3.592 \(\int \frac{(A+C \cos ^2(c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=522 \[ \frac{\left (-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b C+20 A b^9\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{b \left (a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}+\frac{\left (a^2 (A+2 C)+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{\left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 C+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\left (-a^2 b^2 (10 A+C)-4 a^4 C+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

[Out]

((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)*ArcTan[(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) + ((20*A*b^2 + a^2*(A + 2*C))*
ArcTanh[Sin[c + d*x]])/(2*a^6*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))*Tan[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*(2
7*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*
a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])
/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[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))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.65862, antiderivative size = 522, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3056, 3055, 3001, 3770, 2659, 205} \[ \frac{\left (-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b C+20 A b^9\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{b \left (a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right ) \tan (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3}+\frac{\left (a^2 (A+2 C)+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+10 A b^6\right ) \tan (c+d x) \sec (c+d x)}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{\left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 C+20 A b^6\right ) \tan (c+d x) \sec (c+d x)}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\left (-a^2 b^2 (10 A+C)-4 a^4 C+5 A b^4\right ) \tan (c+d x) \sec (c+d x)}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\left (a^2 C+A b^2\right ) \tan (c+d x) \sec (c+d x)}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

((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)*ArcTan[(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) + ((20*A*b^2 + a^2*(A + 2*C))*
ArcTanh[Sin[c + d*x]])/(2*a^6*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))*Tan[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*(2
7*A - C))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sec[c + d*x]*Tan[c + d*x])/(3*
a*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Sec[c + d*x]*Tan[c + d*x])
/(6*a^2*(a^2 - b^2)^2*d*(a + b*Cos[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))*Sec[c + d*x]*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 3056

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 + a^2*C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c +
d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), I
nt[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*
(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d*(A*b^2 + a^2*C)*(m + n + 3)
*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ
[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3055

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((A*b^2 - a*b*B + a^2*C)*Cos[e +
f*x]*(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2)), x] + Dis
t[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*(b
*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(b*
c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /;
FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3001

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3770

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

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 205

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

Rubi steps

\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx &=\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\int \frac{\left (-5 A b^2+a^2 (3 A-2 C)-3 a b (A+C) \cos (c+d x)+4 \left (A b^2+a^2 C\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^3} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{\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 ) \cos (c+d x)-3 \left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^2} \, dx}{6 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\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 ) \cos (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 ) \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\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 ) \cos (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 ) \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{a+b \cos (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 ) \tan (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 ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{\left (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 ) \cos (c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx}{12 a^5 \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 ) \tan (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 ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (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{1}{a+b \cos (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 ) \int \sec (c+d x) \, dx}{2 a^6}\\ &=\frac{\left (20 A b^2+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 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 ) \tan (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 ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (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 ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) 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^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 ) \tan ^{-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 (20 A b^2+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 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 ) \tan (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 ) \sec (c+d x) \tan (c+d x)}{2 a^4 \left (a^2-b^2\right )^3 d}+\frac{\left (A b^2+a^2 C\right ) \sec (c+d x) \tan (c+d x)}{3 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\left (5 A b^4-4 a^4 C-a^2 b^2 (10 A+C)\right ) \sec (c+d x) \tan (c+d x)}{6 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (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 ) \sec (c+d x) \tan (c+d x)}{6 a^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 5.91649, size = 740, normalized size = 1.42 \[ \frac{\left (A \sec ^2(c+d x)+C\right ) \left (\frac{2 a \sin (c+d x) \left (-6 a b \left (3 a^6 b^2 (3 A-20 C)+3 a^4 b^4 (15 C-103 A)+5 a^2 b^6 (80 A-3 C)+20 a^8 A-150 A b^8\right ) \cos (c+d x)+12 b^2 \left (a^6 b^2 (85 A-2 C)-a^4 b^4 (55 A+2 C)+a^2 b^6 (2 C-19 A)-3 a^8 (7 A-4 C)+20 A b^8\right ) \cos (2 (c+d x))-138 a^7 A b^3 \cos (3 (c+d x))-24 a^6 A b^4 \cos (4 (c+d x))+738 a^5 A b^5 \cos (3 (c+d x))+146 a^4 A b^6 \cos (4 (c+d x))-840 a^3 A b^7 \cos (3 (c+d x))-167 a^2 A b^8 \cos (4 (c+d x))-324 a^8 A b^2+1116 a^6 A b^4-830 a^4 A b^6-61 a^2 A b^8+24 a^{10} A+120 a^7 b^3 C \cos (3 (c+d x))+26 a^6 b^4 C \cos (4 (c+d x))-90 a^5 b^5 C \cos (3 (c+d x))-17 a^4 b^6 C \cos (4 (c+d x))+30 a^3 b^7 C \cos (3 (c+d x))+6 a^2 b^8 C \cos (4 (c+d x))+144 a^8 b^2 C-50 a^6 b^4 C-7 a^4 b^6 C+18 a^2 b^8 C+300 a A b^9 \cos (3 (c+d x))+60 A b^{10} \cos (4 (c+d x))+180 A b^{10}\right )}{\left (a^2-b^2\right )^3 (a+b \cos (c+d x))^3}-48 \left (a^2 (A+2 C)+20 A b^2\right ) \cos ^2(c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+48 \left (a^2 (A+2 C)+20 A b^2\right ) \cos ^2(c+d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{96 b \left (8 a^6 b^2 (C-5 A)+7 a^4 b^4 (12 A-C)+a^2 b^6 (2 C-69 A)-8 a^8 C+20 A b^8\right ) \cos ^2(c+d x) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{7/2}}\right )}{48 a^6 d (2 A+C \cos (2 (c+d x))+C)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^3)/(a + b*Cos[c + d*x])^4,x]

