3.929 \(\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+b \sec (c+d x))^4} \, dx\)

Optimal. Leaf size=648 \[ \frac{\sin (c+d x) \left (a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}-\frac{\sin (c+d x) \cos (c+d x) \left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{\sin (c+d x) \cos (c+d x) \left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{x \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{2 a^6} \]

[Out]

((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*x)/(2*a^6) + (b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8
*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTanh[(Sqrt[a - b]*T
an[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b
^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Sin
[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*
b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B -
a*C))*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3
*B - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) +
((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Co
s[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 12.5141, antiderivative size = 648, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {4100, 4104, 3919, 3831, 2659, 208} \[ \frac{\sin (c+d x) \left (a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}-\frac{\sin (c+d x) \cos (c+d x) \left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{b \left (-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{\sin (c+d x) \cos (c+d x) \left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left (-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \cos (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{x \left (a^2 (A+2 C)-8 a b B+20 A b^2\right )}{2 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*x)/(2*a^6) + (b*(20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8
*a*b^7*B - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTanh[(Sqrt[a - b]*T
an[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b
^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Sin
[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*
b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B -
a*C))*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3
*B - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) +
((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Co
s[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 4100

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*b^2 - a*b*B + a^2*C)*Cot[e + f*x]*(a +
b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n)/(a*f*(m + 1)*(a^2 - b^2)), x] + Dist[1/(a*(m + 1)*(a^2 - b^2)), I
nt[(a + b*Csc[e + f*x])^(m + 1)*(d*Csc[e + f*x])^n*Simp[a*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C)*
(m + n + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x] + (A*b^2 - a*b*B + a^2*C)*(m + n + 2)*Csc[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, n}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] &&  !(ILtQ[m + 1/2, 0] &
& ILtQ[n, 0])

Rule 4104

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

Rule 3919

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(c*x)/a,
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0]

Rule 3831

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\cos ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\int \frac{\cos ^2(c+d x) \left (5 A b^2-2 a b B-a^2 (3 A-2 C)+3 a (A b-a B+b C) \sec (c+d x)-4 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\int \frac{\cos ^2(c+d x) \left (2 \left (10 A b^4+9 a^3 b B-4 a b^3 B+3 a^4 (A-2 C)-a^2 b^2 (18 A-C)\right )+2 a \left (A b^3+3 a^3 B+2 a b^2 B-a^2 b (6 A+5 C)\right ) \sec (c+d x)-3 \left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\int \frac{\cos ^2(c+d x) \left (6 \left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right )+a \left (5 A b^5-6 a^5 B-7 a^3 b^2 B-2 a b^4 B-a^2 b^3 (8 A-5 C)+2 a^4 b (9 A+5 C)\right ) \sec (c+d x)-2 \left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\int \frac{\cos (c+d x) \left (2 \left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right )+2 a \left (10 A b^6-18 a^5 b B+7 a^3 b^3 B-4 a b^5 B-a^2 b^4 (25 A-C)+3 a^6 (A+2 C)+a^4 b^2 (27 A+8 C)\right ) \sec (c+d x)-6 b \left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec ^2(c+d x)\right )}{a+b \sec (c+d x)} \, dx}{12 a^4 \left (a^2-b^2\right )^3}\\ &=\frac{\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}-\frac{\int \frac{-6 \left (a^2-b^2\right )^3 \left (20 A b^2-8 a b B+a^2 (A+2 C)\right )+6 a b \left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \sec (c+d x)}{a+b \sec (c+d x)} \, dx}{12 a^5 \left (a^2-b^2\right )^3}\\ &=\frac{\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\left (b \left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right )\right ) \int \frac{\sec (c+d x)}{a+b \sec (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac{\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \int \frac{1}{1+\frac{a \cos (c+d x)}{b}} \, dx}{2 a^6 \left (a^2-b^2\right )^3}\\ &=\frac{\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}+\frac{\left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^6 \left (a^2-b^2\right )^3 d}\\ &=\frac{\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{b \left (20 A b^8+20 a^7 b B-35 a^5 b^3 B+28 a^3 b^5 B-8 a b^7 B-a^2 b^6 (69 A-2 C)-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-8 a^8 C\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3 d}+\frac{\left (60 A b^7+6 a^7 B-65 a^5 b^2 B+68 a^3 b^4 B-24 a b^6 B+a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-12 a^5 b B+11 a^3 b^3 B-4 a b^5 B-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right ) \cos (c+d x) \sin (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2-a (b B-a C)\right ) \cos (c+d x) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^3}-\frac{\left (5 A b^4+7 a^3 b B-2 a b^3 B-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \sec (c+d x))^2}+\frac{\left (20 A b^6-27 a^5 b B+20 a^3 b^3 B-8 a b^5 B-a^2 b^4 (53 A-2 C)+12 a^6 C+a^4 b^2 (48 A+C)\right ) \cos (c+d x) \sin (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \sec (c+d x))}\\ \end{align*}

