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

Optimal. Leaf size=513 \[ \frac{b \left (a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right ) \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}-\frac{\left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{\left (-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b C+20 A b^9\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{\left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 C+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{\left (-a^2 b^2 (10 A+C)-4 a^4 C+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{x \left (a^2 (A+2 C)+20 A b^2\right )}{2 a^6} \]

[Out]

((20*A*b^2 + a^2*(A + 2*C))*x)/(2*a^6) + ((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(
12*A - C) - 8*a^8*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2
- b^2)^3*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Sin[c + d*x])
/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 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^2*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 - 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 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*S
in[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.3451, antiderivative size = 513, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4101, 4100, 4104, 3919, 3831, 2659, 208} \[ \frac{b \left (a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right ) \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}-\frac{\left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+10 A b^6\right ) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{\left (-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b C+20 A b^9\right ) \tanh ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{\left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 C+20 A b^6\right ) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \sec (c+d x))}-\frac{\left (-a^2 b^2 (10 A+C)-4 a^4 C+5 A b^4\right ) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \sec (c+d x))^2}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \sec (c+d x))^3}+\frac{x \left (a^2 (A+2 C)+20 A b^2\right )}{2 a^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((20*A*b^2 + a^2*(A + 2*C))*x)/(2*a^6) + ((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(
12*A - C) - 8*a^8*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2
- b^2)^3*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Sin[c + d*x])
/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 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^2*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 - 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 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*S
in[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))

