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

Optimal. Leaf size=314 \[ -\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {\sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{2 b^3 d \left (a^2-b^2\right )}-\frac {a \sin (c+d x) \left (-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right )}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {\left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {x (b B-3 a C)}{b^4} \]

[Out]

(B*b-3*C*a)*x/b^4+(2*A*b^6-2*a^5*b*B+5*a^3*b^3*B-6*a*b^5*B+6*a^6*C-15*a^4*b^2*C+a^2*b^4*(A+12*C))*arctan((a-b)
^(1/2)*tan(1/2*d*x+1/2*c)/(a+b)^(1/2))/(a-b)^(5/2)/b^4/(a+b)^(5/2)/d+1/2*(A*b^2-B*a*b+3*C*a^2-2*C*b^2)*sin(d*x
+c)/b^3/(a^2-b^2)/d-1/2*(A*b^2-a*(B*b-C*a))*cos(d*x+c)^2*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^2-1/2*a*(2*
A*b^4+a^3*b*B-4*a*b^3*B-3*a^4*C+a^2*b^2*(A+6*C))*sin(d*x+c)/b^3/(a^2-b^2)^2/d/(a+b*cos(d*x+c))

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Rubi [A]  time = 2.82, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3047, 3031, 3023, 2735, 2659, 205} \[ \frac {\sin (c+d x) \left (3 a^2 C-a b B+A b^2-2 b^2 C\right )}{2 b^3 d \left (a^2-b^2\right )}+\frac {\left (a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 C-6 a b^5 B+2 A b^6\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac {\sin (c+d x) \cos ^2(c+d x) \left (A b^2-a (b B-a C)\right )}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {a \sin (c+d x) \left (a^2 b^2 (A+6 C)+a^3 b B-3 a^4 C-4 a b^3 B+2 A b^4\right )}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {x (b B-3 a C)}{b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*B - 3*a*C)*x)/b^4 + ((2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A
+ 12*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 - a
*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c
+ d*x])/(2*b*(a^2 - b^2)*d*(a + b*Cos[c + d*x])^2) - (a*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A
+ 6*C))*Sin[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x]))

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]

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 2735

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

Rule 3023

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[1/(b*
(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x]
, x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rule 3031

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

Rule 3047

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

Rubi steps

\begin {align*} \int \frac {\cos ^2(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (2 \left (A b^2-a (b B-a C)\right )+2 b (b B-a (A+C)) \cos (c+d x)-\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\int \frac {-b \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right )-\left (a^2-b^2\right ) \left (a^2 b B-2 b^3 B-3 a^3 C+a b^2 (A+4 C)\right ) \cos (c+d x)-b \left (a^2-b^2\right ) \left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\int \frac {-b^2 \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right )-2 b \left (a^2-b^2\right )^2 (b B-3 a C) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {(b B-3 a C) x}{b^4}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {(b B-3 a C) x}{b^4}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac {\left (2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 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 )}{b^4 \left (a^2-b^2\right )^2 d}\\ &=\frac {(b B-3 a C) x}{b^4}+\frac {\left (a^2 A b^4+2 A b^6-2 a^5 b B+5 a^3 b^3 B-6 a b^5 B+6 a^6 C-15 a^4 b^2 C+12 a^2 b^4 C\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}+\frac {\left (A b^2-a b B+3 a^2 C-2 b^2 C\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\left (A b^2-a (b B-a C)\right ) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a \left (2 A b^4+a^3 b B-4 a b^3 B-3 a^4 C+a^2 b^2 (A+6 C)\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}