Mathematica [C]  time = 6.99992, size = 658, normalized size = 1.02 \[ \frac{(c+d x) \left (a^2 A+2 a^2 C-8 a b B+20 A b^2\right )}{2 a^6 d}+\frac{a^2 b^4 C \sin (c+d x)-a b^5 B \sin (c+d x)+A b^6 \sin (c+d x)}{3 a^5 d \left (a^2-b^2\right ) (a \cos (c+d x)+b)^3}+\frac{-18 a^2 A b^5 \sin (c+d x)+15 a^3 b^4 B \sin (c+d x)+7 a^2 b^5 C \sin (c+d x)-12 a^4 b^3 C \sin (c+d x)-10 a b^6 B \sin (c+d x)+13 A b^7 \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac{-122 a^2 A b^6 \sin (c+d x)+90 a^4 A b^4 \sin (c+d x)+71 a^3 b^5 B \sin (c+d x)-60 a^5 b^3 B \sin (c+d x)+11 a^2 b^6 C \sin (c+d x)-32 a^4 b^4 C \sin (c+d x)+36 a^6 b^2 C \sin (c+d x)-26 a b^7 B \sin (c+d x)+47 A b^8 \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac{b \left (-40 a^6 A b^2+84 a^4 A b^4-69 a^2 A b^6-35 a^5 b^3 B+28 a^3 b^5 B+8 a^6 b^2 C-7 a^4 b^4 C+2 a^2 b^6 C+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tanh ^{-1}\left (\frac{(b-a) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^6 d \sqrt{a^2-b^2} \left (b^2-a^2\right )^3}+\frac{(4 A b-a B) \left (-\frac{\sin (c+d x)}{2 a^5}-\frac{i \cos (c+d x)}{2 a^5}\right )}{d}+\frac{(4 A b-a B) \left (-\frac{\sin (c+d x)}{2 a^5}+\frac{i \cos (c+d x)}{2 a^5}\right )}{d}+\frac{A \sin (2 (c+d x))}{4 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*(c + d*x))/(2*a^6*d) + (b*(-40*a^6*A*b^2 + 84*a^4*A*b^4 - 69*a^2*A*b^6
+ 20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 8*a^6*b^2*C - 7*a^4*b^4*C + 2*a
^2*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d) + ((4*A
*b - a*B)*(((-I/2)*Cos[c + d*x])/a^5 - Sin[c + d*x]/(2*a^5)))/d + ((4*A*b - a*B)*(((I/2)*Cos[c + d*x])/a^5 - S
in[c + d*x]/(2*a^5)))/d + (A*b^6*Sin[c + d*x] - a*b^5*B*Sin[c + d*x] + a^2*b^4*C*Sin[c + d*x])/(3*a^5*(a^2 - b
^2)*d*(b + a*Cos[c + d*x])^3) + (-18*a^2*A*b^5*Sin[c + d*x] + 13*A*b^7*Sin[c + d*x] + 15*a^3*b^4*B*Sin[c + d*x
] - 10*a*b^6*B*Sin[c + d*x] - 12*a^4*b^3*C*Sin[c + d*x] + 7*a^2*b^5*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^2*d*(b
+ a*Cos[c + d*x])^2) + (90*a^4*A*b^4*Sin[c + d*x] - 122*a^2*A*b^6*Sin[c + d*x] + 47*A*b^8*Sin[c + d*x] - 60*a^
5*b^3*B*Sin[c + d*x] + 71*a^3*b^5*B*Sin[c + d*x] - 26*a*b^7*B*Sin[c + d*x] + 36*a^6*b^2*C*Sin[c + d*x] - 32*a^
4*b^4*C*Sin[c + d*x] + 11*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + (A*Sin[2*(c +
d*x)])/(4*a^4*d)