Rule 4101

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

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+C \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^4} \, dx &=\frac{\left (A b^2+a^2 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-a^2 (3 A-2 C)+3 a b (A+C) \sec (c+d x)-4 \left (A b^2+a^2 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^2 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^4-4 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+3 a^4 (A-2 C)-a^2 b^2 (18 A-C)\right )+2 a b \left (A b^2-a^2 (6 A+5 C)\right ) \sec (c+d x)-3 \left (5 A b^4-4 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^2 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^4-4 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-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-a^6 (A-6 C)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)\right )+a b \left (5 A b^4-a^2 b^2 (8 A-5 C)+2 a^4 (9 A+5 C)\right ) \sec (c+d x)-2 \left (20 A b^6-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-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^2 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^4-4 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-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+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-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-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{b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-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^2 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^4-4 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-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+a^2 (A+2 C)\right )+6 a b \left (10 A b^6-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+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-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^2 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^4-4 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-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^9-a^2 b^7 (69 A-2 C)-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-8 a^8 b C\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+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-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^2 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^4-4 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-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-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+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-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^2 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^4-4 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-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-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+a^2 (A+2 C)\right ) x}{2 a^6}+\frac{b \left (20 A b^8-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{b \left (60 A b^6-a^6 (24 A-26 C)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)\right ) \sin (c+d x)}{6 a^5 \left (a^2-b^2\right )^3 d}-\frac{\left (10 A b^6-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^2 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^4-4 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-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 [B]  time = 5.52352, size = 1314, normalized size = 2.56 \[ \frac{\frac{12 A c \cos (3 (c+d x)) a^{11}+24 c C \cos (3 (c+d x)) a^{11}+12 A d x \cos (3 (c+d x)) a^{11}+24 C d x \cos (3 (c+d x)) a^{11}+6 A \sin (c+d x) a^{11}+9 A \sin (3 (c+d x)) a^{11}+3 A \sin (5 (c+d x)) a^{11}+72 A b c a^{10}+144 b c C a^{10}+72 A b d x a^{10}+144 b C d x a^{10}-60 A b \sin (2 (c+d x)) a^{10}-30 A b \sin (4 (c+d x)) a^{10}+204 A b^2 c \cos (3 (c+d x)) a^9-72 b^2 c C \cos (3 (c+d x)) a^9+204 A b^2 d x \cos (3 (c+d x)) a^9-72 b^2 C d x \cos (3 (c+d x)) a^9-270 A b^2 \sin (c+d x) a^9+144 b^2 C \sin (c+d x) a^9-279 A b^2 \sin (3 (c+d x)) a^9+144 b^2 C \sin (3 (c+d x)) a^9-9 A b^2 \sin (5 (c+d x)) a^9+1272 A b^3 c a^8-336 b^3 c C a^8+1272 A b^3 d x a^8-336 b^3 C d x a^8-372 A b^3 \sin (2 (c+d x)) a^8+480 b^3 C \sin (2 (c+d x)) a^8+90 A b^3 \sin (4 (c+d