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Mathematica [A]  time = 2.95, size = 573, normalized size = 1.82 \[ \frac {\frac {-12 a^7 c C-12 a^7 C d x+4 a^6 b B c+4 a^6 b B d x+12 a^6 b C \sin (c+d x)-4 a^5 b^2 B \sin (c+d x)+9 a^5 b^2 C \sin (2 (c+d x))+18 a^5 b^2 c C+18 a^5 b^2 C d x-3 a^4 b^3 B \sin (2 (c+d x))-6 a^4 b^3 B c-6 a^4 b^3 B d x-21 a^4 b^3 C \sin (c+d x)+a^4 b^3 C \sin (3 (c+d x))+a^3 A b^4 \sin (2 (c+d x))+10 a^3 b^4 B \sin (c+d x)-16 a^3 b^4 C \sin (2 (c+d x))-6 a^2 A b^5 \sin (c+d x)+6 a^2 b^5 B \sin (2 (c+d x))+2 a^2 b^5 C \sin (c+d x)-2 a^2 b^5 C \sin (3 (c+d x))+2 \left (b^3-a^2 b\right )^2 (c+d x) (b B-3 a C) \cos (2 (c+d x))-8 a b \left (a^2-b^2\right )^2 (c+d x) (3 a C-b B) \cos (c+d x)-4 a A b^6 \sin (2 (c+d x))+4 a b^6 C \sin (2 (c+d x))-6 a b^6 c C-6 a b^6 C d x+2 b^7 B c+2 b^7 B d x+b^7 C \sin (c+d x)+b^7 C \sin (3 (c+d x))}{\left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac {4 \left (6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}}{4 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((a
- b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (4*a^6*b*B*c - 6*a^4*b^3*B*c + 2*b^7*B*c - 12*
a^7*c*C + 18*a^5*b^2*c*C - 6*a*b^6*c*C + 4*a^6*b*B*d*x - 6*a^4*b^3*B*d*x + 2*b^7*B*d*x - 12*a^7*C*d*x + 18*a^5
*b^2*C*d*x - 6*a*b^6*C*d*x - 8*a*b*(a^2 - b^2)^2*(-(b*B) + 3*a*C)*(c + d*x)*Cos[c + d*x] + 2*(-(a^2*b) + b^3)^
2*(b*B - 3*a*C)*(c + d*x)*Cos[2*(c + d*x)] - 6*a^2*A*b^5*Sin[c + d*x] - 4*a^5*b^2*B*Sin[c + d*x] + 10*a^3*b^4*
B*Sin[c + d*x] + 12*a^6*b*C*Sin[c + d*x] - 21*a^4*b^3*C*Sin[c + d*x] + 2*a^2*b^5*C*Sin[c + d*x] + b^7*C*Sin[c
+ d*x] + a^3*A*b^4*Sin[2*(c + d*x)] - 4*a*A*b^6*Sin[2*(c + d*x)] - 3*a^4*b^3*B*Sin[2*(c + d*x)] + 6*a^2*b^5*B*
Sin[2*(c + d*x)] + 9*a^5*b^2*C*Sin[2*(c + d*x)] - 16*a^3*b^4*C*Sin[2*(c + d*x)] + 4*a*b^6*C*Sin[2*(c + d*x)] +
a^4*b^3*C*Sin[3*(c + d*x)] - 2*a^2*b^5*C*Sin[3*(c + d*x)] + b^7*C*Sin[3*(c + d*x)])/((a^2 - b^2)^2*(a + b*Cos
[c + d*x])^2))/(4*b^4*d)