________________________________________________________________________________________

Maple [B]  time = 0.162, size = 4523, normalized size = 7. \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*B+20/d*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))
^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B*a-30/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/
2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^4-6/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan
(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^5+34/d/a^3/(tan(1/2*d*x+1/
2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A*b^6-212/3/d/a^3/
(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A*b
^6+24/d*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/
2*c)^3*b^2*C-12/d*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(
1/2*d*x+1/2*c)*b^2*C+60/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b
^2)*tan(1/2*d*x+1/2*c)^3*A*b^4-12/d*a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b
+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*b^2*C+6/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(
a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^5-30/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3
/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^4+34/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A*b^6+2/d*b^7/a^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/
((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C+20/d*b^9/a^6/(a^6-3*a^4*b^2+3*a^2*
b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-7/d*b^5/a^2/(a^6-3*a^4*b^
2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-3/d/a^4/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^7/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+3/d/a^4/
(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^7/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5
*A-8/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*A*b-8/d/a^5/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+
1/2*c)*A*b-44/3/d*b^4/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*
tan(1/2*d*x+1/2*c)^3*C+4/d*b^6/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+
2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+6/d*b^4/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3
*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C+1/d*b^5/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(
a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-12/d*b^8/a^5/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^
2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A+6/d*b^4/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+
1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+24/d*b^8/a^5/(tan(1/2*d*x+1/2*c)^2*a-ta
n(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d*b^6/a^3/(tan(1/2*d*x+1/
2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C-1/d*b^5/a^2/(tan
(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C-2/d*b^6
/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*
C-12/d*b^8/a^5/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d
*x+1/2*c)^5*A+1/d*A/a^4*arctan(tan(1/2*d*x+1/2*c))+8/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*a
rctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-8/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)
*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C*a^2-40/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-
b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-12/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2
*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+6/d*b^7/a^4/(tan(1/2*d*x+1/2*c)^
2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+6/d*b^7/a^4/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+116/3/d*b^5/
a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3
*B+28/d*b^6/a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b
))^(1/2))*B-8/d*b^8/a^5/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a
+b)*(a-b))^(1/2))*B-35/d*b^4/a/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2
*c)/((a+b)*(a-b))^(1/2))*B+2/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*B-8/d/a^5*arctan(tan(1/2*d*x+
1/2*c))*B*b+20/d/a^6*arctan(tan(1/2*d*x+1/2*c))*A*b^2-1/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)^3*
A+1/d/a^4/(1+tan(1/2*d*x+1/2*c)^2)^2*tan(1/2*d*x+1/2*c)*A+84/d/a^2*b^5/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a
-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-69/d/a^4*b^7/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/
((a+b)*(a-b))^(1/2)*arctanh((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-4/d/(tan(1/2*d*x+1/2*c)^2*a-tan(1/
2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*C*b^3+4/d/(tan(1/2*d*x+1/2*c)^2*a
-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C*b^3+20/d/(tan(1/2*d*x+1/2*
c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B*b^3+20/d/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B*b^3-40/d/(ta
n(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3/(a^2-2*a*b+b^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B*b^3+
5/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^4/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/
2*c)^5*B-18/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^5/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*ta
n(1/2*d*x+1/2*c)^5*B-18/d/a^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^5/(a+b)/(a^3-3*a^2*b+3*a
*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+2/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^6/(a+b)/(a^3-3*
a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-5/d/a/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b^4/(a+b)/
(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-2/d/a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b-a-b)^3*b
^6/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+2/d/a^4*arctan(tan(1/2*d*x+1/2*c))*C