x)) a^8-684 A b^4 c \cos (3 (c+d x)) a^7+72 b^4 c C \cos (3 (c+d x)) a^7-684 A b^4 d x \cos (3 (c+d x)) a^7+72 b^4 C d x \cos (3 (c+d x)) a^7+750 A b^4 \sin (c+d x) a^7+288 b^4 C \sin (c+d x) a^7+1143 A b^4 \sin (3 (c+d x)) a^7-128 b^4 C \sin (3 (c+d x)) a^7+9 A b^4 \sin (5 (c+d x)) a^7-3288 A b^5 c a^6+144 b^5 c C a^6-3288 A b^5 d x a^6+144 b^5 C d x a^6+2772 A b^5 \sin (2 (c+d x)) a^6-360 b^5 C \sin (2 (c+d x)) a^6-90 A b^5 \sin (4 (c+d x)) a^6+708 A b^6 c \cos (3 (c+d x)) a^5-24 b^6 c C \cos (3 (c+d x)) a^5+708 A b^6 d x \cos (3 (c+d x)) a^5-24 b^6 C d x \cos (3 (c+d x)) a^5+1086 A b^6 \sin (c+d x) a^5-228 b^6 C \sin (c+d x) a^5-1253 A b^6 \sin (3 (c+d x)) a^5+44 b^6 C \sin (3 (c+d x)) a^5-3 A b^6 \sin (5 (c+d x)) a^5+1512 A b^7 c a^4+144 b^7 c C a^4+1512 A b^7 d x a^4+144 b^7 C d x a^4-3300 A b^7 \sin (2 (c+d x)) a^4+120 b^7 C \sin (2 (c+d x)) a^4+30 A b^7 \sin (4 (c+d x)) a^4-240 A b^8 c \cos (3 (c+d x)) a^3-240 A b^8 d x \cos (3 (c+d x)) a^3-2232 A b^8 \sin (c+d x) a^3+96 b^8 C \sin (c+d x) a^3+440 A b^8 \sin (3 (c+d x)) a^3+1392 A b^9 c a^2-96 b^9 c C a^2+1392 A b^9 d x a^2-96 b^9 C d x a^2+72 b \left (a^2-b^2\right )^3 \left ((A+2 C) a^2+20 A b^2\right ) (c+d x) \cos (2 (c+d x)) a^2+1200 A b^9 \sin (2 (c+d x)) a^2+36 \left (a^2-b^2\right )^3 \left (a^2+4 b^2\right ) \left ((A+2 C) a^2+20 A b^2\right ) (c+d x) \cos (c+d x) a+960 A b^{10} \sin (c+d x) a-960 A b^{11} c-960 A b^{11} d x}{\left (a^2-b^2\right )^3 (b+a \cos (c+d x))^3}-\frac{96 b \left (-8 C a^8+8 b^2 (C-5 A) a^6+7 b^4 (12 A-C) a^4+b^6 (2 C-69 A) a^2+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 )}{\left (a^2-b^2\right )^{7/2}}}{96 a^6 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-96*b*(20*A*b^8 + 7*a^4*b^4*(12*A - C) - 8*a^8*C + 8*a^6*b^2*(-5*A + C) + a^2*b^6*(-69*A + 2*C))*ArcTanh[((-
a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (72*a^10*A*b*c + 1272*a^8*A*b^3*c - 3288*a^6*A*
b^5*c + 1512*a^4*A*b^7*c + 1392*a^2*A*b^9*c - 960*A*b^11*c + 144*a^10*b*c*C - 336*a^8*b^3*c*C + 144*a^6*b^5*c*
C + 144*a^4*b^7*c*C - 96*a^2*b^9*c*C + 72*a^10*A*b*d*x + 1272*a^8*A*b^3*d*x - 3288*a^6*A*b^5*d*x + 1512*a^4*A*
b^7*d*x + 1392*a^2*A*b^9*d*x - 960*A*b^11*d*x + 144*a^10*b*C*d*x - 336*a^8*b^3*C*d*x + 144*a^6*b^5*C*d*x + 144
*a^4*b^7*C*d*x - 96*a^2*b^9*C*d*x + 36*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(20*A*b^2 + a^2*(A + 2*C))*(c + d*x)*Cos[
c + d*x] + 72*a^2*b*(a^2 - b^2)^3*(20*A*b^2 + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] + 12*a^11*A*c*Cos[3*(c
+ d*x)] + 204*a^9*A*b^2*c*Cos[3*(c + d*x)] - 684*a^7*A*b^4*c*Cos[3*(c + d*x)] + 708*a^5*A*b^6*c*Cos[3*(c + d*
x)] - 240*a^3*A*b^8*c*Cos[3*(c + d*x)] + 24*a^11*c*C*Cos[3*(c + d*x)] - 72*a^9*b^2*c*C*Cos[3*(c + d*x)] + 72*a
^7*b^4*c*C*Cos[3*(c + d*x)] - 24*a^5*b^6*c*C*Cos[3*(c + d*x)] + 12*a^11*A*d*x*Cos[3*(c + d*x)] + 204*a^9*A*b^2
*d*x*Cos[3*(c + d*x)] - 684*a^7*A*b^4*d*x*Cos[3*(c + d*x)] + 708*a^5*A*b^6*d*x*Cos[3*(c + d*x)] - 240*a^3*A*b^
8*d*x*Cos[3*(c + d*x)] + 24*a^11*C*d*x*Cos[3*(c + d*x)] - 72*a^9*b^2*C*d*x*Cos[3*(c + d*x)] + 72*a^7*b^4*C*d*x
*Cos[3*(c + d*x)] - 24*a^5*b^6*C*d*x*Cos[3*(c + d*x)] + 6*a^11*A*Sin[c + d*x] - 270*a^9*A*b^2*Sin[c + d*x] + 7
50*a^7*A*b^4*Sin[c + d*x] + 1086*a^5*A*b^6*Sin[c + d*x] - 2232*a^3*A*b^8*Sin[c + d*x] + 960*a*A*b^10*Sin[c + d
*x] + 144*a^9*b^2*C*Sin[c + d*x] + 288*a^7*b^4*C*Sin[c + d*x] - 228*a^5*b^6*C*Sin[c + d*x] + 96*a^3*b^8*C*Sin[
c + d*x] - 60*a^10*A*b*Sin[2*(c + d*x)] - 372*a^8*A*b^3*Sin[2*(c + d*x)] + 2772*a^6*A*b^5*Sin[2*(c + d*x)] - 3
300*a^4*A*b^7*Sin[2*(c + d*x)] + 1200*a^2*A*b^9*Sin[2*(c + d*x)] + 480*a^8*b^3*C*Sin[2*(c + d*x)] - 360*a^6*b^
5*C*Sin[2*(c + d*x)] + 120*a^4*b^7*C*Sin[2*(c + d*x)] + 9*a^11*A*Sin[3*(c + d*x)] - 279*a^9*A*b^2*Sin[3*(c + d
*x)] + 1143*a^7*A*b^4*Sin[3*(c + d*x)] - 1253*a^5*A*b^6*Sin[3*(c + d*x)] + 440*a^3*A*b^8*Sin[3*(c + d*x)] + 14
4*a^9*b^2*C*Sin[3*(c + d*x)] - 128*a^7*b^4*C*Sin[3*(c + d*x)] + 44*a^5*b^6*C*Sin[3*(c + d*x)] - 30*a^10*A*b*Si
n[4*(c + d*x)] + 90*a^8*A*b^3*Sin[4*(c + d*x)] - 90*a^6*A*b^5*Sin[4*(c + d*x)] + 30*a^4*A*b^7*Sin[4*(c + d*x)]
+ 3*a^11*A*Sin[5*(c + d*x)] - 9*a^9*A*b^2*Sin[5*(c + d*x)] + 9*a^7*A*b^4*Sin[5*(c + d*x)] - 3*a^5*A*b^6*Sin[5
*(c + d*x)])/((a^2 - b^2)^3*(b + a*Cos[c + d*x])^3))/(96*a^6*d)