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fricas [B]  time = 1.28, size = 1666, normalized size = 5.31 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*(3*C*a^7*b^2 - B*a^6*b^3 - 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)
*d*x*cos(d*x + c)^2 + 8*(3*C*a^8*b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a
^2*b^7 + B*a*b^8)*d*x*cos(d*x + c) + 4*(3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*
b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*d*x + (6*C*a^8 - 2*B*a^7*b - 15*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 -
6*B*a^3*b^5 + 2*A*a^2*b^6 + (6*C*a^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^3*b^5 + (A + 12*C)*a^2*b^6 - 6*B
*a*b^7 + 2*A*b^8)*cos(d*x + c)^2 + 2*(6*C*a^7*b - 2*B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^
5 - 6*B*a^2*b^6 + 2*A*a*b^7)*cos(d*x + c))*sqrt(-a^2 + b^2)*log((2*a*b*cos(d*x + c) + (2*a^2 - b^2)*cos(d*x +
c)^2 + 2*sqrt(-a^2 + b^2)*(a*cos(d*x + c) + b)*sin(d*x + c) - a^2 + 2*b^2)/(b^2*cos(d*x + c)^2 + 2*a*b*cos(d*x
+ c) + a^2)) - 2*(6*C*a^8*b - 2*B*a^7*b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4 - (3*A - 13*C)*a^4*b^5 - 5*B*a^3*b^6 +
(3*A - 2*C)*a^2*b^7 + 2*(C*a^6*b^3 - 3*C*a^4*b^5 + 3*C*a^2*b^7 - C*b^9)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a
^6*b^3 + (A - 25*C)*a^5*b^4 + 9*B*a^4*b^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))
*sin(d*x + c))/((a^6*b^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*d*cos(d*x + c)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^
9 - a*b^11)*d*cos(d*x + c) + (a^8*b^4 - 3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d), -1/2*(2*(3*C*a^7*b^2 - B*a^6*b^3
- 9*C*a^5*b^4 + 3*B*a^4*b^5 + 9*C*a^3*b^6 - 3*B*a^2*b^7 - 3*C*a*b^8 + B*b^9)*d*x*cos(d*x + c)^2 + 4*(3*C*a^8*
b - B*a^7*b^2 - 9*C*a^6*b^3 + 3*B*a^5*b^4 + 9*C*a^4*b^5 - 3*B*a^3*b^6 - 3*C*a^2*b^7 + B*a*b^8)*d*x*cos(d*x + c
) + 2*(3*C*a^9 - B*a^8*b - 9*C*a^7*b^2 + 3*B*a^6*b^3 + 9*C*a^5*b^4 - 3*B*a^4*b^5 - 3*C*a^3*b^6 + B*a^2*b^7)*d*
x - (6*C*a^8 - 2*B*a^7*b - 15*C*a^6*b^2 + 5*B*a^5*b^3 + (A + 12*C)*a^4*b^4 - 6*B*a^3*b^5 + 2*A*a^2*b^6 + (6*C*
a^6*b^2 - 2*B*a^5*b^3 - 15*C*a^4*b^4 + 5*B*a^3*b^5 + (A + 12*C)*a^2*b^6 - 6*B*a*b^7 + 2*A*b^8)*cos(d*x + c)^2
+ 2*(6*C*a^7*b - 2*B*a^6*b^2 - 15*C*a^5*b^3 + 5*B*a^4*b^4 + (A + 12*C)*a^3*b^5 - 6*B*a^2*b^6 + 2*A*a*b^7)*cos(
d*x + c))*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (6*C*a^8*b - 2*B*a^7*
b^2 - 17*C*a^6*b^3 + 7*B*a^5*b^4 - (3*A - 13*C)*a^4*b^5 - 5*B*a^3*b^6 + (3*A - 2*C)*a^2*b^7 + 2*(C*a^6*b^3 - 3
*C*a^4*b^5 + 3*C*a^2*b^7 - C*b^9)*cos(d*x + c)^2 + (9*C*a^7*b^2 - 3*B*a^6*b^3 + (A - 25*C)*a^5*b^4 + 9*B*a^4*b
^5 - 5*(A - 4*C)*a^3*b^6 - 6*B*a^2*b^7 + 4*(A - C)*a*b^8)*cos(d*x + c))*sin(d*x + c))/((a^6*b^6 - 3*a^4*b^8 +
3*a^2*b^10 - b^12)*d*cos(d*x + c)^2 + 2*(a^7*b^5 - 3*a^5*b^7 + 3*a^3*b^9 - a*b^11)*d*cos(d*x + c) + (a^8*b^4 -
3*a^6*b^6 + 3*a^4*b^8 - a^2*b^10)*d)]

