3.1008 \(\int \frac{(A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=657 \[ \frac{b \left (-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{\tan (c+d x) \left (a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}+\frac{\left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\tan (c+d x) \sec (c+d x) \left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{\tan (c+d x) \sec (c+d x) \left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left (-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

[Out]

(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)*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 - 8*a*b*B + a^2*(A + 2*C))*ArcTanh[Sin[c + d*x]])/(2*a^6*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))*Tan[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))*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*(a^2 - b^2)^3*d)
+ ((A*b^2 - a*(b*B - a*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
+ 7*a^3*b*B - 2*a*b^3*B - 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 - 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))*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 = 12.8586, antiderivative size = 657, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.122, Rules used = {3055, 3001, 3770, 2659, 205} \[ \frac{b \left (-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 C-8 a b^7 B+20 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}+\frac{\tan (c+d x) \left (a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right )}{6 a^5 d \left (a^2-b^2\right )^3}+\frac{\left (a^2 (A+2 C)-8 a b B+20 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 d}-\frac{\tan (c+d x) \sec (c+d x) \left (a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right )}{2 a^4 d \left (a^2-b^2\right )^3}+\frac{\tan (c+d x) \sec (c+d x) \left (a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 C-8 a b^5 B+20 A b^6\right )}{6 a^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left (-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 C-2 a b^3 B+5 A b^4\right )}{6 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{\tan (c+d x) \sec (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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+B \cos (c+d x)+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 (b B-a 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+2 a b B+a^2 (3 A-2 C)-3 a (A b-a B+b C) \cos (c+d x)+4 \left (A b^2-a (b B-a 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 (b B-a 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+7 a^3 b B-2 a b^3 B-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+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 ) \cos (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 ) \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 (b B-a 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+7 a^3 b B-2 a b^3 B-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^6-27 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 (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^6-12 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 ) \cos (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 ) \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^6-12 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) \tan (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 ) \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+7 a^3 b B-2 a b^3 B-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^6-27 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 (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+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 ) \cos (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 ) \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{\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 ) \tan (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 ) \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 (b B-a 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+7 a^3 b B-2 a b^3 B-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^6-27 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 (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^2-8 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 ) \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{\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 ) \tan (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 ) \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 (b B-a 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+7 a^3 b B-2 a b^3 B-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^6-27 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 (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 (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{1}{a+b \cos (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 ) \int \sec (c+d x) \, dx}{2 a^6}\\ &=\frac{\left (20 A b^2-8 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 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 ) \tan (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 ) \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 (b B-a 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+7 a^3 b B-2 a b^3 B-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^6-27 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 (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 (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 ) \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{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 ) \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^2-8 a b B+a^2 (A+2 C)\right ) \tanh ^{-1}(\sin (c+d x))}{2 a^6 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 ) \tan (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 ) \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 (b B-a 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+7 a^3 b B-2 a b^3 B-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^6-27 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 (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 = 6.51213, size = 686, normalized size = 1.04 \[ \frac{a^2 b^2 C \sin (c+d x)-a b^3 B \sin (c+d x)+A b^4 \sin (c+d x)}{3 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{14 a^2 A b^4 \sin (c+d x)-11 a^3 b^3 B \sin (c+d x)-3 a^2 b^4 C \sin (c+d x)+8 a^4 b^2 C \sin (c+d x)+6 a b^5 B \sin (c+d x)-9 A b^6 \sin (c+d x)}{6 a^4 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{-95 a^2 A b^6 \sin (c+d x)+74 a^4 A b^4 \sin (c+d x)+50 a^3 b^5 B \sin (c+d x)-47 a^5 b^3 B \sin (c+d x)+6 a^2 b^6 C \sin (c+d x)-17 a^4 b^4 C \sin (c+d x)+26 a^6 b^2 C \sin (c+d x)-18 a b^7 B \sin (c+d x)+36 A b^8 \sin (c+d x)}{6 a^5 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{b \left (40 a^6 A b^2-84 a^4 A b^4+69 a^2 A b^6+35 a^5 b^3 B-28 a^3 b^5 B-8 a^6 b^2 C+7 a^4 b^4 C-2 a^2 b^6 C-20 a^7 b B+8 a^8 C+8 a b^7 B-20 A b^8\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{a^6 d \left (a^2-b^2\right )^3 \sqrt{b^2-a^2}}+\frac{\left (a^2 (-A)-2 a^2 C+8 a b B-20 A b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac{\left (a^2 A+2 a^2 C-8 a b B+20 A b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac{\sec (c+d x) (a B \sin (c+d x)-4 A b \sin (c+d x))}{a^5 d}+\frac{A \tan (c+d x) \sec (c+d x)}{2 a^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + ((-(a^2*A) - 20*A*b^2 + 8*a*b*B - 2*a^2*C)*Log[Cos[(c + d*x)/2] - Sin
[(c + d*x)/2]])/(2*a^6*d) + ((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/
(2*a^6*d) + (Sec[c + d*x]*(-4*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(a^5*d) + (A*b^4*Sin[c + d*x] - a*b^3*B*Si
n[c + d*x] + a^2*b^2*C*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^3) + (14*a^2*A*b^4*Sin[c + d*x]
- 9*A*b^6*Sin[c + d*x] - 11*a^3*b^3*B*Sin[c + d*x] + 6*a*b^5*B*Sin[c + d*x] + 8*a^4*b^2*C*Sin[c + d*x] - 3*a^
2*b^4*C*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (74*a^4*A*b^4*Sin[c + d*x] - 95*a^2*A*b
^6*Sin[c + d*x] + 36*A*b^8*Sin[c + d*x] - 47*a^5*b^3*B*Sin[c + d*x] + 50*a^3*b^5*B*Sin[c + d*x] - 18*a*b^7*B*S
in[c + d*x] + 26*a^6*b^2*C*Sin[c + d*x] - 17*a^4*b^4*C*Sin[c + d*x] + 6*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 -
b^2)^3*d*(a + b*Cos[c + d*x])) + (A*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d)