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.80355, size = 7804, normalized size = 12.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(6*((A + 2*C)*a^13 - 8*B*a^12*b + 8*(2*A - C)*a^11*b^2 + 32*B*a^10*b^3 - 2*(37*A - 6*C)*a^9*b^4 - 48*B*a
^8*b^5 + 4*(29*A - 2*C)*a^7*b^6 + 32*B*a^6*b^7 - (79*A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)*d*x*cos(d
*x + c)^3 + 18*((A + 2*C)*a^12*b - 8*B*a^11*b^2 + 8*(2*A - C)*a^10*b^3 + 32*B*a^9*b^4 - 2*(37*A - 6*C)*a^8*b^5
- 48*B*a^7*b^6 + 4*(29*A - 2*C)*a^6*b^7 + 32*B*a^5*b^8 - (79*A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)
*d*x*cos(d*x + c)^2 + 18*((A + 2*C)*a^11*b^2 - 8*B*a^10*b^3 + 8*(2*A - C)*a^9*b^4 + 32*B*a^8*b^5 - 2*(37*A - 6
*C)*a^7*b^6 - 48*B*a^6*b^7 + 4*(29*A - 2*C)*a^5*b^8 + 32*B*a^4*b^9 - (79*A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + 20
*A*a*b^12)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^10*b^3 - 8*B*a^9*b^4 + 8*(2*A - C)*a^8*b^5 + 32*B*a^7*b^6 - 2*(37
*A - 6*C)*a^6*b^7 - 48*B*a^5*b^8 + 4*(29*A - 2*C)*a^4*b^9 + 32*B*a^3*b^10 - (79*A - 2*C)*a^2*b^11 - 8*B*a*b^12
+ 20*A*b^13)*d*x + 3*(8*C*a^8*b^4 - 20*B*a^7*b^5 + 8*(5*A - C)*a^6*b^6 + 35*B*a^5*b^7 - 7*(12*A - C)*a^4*b^8
- 28*B*a^3*b^9 + (69*A - 2*C)*a^2*b^10 + 8*B*a*b^11 - 20*A*b^12 + (8*C*a^11*b - 20*B*a^10*b^2 + 8*(5*A - C)*a^
9*b^3 + 35*B*a^8*b^4 - 7*(12*A - C)*a^7*b^5 - 28*B*a^6*b^6 + (69*A - 2*C)*a^5*b^7 + 8*B*a^4*b^8 - 20*A*a^3*b^9
)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 - 20*B*a^9*b^3 + 8*(5*A - C)*a^8*b^4 + 35*B*a^7*b^5 - 7*(12*A - C)*a^6*b^6
- 28*B*a^5*b^7 + (69*A - 2*C)*a^4*b^8 + 8*B*a^3*b^9 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 - 20*B*a^
8*b^4 + 8*(5*A - C)*a^7*b^5 + 35*B*a^6*b^6 - 7*(12*A - C)*a^5*b^7 - 28*B*a^4*b^8 + (69*A - 2*C)*a^3*b^9 + 8*B*
a^2*b^10 - 20*A*a*b^11)*cos(d*x + c))*sqrt(a^2 - b^2)*log((2*a*b*cos(d*x + c) - (a^2 - 2*b^2)*cos(d*x + c)^2 -
2*sqrt(a^2 - b^2)*(b*cos(d*x + c) + a)*sin(d*x + c) + 2*a^2 - b^2)/(a^2*cos(d*x + c)^2 + 2*a*b*cos(d*x + c) +
b^2)) + 2*(6*B*a^10*b^3 - 2*(12*A - 13*C)*a^9*b^4 - 71*B*a^8*b^5 + (170*A - 43*C)*a^7*b^6 + 133*B*a^6*b^7 - (
313*A - 23*C)*a^5*b^8 - 92*B*a^4*b^9 + (227*A - 6*C)*a^3*b^10 + 24*B*a^2*b^11 - 60*A*a*b^12 + 3*(A*a^13 - 4*A*
a^11*b^2 + 6*A*a^9*b^4 - 4*A*a^7*b^6 + A*a^5*b^8)*cos(d*x + c)^4 + 3*(2*B*a^13 - 5*A*a^12*b - 8*B*a^11*b^2 + 2