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giac [B]  time = 0.33, size = 666, normalized size = 2.12 \[ -\frac {\frac {{\left (6 \, C a^{6} - 2 \, B a^{5} b - 15 \, C a^{4} b^{2} + 5 \, B a^{3} b^{3} + A a^{2} b^{4} + 12 \, C a^{2} b^{4} - 6 \, B a b^{5} + 2 \, A b^{6}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, C a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, C a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, C a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, C a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, B a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, A a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}} + \frac {{\left (3 \, C a - B b\right )} {\left (d x + c\right )}}{b^{4}} - \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} b^{3}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^3,x, algorithm="giac")

[Out]

-((6*C*a^6 - 2*B*a^5*b - 15*C*a^4*b^2 + 5*B*a^3*b^3 + A*a^2*b^4 + 12*C*a^2*b^4 - 6*B*a*b^5 + 2*A*b^6)*(pi*floo
r(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^4*b^4 - 2*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) - (4*C*a^6*tan(1/2*d*x + 1/2*c)^3 - 2*B*a^5*b*tan(1/2*
d*x + 1/2*c)^3 - 5*C*a^5*b*tan(1/2*d*x + 1/2*c)^3 + 3*B*a^4*b^2*tan(1/2*d*x + 1/2*c)^3 - 7*C*a^4*b^2*tan(1/2*d
*x + 1/2*c)^3 - A*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 5*B*a^3*b^3*tan(1/2*d*x + 1/2*c)^3 + 8*C*a^3*b^3*tan(1/2*d*
x + 1/2*c)^3 - 3*A*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 - 6*B*a^2*b^4*tan(1/2*d*x + 1/2*c)^3 + 4*A*a*b^5*tan(1/2*d*x
+ 1/2*c)^3 + 4*C*a^6*tan(1/2*d*x + 1/2*c) - 2*B*a^5*b*tan(1/2*d*x + 1/2*c) + 5*C*a^5*b*tan(1/2*d*x + 1/2*c) -
3*B*a^4*b^2*tan(1/2*d*x + 1/2*c) - 7*C*a^4*b^2*tan(1/2*d*x + 1/2*c) + A*a^3*b^3*tan(1/2*d*x + 1/2*c) + 5*B*a^
3*b^3*tan(1/2*d*x + 1/2*c) - 8*C*a^3*b^3*tan(1/2*d*x + 1/2*c) - 3*A*a^2*b^4*tan(1/2*d*x + 1/2*c) + 6*B*a^2*b^4
*tan(1/2*d*x + 1/2*c) - 4*A*a*b^5*tan(1/2*d*x + 1/2*c))/((a^4*b^3 - 2*a^2*b^5 + b^7)*(a*tan(1/2*d*x + 1/2*c)^2
- b*tan(1/2*d*x + 1/2*c)^2 + a + b)^2) + (3*C*a - B*b)*(d*x + c)/b^4 - 2*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x
+ 1/2*c)^2 + 1)*b^3))/d