[Out]

((C + A*Sec[c + d*x]^2)*((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*(-6
9*A + 2*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]]*Cos[c + d*x]^2)/(-a^2 + b^2)^(7/2) - 48*(20*A
*b^2 + a^2*(A + 2*C))*Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(20*A*b^2 + a^2*(A + 2*C))*
Cos[c + d*x]^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(24*a^10*A - 324*a^8*A*b^2 + 1116*a^6*A*b^4 - 8
30*a^4*A*b^6 - 61*a^2*A*b^8 + 180*A*b^10 + 144*a^8*b^2*C - 50*a^6*b^4*C - 7*a^4*b^6*C + 18*a^2*b^8*C - 6*a*b*(
20*a^8*A - 150*A*b^8 + 3*a^6*b^2*(3*A - 20*C) + 5*a^2*b^6*(80*A - 3*C) + 3*a^4*b^4*(-103*A + 15*C))*Cos[c + d*
x] + 12*b^2*(20*A*b^8 - 3*a^8*(7*A - 4*C) + a^6*b^2*(85*A - 2*C) + a^2*b^6*(-19*A + 2*C) - a^4*b^4*(55*A + 2*C
))*Cos[2*(c + d*x)] - 138*a^7*A*b^3*Cos[3*(c + d*x)] + 738*a^5*A*b^5*Cos[3*(c + d*x)] - 840*a^3*A*b^7*Cos[3*(c
+ d*x)] + 300*a*A*b^9*Cos[3*(c + d*x)] + 120*a^7*b^3*C*Cos[3*(c + d*x)] - 90*a^5*b^5*C*Cos[3*(c + d*x)] + 30*
a^3*b^7*C*Cos[3*(c + d*x)] - 24*a^6*A*b^4*Cos[4*(c + d*x)] + 146*a^4*A*b^6*Cos[4*(c + d*x)] - 167*a^2*A*b^8*Co
s[4*(c + d*x)] + 60*A*b^10*Cos[4*(c + d*x)] + 26*a^6*b^4*C*Cos[4*(c + d*x)] - 17*a^4*b^6*C*Cos[4*(c + d*x)] +
6*a^2*b^8*C*Cos[4*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^3*(a + b*Cos[c + d*x])^3)))/(48*a^6*d*(2*A + C + C*Co
s[2*(c + d*x)]))

________________________________________________________________________________________

Maple [B]  time = 0.119, size = 2988, normalized size = 5.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x)

[Out]

4/d/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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^3-40/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*
(a-b))^(1/2))*A+8/d*b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+
b)*(a-b))^(1/2))*C-1/d/a^4*ln(tan(1/2*d*x+1/2*c)-1)*C+1/2/d/a^4*A/(tan(1/2*d*x+1/2*c)-1)^2+1/d/a^4*ln(tan(1/2*
d*x+1/2*c)+1)*C-1/2/d/a^4*A/(tan(1/2*d*x+1/2*c)+1)^2+1/2/d/a^4*A/(tan(1/2*d*x+1/2*c)-1)+1/2/d/a^4*A/(tan(1/2*d
*x+1/2*c)+1)+1/2/d/a^4*A*ln(tan(1/2*d*x+1/2*c)+1)-1/2/d/a^4*A*ln(tan(1/2*d*x+1/2*c)-1)+4/d*A/a^5/(tan(1/2*d*x+
1/2*c)-1)*b+10/d/a^6*ln(tan(1/2*d*x+1/2*c)+1)*A*b^2+4/d*A/a^5/(tan(1/2*d*x+1/2*c)+1)*b-10/d/a^6*ln(tan(1/2*d*x
+1/2*c)-1)*A*b^2+12/d/(a*tan(1/2*d*x+1/2*c)^2-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)*ta
n(1/2*d*x+1/2*c)^5*C*a*b^2+12/d/(a*tan(1/2*d*x+1/2*c)^2-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*a*b^2+24/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3*a/(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-6/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-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+1/d*b^5/a^2/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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-6/d*b^4/a/(a*tan(1/2*d*x+1/2
*c)^2-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/(a*tan(
1/2*d*x+1/2*c)^2-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+2/d*b^6/
a^3/(a*tan(1/2*d*x+1/2*c)^2-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-8/d*b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2)
)*C*a^2+24/d*b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-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-44/3/d*b^4/a/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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+12/d*b^8/a^5/(a*tan(1/2*d*x+1/2*c)^2-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-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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(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)*arctan((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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*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)*arctan((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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A-34/d/a^3/(a*tan(1/2
*d*x+1/2*c)^2-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+60/d/a/
(a*tan(1/2*d*x+1/2*c)^2-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
^4-212/3/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-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+3/d/a^4/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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+30/d/a/(a*tan(1/2*d*x+1/2*c)^2-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^4-6/d/a^2/(a*tan(1/2*d*x+1/2*c)^2-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^5-34/d/a^3/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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/(a*tan(1/2*d*x+1/2*c)^2-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

________________________________________________________________________________________

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((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [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((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

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((A+C*cos(d*x+c)**2)*sec(d*x+c)**3/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.93487, size = 1445, normalized size = 2.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^3/(a+b*cos(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*tan(
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) - 21
3*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)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^6 - 3*(A*a^2 + 2*C*a^2 + 20*A*b^2)*log(abs(tan(1/2*d*x
+ 1/2*c) - 1))/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