0*A*a^10*b^3 + 12*B*a^9*b^4 - 30*A*a^8*b^5 - 8*B*a^7*b^6 + 20*A*a^6*b^7 + 2*B*a^5*b^8 - 5*A*a^4*b^9)*cos(d*x +
c)^3 + (18*B*a^12*b - 9*(7*A - 4*C)*a^11*b^2 - 132*B*a^10*b^3 + 2*(171*A - 34*C)*a^9*b^4 + 239*B*a^8*b^5 - (5
90*A - 43*C)*a^7*b^6 - 169*B*a^6*b^7 + (421*A - 11*C)*a^5*b^8 + 44*B*a^4*b^9 - 110*A*a^3*b^10)*cos(d*x + c)^2
+ 3*(6*B*a^11*b^2 - (23*A - 20*C)*a^10*b^3 - 59*B*a^9*b^4 + (146*A - 35*C)*a^8*b^5 + 110*B*a^7*b^6 - (263*A -
20*C)*a^6*b^7 - 77*B*a^5*b^8 + 5*(38*A - C)*a^4*b^9 + 20*B*a^3*b^10 - 50*A*a^2*b^11)*cos(d*x + c))*sin(d*x + c
))/((a^17 - 4*a^15*b^2 + 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*cos(d*x + c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12
*b^5 - 4*a^10*b^7 + a^8*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*
d*cos(d*x + c) + (a^14*b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6*b^11)*d), 1/6*(3*((A + 2*C)*a^13 - 8*B*
a^12*b + 8*(2*A - C)*a^11*b^2 + 32*B*a^10*b^3 - 2*(37*A - 6*C)*a^9*b^4 - 48*B*a^8*b^5 + 4*(29*A - 2*C)*a^7*b^6
+ 32*B*a^6*b^7 - (79*A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 9*((A + 2*C)*a^12*b
- 8*B*a^11*b^2 + 8*(2*A - C)*a^10*b^3 + 32*B*a^9*b^4 - 2*(37*A - 6*C)*a^8*b^5 - 48*B*a^7*b^6 + 4*(29*A - 2*C)
*a^6*b^7 + 32*B*a^5*b^8 - (79*A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 9*((A + 2*
C)*a^11*b^2 - 8*B*a^10*b^3 + 8*(2*A - C)*a^9*b^4 + 32*B*a^8*b^5 - 2*(37*A - 6*C)*a^7*b^6 - 48*B*a^6*b^7 + 4*(2
9*A - 2*C)*a^5*b^8 + 32*B*a^4*b^9 - (79*A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + 20*A*a*b^12)*d*x*cos(d*x + c) + 3*(
(A + 2*C)*a^10*b^3 - 8*B*a^9*b^4 + 8*(2*A - C)*a^8*b^5 + 32*B*a^7*b^6 - 2*(37*A - 6*C)*a^6*b^7 - 48*B*a^5*b^8
+ 4*(29*A - 2*C)*a^4*b^9 + 32*B*a^3*b^10 - (79*A - 2*C)*a^2*b^11 - 8*B*a*b^12 + 20*A*b^13)*d*x - 3*(8*C*a^8*b^
4 - 20*B*a^7*b^5 + 8*(5*A - C)*a^6*b^6 + 35*B*a^5*b^7 - 7*(12*A - C)*a^4*b^8 - 28*B*a^3*b^9 + (69*A - 2*C)*a^2
*b^10 + 8*B*a*b^11 - 20*A*b^12 + (8*C*a^11*b - 20*B*a^10*b^2 + 8*(5*A - C)*a^9*b^3 + 35*B*a^8*b^4 - 7*(12*A -
C)*a^7*b^5 - 28*B*a^6*b^6 + (69*A - 2*C)*a^5*b^7 + 8*B*a^4*b^8 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^
2 - 20*B*a^9*b^3 + 8*(5*A - C)*a^8*b^4 + 35*B*a^7*b^5 - 7*(12*A - C)*a^6*b^6 - 28*B*a^5*b^7 + (69*A - 2*C)*a^4