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maple [B]  time = 0.13, size = 1693, normalized size = 5.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C-1/d*a^3/b
/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-8/d/b/(a*tan(1/2*d*x
+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a^3/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-4/d*b/(a*tan(1/2*d*x+
1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-4/d*b/(a*tan(1/2*d*x+1/2*c)^2-tan(
1/2*d*x+1/2*c)^2*b+a+b)^2*a/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-8/d/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2
*d*x+1/2*c)^2*b+a+b)^2*a^3/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+1/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+
1/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*C+5/d*a^3/b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(
tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-2/d*a^5/b^3/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan
(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+6/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a^2/(a-
b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+6/d/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^2*a^2/(a+b)/
(a-b)^2*tan(1/2*d*x+1/2*c)*B+6/d/b^4/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/(
(a-b)*(a+b))^(1/2))*a^6*C-15/d/b^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a
-b)*(a+b))^(1/2))*a^4*C+2/d/b^3*arctan(tan(1/2*d*x+1/2*c))*B-2/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1
/2*c)^2*b+a+b)^2/(a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*B-2/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b
+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+4/d*a^5/b^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*
b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C-1/d*a^4/b^2/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2
*b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C+1/d*a^3/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*
b+a+b)^2/(a-b)/(a^2+2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-6/d*a*b/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(t
an(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+1/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)^2/(
a+b)/(a-b)^2*tan(1/2*d*x+1/2*c)*A-1/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)^2/(a-b)/(a^2+2*a
*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+2/d/b^3*C*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-6/d/b^4*C*arctan(tan(1/2*
d*x+1/2*c))*a+12/d*a^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(
1/2))*C+2/d*b^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A
+1/d*a^2/(a^4-2*a^2*b^2+b^4)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 7.97, size = 6721, normalized size = 21.40 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(tan(c/2 + (d*x)/2) + 1i)*(B*b - 3*C*a)*1i)/(b^4*d) - ((tan(c/2 + (d*x)/2)^5*(2*C*b^5 - 6*C*a^5 + A*a^2*b^
3 - 6*B*a^2*b^3 - B*a^3*b^2 - 4*C*a^2*b^3 + 12*C*a^3*b^2 + 4*A*a*b^4 + 2*B*a^4*b - 2*C*a*b^4 + 3*C*a^4*b))/((a