________________________________________________________________________________________

Maple [B]  time = 0.161, size = 3023, normalized size = 5.9 \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+C*sec(d*x+c)^2)/(a+b*sec(d*x+c))^4,x)

[Out]

-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-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-2/d*b^6/a^3/(tan(1/2*d*x+1/2*c)^2*a-t
an(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-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^3-1/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-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-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+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+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^6-30/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+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+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)*C*b^3-8/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+1/d*A/a^4*arctan(tan(1/2*d*x+1/2*c))+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+2/d/a^4*arctan(tan(1/2*d*x+1/2*c))*C-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-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+8/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))*C+20/d/a^6*arctan(tan(1/2*d*x+1/2*c))*A
*b^2

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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+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 = 1.44403, size = 5536, normalized size = 10.79 \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+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*(2*A - C)*a^11*b^2 - 2*(37*A - 6*C)*a^9*b^4 + 4*(29*A - 2*C)*a^7*b^6 - (79*A - 2*
C)*a^5*b^8 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 18*((A + 2*C)*a^12*b + 8*(2*A - C)*a^10*b^3 - 2*(37*A - 6*C)*
a^8*b^5 + 4*(29*A - 2*C)*a^6*b^7 - (79*A - 2*C)*a^4*b^9 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 18*((A + 2*C)*a^
11*b^2 + 8*(2*A - C)*a^9*b^4 - 2*(37*A - 6*C)*a^7*b^6 + 4*(29*A - 2*C)*a^5*b^8 - (79*A - 2*C)*a^3*b^10 + 20*A*
a*b^12)*d*x*cos(d*x + c) + 6*((A + 2*C)*a^10*b^3 + 8*(2*A - C)*a^8*b^5 - 2*(37*A - 6*C)*a^6*b^7 + 4*(29*A - 2*
C)*a^4*b^9 - (79*A - 2*C)*a^2*b^11 + 20*A*b^13)*d*x + 3*(8*C*a^8*b^4 + 8*(5*A - C)*a^6*b^6 - 7*(12*A - C)*a^4*
b^8 + (69*A - 2*C)*a^2*b^10 - 20*A*b^12 + (8*C*a^11*b + 8*(5*A - C)*a^9*b^3 - 7*(12*A - C)*a^7*b^5 + (69*A - 2
*C)*a^5*b^7 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 + 8*(5*A - C)*a^8*b^4 - 7*(12*A - C)*a^6*b^6 + (6
9*A - 2*C)*a^4*b^8 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 + 8*(5*A - C)*a^7*b^5 - 7*(12*A - C)*a^5*b
^7 + (69*A - 2*C)*a^3*b^9 - 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*(2*(12*A - 13*C)*a^9*b^4 - (170*A - 43*C)*a^7*b^6 + (313*A - 23*C)*a^5*b^8 - (227*
A - 6*C)*a^3*b^10 + 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 + 15*(A*a^12*b - 4*A*a^10*b^3 + 6*A*a^8*b^5 - 4*A*a^6*b^7 + A*a^4*b^9)*cos(d*x + c)^3 + (9*(7*A - 4*C)*a^
11*b^2 - 2*(171*A - 34*C)*a^9*b^4 + (590*A - 43*C)*a^7*b^6 - (421*A - 11*C)*a^5*b^8 + 110*A*a^3*b^10)*cos(d*x
+ c)^2 + 3*((23*A - 20*C)*a^10*b^3 - (146*A - 35*C)*a^8*b^5 + (263*A - 20*C)*a^6*b^7 - 5*(38*A - C)*a^4*b^9 +