*b^8 + 8*B*a^3*b^9 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 - 20*B*a^8*b^4 + 8*(5*A - C)*a^7*b^5 + 35*
B*a^6*b^6 - 7*(12*A - C)*a^5*b^7 - 28*B*a^4*b^8 + (69*A - 2*C)*a^3*b^9 + 8*B*a^2*b^10 - 20*A*a*b^11)*cos(d*x +
c))*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(d*x + c) + a)/((a^2 - b^2)*sin(d*x + c))) + (6*B*a^10*b^
3 - 2*(12*A - 13*C)*a^9*b^4 - 71*B*a^8*b^5 + (170*A - 43*C)*a^7*b^6 + 133*B*a^6*b^7 - (313*A - 23*C)*a^5*b^8 -
92*B*a^4*b^9 + (227*A - 6*C)*a^3*b^10 + 24*B*a^2*b^11 - 60*A*a*b^12 + 3*(A*a^13 - 4*A*a^11*b^2 + 6*A*a^9*b^4
- 4*A*a^7*b^6 + A*a^5*b^8)*cos(d*x + c)^4 + 3*(2*B*a^13 - 5*A*a^12*b - 8*B*a^11*b^2 + 20*A*a^10*b^3 + 12*B*a^9
*b^4 - 30*A*a^8*b^5 - 8*B*a^7*b^6 + 20*A*a^6*b^7 + 2*B*a^5*b^8 - 5*A*a^4*b^9)*cos(d*x + c)^3 + (18*B*a^12*b -
9*(7*A - 4*C)*a^11*b^2 - 132*B*a^10*b^3 + 2*(171*A - 34*C)*a^9*b^4 + 239*B*a^8*b^5 - (590*A - 43*C)*a^7*b^6 -
169*B*a^6*b^7 + (421*A - 11*C)*a^5*b^8 + 44*B*a^4*b^9 - 110*A*a^3*b^10)*cos(d*x + c)^2 + 3*(6*B*a^11*b^2 - (23
*A - 20*C)*a^10*b^3 - 59*B*a^9*b^4 + (146*A - 35*C)*a^8*b^5 + 110*B*a^7*b^6 - (263*A - 20*C)*a^6*b^7 - 77*B*a^
5*b^8 + 5*(38*A - C)*a^4*b^9 + 20*B*a^3*b^10 - 50*A*a^2*b^11)*cos(d*x + c))*sin(d*x + c))/((a^17 - 4*a^15*b^2
+ 6*a^13*b^4 - 4*a^11*b^6 + a^9*b^8)*d*cos(d*x + c)^3 + 3*(a^16*b - 4*a^14*b^3 + 6*a^12*b^5 - 4*a^10*b^7 + a^8
*b^9)*d*cos(d*x + c)^2 + 3*(a^15*b^2 - 4*a^13*b^4 + 6*a^11*b^6 - 4*a^9*b^8 + a^7*b^10)*d*cos(d*x + c) + (a^14*
b^3 - 4*a^12*b^5 + 6*a^10*b^7 - 4*a^8*b^9 + a^6*b^11)*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)**2*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.52875, size = 1941, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(6*(8*C*a^8*b - 20*B*a^7*b^2 + 40*A*a^6*b^3 - 8*C*a^6*b^3 + 35*B*a^5*b^4 - 84*A*a^4*b^5 + 7*C*a^4*b^5 - 2
8*B*a^3*b^6 + 69*A*a^2*b^7 - 2*C*a^2*b^7 + 8*B*a*b^8 - 20*A*b^9)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(-2*a +
2*b) + arctan(-(a*tan(1/2*d*x + 1/2*c) - b*tan(1/2*d*x + 1/2*c))/sqrt(-a^2 + b^2)))/((a^12 - 3*a^10*b^2 + 3*a^
8*b^4 - a^6*b^6)*sqrt(-a^2 + b^2)) + 2*(36*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c
)^5 - 60*C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 105*B*a^6*b^4*tan(1/2*d*x +
1/2*c)^5 - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 - 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^5*b^5*tan(1/2*d*