*b^3 - b^4)*(a + b)^2) - (tan(c/2 + (d*x)/2)*(6*C*a^5 + 2*C*b^5 + A*a^2*b^3 + 6*B*a^2*b^3 - B*a^3*b^2 - 4*C*a^
2*b^3 - 12*C*a^3*b^2 - 4*A*a*b^4 - 2*B*a^4*b + 2*C*a*b^4 + 3*C*a^4*b))/((a + b)*(b^5 - 2*a*b^4 + a^2*b^3)) + (
2*tan(c/2 + (d*x)/2)^3*(2*C*b^6 - 6*C*a^6 + 3*A*a^2*b^4 - 5*B*a^3*b^3 - 6*C*a^2*b^4 + 13*C*a^4*b^2 + 2*B*a^5*b
))/(b*(a*b^2 - b^3)*(a + b)^2*(a - b)))/(d*(2*a*b + tan(c/2 + (d*x)/2)^2*(2*a*b + 3*a^2 - b^2) + tan(c/2 + (d*
x)/2)^6*(a^2 - 2*a*b + b^2) + a^2 + b^2 - tan(c/2 + (d*x)/2)^4*(2*a*b - 3*a^2 + b^2))) - (log(tan(c/2 + (d*x)/
2) - 1i)*(B*b*1i - C*a*3i))/(b^4*d) - (atan(((((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 4*B^2*b^12 + 72*C^2*a^12 -
8*B^2*a*b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2*a^2*b^10 + 32*B^2*a^3*b^9 - 52*B^2*a^4*b^
8 - 48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a^7*b^5 - 32*B^2*a^8*b^4 - 8*B^2*a^9*b^3 + 8*B^2*a^10*b^2 + 36*C^
2*a^2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288*C^2*a^6*b^6 - 432*C^2*a^7*b^5 + 441*C^2*a
^8*b^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24*A*B*a*b^11 - 24*B*C*a*b^11 - 48*B*C*a^11*b + 8*A*B*a^3*b^9 +
2*A*B*a^5*b^7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6*A*C*a^6*b^6 + 12*A*C*a^8*b^4 + 48*B*C*a^2
*b^10 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*B*C*a^5*b^7 + 288*B*C*a^6*b^6 - 318*B*C*a^7*b^5 - 192*B*C*a^8*b
^4 + 192*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^
6*b^7 - a^7*b^6) + ((-(a + b)^5*(a - b)^5)^(1/2)*((8*(4*A*b^18 + 4*B*b^18 - 6*A*a^2*b^16 + 6*A*a^3*b^15 + 2*A*
a^6*b^12 - 2*A*a^7*b^11 - 8*B*a^2*b^16 + 34*B*a^3*b^15 + 6*B*a^4*b^14 - 36*B*a^5*b^13 - 4*B*a^6*b^12 + 18*B*a^
7*b^11 + 2*B*a^8*b^10 - 4*B*a^9*b^9 + 24*C*a^2*b^16 + 36*C*a^3*b^15 - 78*C*a^4*b^14 - 42*C*a^5*b^13 + 96*C*a^6
*b^12 + 24*C*a^7*b^11 - 54*C*a^8*b^10 - 6*C*a^9*b^9 + 12*C*a^10*b^8 - 4*A*a*b^17 - 12*B*a*b^17 - 12*C*a*b^17))
/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) - (4*tan(c/2 + (d*x)
/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 -
6*B*a*b^5 - 2*B*a^5*b)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7
*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^
10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(2*A*b^6 + 6*C
*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b))/(2*(b^14 - 5*a^2*b^12 +
10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^
4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b)*1i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10
- 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)) + (((8*tan(c/2 + (d*x)/2)*(4*A^2*b^12 + 4*B^2*b^12 + 72*C^2*a^12 - 8*B^
2*a*b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2*a^2*b^10 + 32*B^2*a^3*b^9 - 52*B^2*a^4*b^8 -
48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a^7*b^5 - 32*B^2*a^8*b^4 - 8*B^2*a^9*b^3 + 8*B^2*a^10*b^2 + 36*C^2*a^
2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288*C^2*a^6*b^6 - 432*C^2*a^7*b^5 + 441*C^2*a^8*b
^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24*A*B*a*b^11 - 24*B*C*a*b^11 - 48*B*C*a^11*b + 8*A*B*a^3*b^9 + 2*A*
B*a^5*b^7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6*A*C*a^6*b^6 + 12*A*C*a^8*b^4 + 48*B*C*a^2*b^1
