4*sin(d*x + c))/d

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mupad [B]  time = 7.80, size = 4118, normalized size = 13.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*(a + b/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

(tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + (5*B*a^4)/4 + 2*C*a^4 + 12*A*a^2*b^2 + 6*B*a^2*b^2 + 12*C*a^2*b^2 + 4
*A*a*b^3 + 5*A*a^3*b + 8*B*a*b^3 + 8*B*a^3*b + 4*C*a^3*b) + tan(c/2 + (d*x)/2)^9*(2*A*a^4 + 2*A*b^4 - (5*B*a^4
)/4 + 2*C*a^4 + 12*A*a^2*b^2 - 6*B*a^2*b^2 + 12*C*a^2*b^2 - 4*A*a*b^3 - 5*A*a^3*b + 8*B*a*b^3 + 8*B*a^3*b - 4*
C*a^3*b) + tan(c/2 + (d*x)/2)^3*((8*A*a^4)/3 + 8*A*b^4 + (B*a^4)/2 + (16*C*a^4)/3 + 32*A*a^2*b^2 + 12*B*a^2*b^
2 + 48*C*a^2*b^2 + 8*A*a*b^3 + 2*A*a^3*b + 32*B*a*b^3 + (64*B*a^3*b)/3 + 8*C*a^3*b) + tan(c/2 + (d*x)/2)^7*((8
*A*a^4)/3 + 8*A*b^4 - (B*a^4)/2 + (16*C*a^4)/3 + 32*A*a^2*b^2 - 12*B*a^2*b^2 + 48*C*a^2*b^2 - 8*A*a*b^3 - 2*A*
a^3*b + 32*B*a*b^3 + (64*B*a^3*b)/3 - 8*C*a^3*b) + tan(c/2 + (d*x)/2)^5*((116*A*a^4)/15 + 12*A*b^4 + (20*C*a^4
)/3 + 40*A*a^2*b^2 + 72*C*a^2*b^2 + 48*B*a*b^3 + (80*B*a^3*b)/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)) + (atan(((tan(c/2 +
(d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B
^2*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*
a*b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 64
0*A*C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + ((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i
+ A*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i)*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*
A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^
3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i)*1i + (tan(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A
^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a
^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480
*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B
*C*a^5*b^3) - ((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i)
*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*((B*a^4
*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i)*1i)/((tan(c/2 + (d*x
)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a
^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^
7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*
C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + ((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A
*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i)*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*A*a*
b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^3*b*
3i)/2 + C*a*b^3*4i + C*a^3*b*2i) - (tan(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*
b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6
+ 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^
3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*
b^3) - ((B*a^4*3i)/8 + B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i)*(12*B*
a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*((B*a^4*3i)/8
+ B*b^4*1i + B*a^2*b^2*3i + A*a*b^3*2i + (A*a^3*b*3i)/2 + C*a*b^3*4i + C*a^3*b*2i) - 64*B*C^2*b^12 + 64*B^2*C*
b^12 - 256*C^3*a*b^11 + 1024*C^3*a^2*b^10 - 128*C^3*a^3*b^9 + 1024*C^3*a^4*b^8 + 256*C^3*a^6*b^6 - 128*A*C^2*a
*b^11 + 512*B*C^2*a*b^11 + 1024*A*C^2*a^2*b^10 - 96*A*C^2*a^3*b^9 + 1280*A*C^2*a^4*b^8 + 384*A*C^2*a^6*b^6 + 2
56*A^2*C*a^2*b^10 + 384*A^2*C*a^4*b^8 + 144*A^2*C*a^6*b^6 - 192*B*C^2*a^2*b^10 + 1792*B*C^2*a^3*b^9 - 24*B*C^2
*a^4*b^8 + 960*B*C^2*a^5*b^7 + 96*B*C^2*a^7*b^5 + 384*B^2*C*a^2*b^10 + 624*B^2*C*a^4*b^8 + 144*B^2*C*a^6*b^6 +
9*B^2*C*a^8*b^4 + 256*A*B*C*a*b^11 + 960*A*B*C*a^3*b^9 + 672*A*B*C*a^5*b^7 + 72*A*B*C*a^7*b^5))*((3*B*a^4)/4
+ 2*B*b^4 + 6*B*a^2*b^2 + 4*A*a*b^3 + 3*A*a^3*b + 8*C*a*b^3 + 4*C*a^3*b))/d - (C*b^4*atan((C*b^4*(tan(c/2 + (d
*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2
*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*
b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*
A*C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + C*b^4*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 9
6*B*a^2*b^2 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*1i + C*b^4*(tan(c/2 + (d*x)/2)*((9*B^2*a^8)
/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 312*B^2*
a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^7 + 36*A*B*a^7*b
+ 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A
*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) - C*b^4*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*A*
a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*1i)/(64*B^2*C*b^12 - 64*B*C^2*b^12 - 256*C^3*a*b^11 + C*b^4*(t
an(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^
2 + 192*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 +
128*A*B*a*b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2
*b^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + C*b^4*(12*B*a^4 + 32*B*b^4 + 3
2*C*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b)) - C*b^4*(tan(c/2 + (d*x)/2)*((9*
B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a^2*b^6 +
312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^7 + 36*A*
B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4
+ 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) - C*b^4*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2
+ 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b)) + 1024*C^3*a^2*b^10 - 128*C^3*a^3*b^9 + 1024*C^3*a^4*b
^8 + 256*C^3*a^6*b^6 - 128*A*C^2*a*b^11 + 512*B*C^2*a*b^11 + 1024*A*C^2*a^2*b^10 - 96*A*C^2*a^3*b^9 + 1280*A*C
^2*a^4*b^8 + 384*A*C^2*a^6*b^6 + 256*A^2*C*a^2*b^10 + 384*A^2*C*a^4*b^8 + 144*A^2*C*a^6*b^6 - 192*B*C^2*a^2*b^
10 + 1792*B*C^2*a^3*b^9 - 24*B*C^2*a^4*b^8 + 960*B*C^2*a^5*b^7 + 96*B*C^2*a^7*b^5 + 384*B^2*C*a^2*b^10 + 624*B
^2*C*a^4*b^8 + 144*B^2*C*a^6*b^6 + 9*B^2*C*a^8*b^4 + 256*A*B*C*a*b^11 + 960*A*B*C*a^3*b^9 + 672*A*B*C*a^5*b^7
+ 72*A*B*C*a^7*b^5))*2i)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*(a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Timed out

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