x + 1/2*c)^5 + 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 117*B*a^4*b^6*tan(1
/2*d*x + 1/2*c)^5 - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 + 24*B*a^3*b^7*t
an(1/2*d*x + 1/2*c)^5 - 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c)^5 - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 42*B*a^2*b
^8*tan(1/2*d*x + 1/2*c)^5 + 6*C*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 - 81*A*a*b^9*tan(1/2*d*x + 1/2*c)^5 - 18*B*a*b^
9*tan(1/2*d*x + 1/2*c)^5 + 36*A*b^10*tan(1/2*d*x + 1/2*c)^5 - 72*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*B*a^7*
b^3*tan(1/2*d*x + 1/2*c)^3 - 180*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 116*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 - 236
*B*a^5*b^5*tan(1/2*d*x + 1/2*c)^3 + 392*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 - 56*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^3
+ 152*B*a^3*b^7*tan(1/2*d*x + 1/2*c)^3 - 284*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 12*C*a^2*b^8*tan(1/2*d*x + 1/
2*c)^3 - 36*B*a*b^9*tan(1/2*d*x + 1/2*c)^3 + 72*A*b^10*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^8*b^2*tan(1/2*d*x + 1/2
*c) - 60*B*a^7*b^3*tan(1/2*d*x + 1/2*c) + 60*C*a^7*b^3*tan(1/2*d*x + 1/2*c) + 90*A*a^6*b^4*tan(1/2*d*x + 1/2*c
) - 105*B*a^6*b^4*tan(1/2*d*x + 1/2*c) - 6*C*a^6*b^4*tan(1/2*d*x + 1/2*c) + 162*A*a^5*b^5*tan(1/2*d*x + 1/2*c)
+ 24*B*a^5*b^5*tan(1/2*d*x + 1/2*c) - 45*C*a^5*b^5*tan(1/2*d*x + 1/2*c) - 48*A*a^4*b^6*tan(1/2*d*x + 1/2*c) +
117*B*a^4*b^6*tan(1/2*d*x + 1/2*c) - 6*C*a^4*b^6*tan(1/2*d*x + 1/2*c) - 213*A*a^3*b^7*tan(1/2*d*x + 1/2*c) +
24*B*a^3*b^7*tan(1/2*d*x + 1/2*c) + 15*C*a^3*b^7*tan(1/2*d*x + 1/2*c) - 48*A*a^2*b^8*tan(1/2*d*x + 1/2*c) - 42
*B*a^2*b^8*tan(1/2*d*x + 1/2*c) + 6*C*a^2*b^8*tan(1/2*d*x + 1/2*c) + 81*A*a*b^9*tan(1/2*d*x + 1/2*c) - 18*B*a*
b^9*tan(1/2*d*x + 1/2*c) + 36*A*b^10*tan(1/2*d*x + 1/2*c))/((a^11 - 3*a^9*b^2 + 3*a^7*b^4 - a^5*b^6)*(a*tan(1/
2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 - a - b)^3) - 3*(A*a^2 + 2*C*a^2 - 8*B*a*b + 20*A*b^2)*(d*x + c)/a
^6 + 6*(A*a*tan(1/2*d*x + 1/2*c)^3 - 2*B*a*tan(1/2*d*x + 1/2*c)^3 + 8*A*b*tan(1/2*d*x + 1/2*c)^3 - A*a*tan(1/2
*d*x + 1/2*c) - 2*B*a*tan(1/2*d*x + 1/2*c) + 8*A*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*a^5))
/d