0 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*B*C*a^5*b^7 + 288*B*C*a^6*b^6 - 318*B*C*a^7*b^5 - 192*B*C*a^8*b^4 +
192*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^
7 - a^7*b^6) - ((-(a + b)^5*(a - b)^5)^(1/2)*((8*(4*A*b^18 + 4*B*b^18 - 6*A*a^2*b^16 + 6*A*a^3*b^15 + 2*A*a^6*
b^12 - 2*A*a^7*b^11 - 8*B*a^2*b^16 + 34*B*a^3*b^15 + 6*B*a^4*b^14 - 36*B*a^5*b^13 - 4*B*a^6*b^12 + 18*B*a^7*b^
11 + 2*B*a^8*b^10 - 4*B*a^9*b^9 + 24*C*a^2*b^16 + 36*C*a^3*b^15 - 78*C*a^4*b^14 - 42*C*a^5*b^13 + 96*C*a^6*b^1
2 + 24*C*a^7*b^11 - 54*C*a^8*b^10 - 6*C*a^9*b^9 + 12*C*a^10*b^8 - 4*A*a*b^17 - 12*B*a*b^17 - 12*C*a*b^17))/(a*
b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (4*tan(c/2 + (d*x)/2)*
(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*
a*b^5 - 2*B*a^5*b)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^1
1 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b
^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6)))*(2*A*b^6 + 6*C*a^6
+ A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b))/(2*(b^14 - 5*a^2*b^12 + 10*
a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 +
5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b)*1i)/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 1
0*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))/((16*(108*C^3*a^12 + 4*A*B^2*b^12 - 4*A^2*B*b^12 - 12*B^3*a*b^11 - 54*C^3*
a^11*b - 24*B^3*a^2*b^10 + 34*B^3*a^3*b^9 + 26*B^3*a^4*b^8 - 36*B^3*a^5*b^7 - 13*B^3*a^6*b^6 + 18*B^3*a^7*b^5
+ 2*B^3*a^8*b^4 - 4*B^3*a^9*b^3 + 216*C^3*a^4*b^8 + 216*C^3*a^5*b^7 - 702*C^3*a^6*b^6 - 378*C^3*a^7*b^5 + 864*
C^3*a^8*b^4 + 243*C^3*a^9*b^3 - 486*C^3*a^10*b^2 + 20*A*B^2*a*b^11 + 12*A^2*C*a*b^11 - 108*B*C^2*a^11*b - 6*A*
B^2*a^2*b^10 - 2*A*B^2*a^3*b^9 - 2*A*B^2*a^5*b^7 + 2*A*B^2*a^6*b^6 + 2*A*B^2*a^7*b^5 - 4*A^2*B*a^2*b^10 - A^2*
B*a^4*b^8 + 36*A*C^2*a^2*b^10 + 108*A*C^2*a^3*b^9 - 54*A*C^2*a^4*b^8 - 54*A*C^2*a^5*b^7 - 18*A*C^2*a^7*b^5 + 1
8*A*C^2*a^8*b^4 + 18*A*C^2*a^9*b^3 + 12*A^2*C*a^3*b^9 + 3*A^2*C*a^5*b^7 - 252*B*C^2*a^3*b^9 - 324*B*C^2*a^4*b^
8 + 774*B*C^2*a^5*b^7 + 486*B*C^2*a^6*b^6 - 900*B*C^2*a^7*b^5 - 279*B*C^2*a^8*b^4 + 486*B*C^2*a^9*b^3 + 54*B*C
^2*a^10*b^2 + 96*B^2*C*a^2*b^10 + 156*B^2*C*a^3*b^9 - 282*B^2*C*a^4*b^8 - 198*B^2*C*a^5*b^7 + 312*B^2*C*a^6*b^
6 + 105*B^2*C*a^7*b^5 - 162*B^2*C*a^8*b^4 - 18*B^2*C*a^9*b^3 + 36*B^2*C*a^10*b^2 - 24*A*B*C*a*b^11 - 96*A*B*C*
a^2*b^10 + 36*A*B*C*a^3*b^9 + 24*A*B*C*a^4*b^8 + 12*A*B*C*a^6*b^6 - 12*A*B*C*a^7*b^5 - 12*A*B*C*a^8*b^4))/(a*b
^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b^10 - a^7*b^9) + (((8*tan(c/2 + (d*x)/2)
*(4*A^2*b^12 + 4*B^2*b^12 + 72*C^2*a^12 - 8*B^2*a*b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2
*a^2*b^10 + 32*B^2*a^3*b^9 - 52*B^2*a^4*b^8 - 48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a^7*b^5 - 32*B^2*a^8*b^
4 - 8*B^2*a^9*b^3 + 8*B^2*a^10*b^2 + 36*C^2*a^2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288
*C^2*a^6*b^6 - 432*C^2*a^7*b^5 + 441*C^2*a^8*b^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24*A*B*a*b^11 - 24*B*C
*a*b^11 - 48*B*C*a^11*b + 8*A*B*a^3*b^9 + 2*A*B*a^5*b^7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6