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*(2*A - C)*a^11*b^2 - 2*(37*A - 6*C)*a^9*b^4 + 4*(29*A - 2*C)*a^7*b^6 - (
79*A - 2*C)*a^5*b^8 + 20*A*a^3*b^10)*d*x*cos(d*x + c)^3 + 9*((A + 2*C)*a^12*b + 8*(2*A - C)*a^10*b^3 - 2*(37*A
- 6*C)*a^8*b^5 + 4*(29*A - 2*C)*a^6*b^7 - (79*A - 2*C)*a^4*b^9 + 20*A*a^2*b^11)*d*x*cos(d*x + c)^2 + 9*((A +
2*C)*a^11*b^2 + 8*(2*A - C)*a^9*b^4 - 2*(37*A - 6*C)*a^7*b^6 + 4*(29*A - 2*C)*a^5*b^8 - (79*A - 2*C)*a^3*b^10
+ 20*A*a*b^12)*d*x*cos(d*x + c) + 3*((A + 2*C)*a^10*b^3 + 8*(2*A - C)*a^8*b^5 - 2*(37*A - 6*C)*a^6*b^7 + 4*(29
*A - 2*C)*a^4*b^9 - (79*A - 2*C)*a^2*b^11 + 20*A*b^13)*d*x - 3*(8*C*a^8*b^4 + 8*(5*A - C)*a^6*b^6 - 7*(12*A -
C)*a^4*b^8 + (69*A - 2*C)*a^2*b^10 - 20*A*b^12 + (8*C*a^11*b + 8*(5*A - C)*a^9*b^3 - 7*(12*A - C)*a^7*b^5 + (6
9*A - 2*C)*a^5*b^7 - 20*A*a^3*b^9)*cos(d*x + c)^3 + 3*(8*C*a^10*b^2 + 8*(5*A - C)*a^8*b^4 - 7*(12*A - C)*a^6*b
^6 + (69*A - 2*C)*a^4*b^8 - 20*A*a^2*b^10)*cos(d*x + c)^2 + 3*(8*C*a^9*b^3 + 8*(5*A - C)*a^7*b^5 - 7*(12*A - C
)*a^5*b^7 + (69*A - 2*C)*a^3*b^9 - 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))) - (2*(12*A - 13*C)*a^9*b^4 - (170*A - 43*C)*a^7*b^6 + (313*A - 23*C
)*a^5*b^8 - (227*A - 6*C)*a^3*b^10 + 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 + 15*(A*a^12*b - 4*A*a^10*b^3 + 6*A*a^8*b^5 - 4*A*a^6*b^7 + A*a^4*b^9)*cos(d*x + c)^3 +
(9*(7*A - 4*C)*a^11*b^2 - 2*(171*A - 34*C)*a^9*b^4 + (590*A - 43*C)*a^7*b^6 - (421*A - 11*C)*a^5*b^8 + 110*A*a
^3*b^10)*cos(d*x + c)^2 + 3*((23*A - 20*C)*a^10*b^3 - (146*A - 35*C)*a^8*b^5 + (263*A - 20*C)*a^6*b^7 - 5*(38*
A - C)*a^4*b^9 + 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+C*sec(d*x+c)**2)/(a+b*sec(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.35943, size = 1392, normalized size = 2.71 \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+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 + 40*A*a^6*b^3 - 8*C*a^6*b^3 - 84*A*a^4*b^5 + 7*C*a^4*b^5 + 69*A*a^2*b^7 - 2*C*a^2*b^7 - 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*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 - 6*C*a^6*b^4*t
an(1/2*d*x + 1/2*c)^5 - 162*A*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 - 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 - 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 + 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 + 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 - 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 + 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 - 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 + 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*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) - 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
) - 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) - 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) + 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) +
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) + 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 + 20*A*b^2)*(d*x + c)/a^6 + 6*(A*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) + 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