________________________________________________________________________________________

Maple [B]  time = 0.135, size = 4436, normalized size = 6.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12/d*b^7/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/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*
x+1/2*c)^3*B+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+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-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+116/3/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^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^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+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-5/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*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+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+
8/d/(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*b
^3-1/d/a^4/(tan(1/2*d*x+1/2*c)-1)*B-1/d/a^4/(tan(1/2*d*x+1/2*c)+1)*B-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)+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)^5*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^2-2*a*b+b^2
)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*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+18/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*b^5/(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/(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)*B-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+2/d/a^3/(a*t
an(1/2*d*x+1/2*c)^2-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+8
4/d/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))
*A*b^5+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+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-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+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*ta
n(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^5-40
/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+4/d/a^5/(tan(1/2*d*x+1/2*c)-1)*A*b-10/d/a^6*ln(tan(1/2*d*x+1/2*c)-1)*A*b^2+4/d/a^5*ln(tan(1/2*d*x+1/2*c)-1)*
b*B+4/d/a^5/(tan(1/2*d*x+1/2*c)+1)*A*b+10/d/a^6*ln(tan(1/2*d*x+1/2*c)+1)*A*b^2-4/d/a^5*ln(tan(1/2*d*x+1/2*c)+1
)*b*B+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^6-69/d/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))*A*b^7-40/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^2-2*a*b+b
^2)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*b^3*B-20/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*b^3*B-20/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)*b^3*B-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+20/d*b^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))*a*B-6/d*b^7/a^4/(a*tan(1/2*d*x+1/2*c)^2-ta
n(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^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+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+18/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*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/(a*tan(1/2*d*x+1/2*c)^2-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-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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a
-b))^(1/2))*B+5/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*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-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+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-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)*arctan((a-b)
*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*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)*arct
an((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*B

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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((A+B*cos(d*x+c)+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

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

[Out]

Timed out

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

[Out]

Timed out

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Giac [B]  time = 1.47239, size = 2001, normalized size = 3.05 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*cos(d*x+c)+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 - 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 - 28
*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*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 + 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*t
an(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) + 11
7*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)*log(abs(tan(1/
2*d*x + 1/2*c) + 1))/a^6 - 3*(A*a^2 + 2*C*a^2 - 8*B*a*b + 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 - 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