*A*C*a^6*b^6 + 12*A*C*a^8*b^4 + 48*B*C*a^2*b^10 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*B*C*a^5*b^7 + 288*B*C
*a^6*b^6 - 318*B*C*a^7*b^5 - 192*B*C*a^8*b^4 + 192*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11
- 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) + ((-(a + b)^5*(a - b)^5)^(1/2)*((8*(4*A*b^18 + 4*B
*b^18 - 6*A*a^2*b^16 + 6*A*a^3*b^15 + 2*A*a^6*b^12 - 2*A*a^7*b^11 - 8*B*a^2*b^16 + 34*B*a^3*b^15 + 6*B*a^4*b^1
4 - 36*B*a^5*b^13 - 4*B*a^6*b^12 + 18*B*a^7*b^11 + 2*B*a^8*b^10 - 4*B*a^9*b^9 + 24*C*a^2*b^16 + 36*C*a^3*b^15
- 78*C*a^4*b^14 - 42*C*a^5*b^13 + 96*C*a^6*b^12 + 24*C*a^7*b^11 - 54*C*a^8*b^10 - 6*C*a^9*b^9 + 12*C*a^10*b^8
- 4*A*a*b^17 - 12*B*a*b^17 - 12*C*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11
- a^6*b^10 - a^7*b^9) - (4*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*
B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4
*b^14 + 48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 +
10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5
*b^8 - a^6*b^7 - a^7*b^6)))*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a
*b^5 - 2*B*a^5*b))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(-(a + b)^5*(a -
b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*
b))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)) - (((8*tan(c/2 + (d*x)/2)*(4*A^2
*b^12 + 4*B^2*b^12 + 72*C^2*a^12 - 8*B^2*a*b^11 - 72*C^2*a^11*b + 4*A^2*a^2*b^10 + A^2*a^4*b^8 + 24*B^2*a^2*b^
10 + 32*B^2*a^3*b^9 - 52*B^2*a^4*b^8 - 48*B^2*a^5*b^7 + 57*B^2*a^6*b^6 + 32*B^2*a^7*b^5 - 32*B^2*a^8*b^4 - 8*B
^2*a^9*b^3 + 8*B^2*a^10*b^2 + 36*C^2*a^2*b^10 - 72*C^2*a^3*b^9 + 36*C^2*a^4*b^8 + 288*C^2*a^5*b^7 - 288*C^2*a^
6*b^6 - 432*C^2*a^7*b^5 + 441*C^2*a^8*b^4 + 288*C^2*a^9*b^3 - 288*C^2*a^10*b^2 - 24*A*B*a*b^11 - 24*B*C*a*b^11
- 48*B*C*a^11*b + 8*A*B*a^3*b^9 + 2*A*B*a^5*b^7 - 4*A*B*a^7*b^5 + 48*A*C*a^2*b^10 - 36*A*C*a^4*b^8 - 6*A*C*a^
6*b^6 + 12*A*C*a^8*b^4 + 48*B*C*a^2*b^10 - 72*B*C*a^3*b^9 - 192*B*C*a^4*b^8 + 252*B*C*a^5*b^7 + 288*B*C*a^6*b^
6 - 318*B*C*a^7*b^5 - 192*B*C*a^8*b^4 + 192*B*C*a^9*b^3 + 48*B*C*a^10*b^2))/(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^
3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 - a^6*b^7 - a^7*b^6) - ((-(a + b)^5*(a - b)^5)^(1/2)*((8*(4*A*b^18 + 4*B*b^18 -
6*A*a^2*b^16 + 6*A*a^3*b^15 + 2*A*a^6*b^12 - 2*A*a^7*b^11 - 8*B*a^2*b^16 + 34*B*a^3*b^15 + 6*B*a^4*b^14 - 36*
B*a^5*b^13 - 4*B*a^6*b^12 + 18*B*a^7*b^11 + 2*B*a^8*b^10 - 4*B*a^9*b^9 + 24*C*a^2*b^16 + 36*C*a^3*b^15 - 78*C*
a^4*b^14 - 42*C*a^5*b^13 + 96*C*a^6*b^12 + 24*C*a^7*b^11 - 54*C*a^8*b^10 - 6*C*a^9*b^9 + 12*C*a^10*b^8 - 4*A*a
*b^17 - 12*B*a*b^17 - 12*C*a*b^17))/(a*b^15 + b^16 - 3*a^2*b^14 - 3*a^3*b^13 + 3*a^4*b^12 + 3*a^5*b^11 - a^6*b
^10 - a^7*b^9) + (4*tan(c/2 + (d*x)/2)*(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b
^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b)*(8*a*b^17 - 8*a^2*b^16 - 32*a^3*b^15 + 32*a^4*b^14 +
48*a^5*b^13 - 48*a^6*b^12 - 32*a^7*b^11 + 32*a^8*b^10 + 8*a^9*b^9 - 8*a^10*b^8))/((b^14 - 5*a^2*b^12 + 10*a^4
*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)*(a*b^12 + b^13 - 3*a^2*b^11 - 3*a^3*b^10 + 3*a^4*b^9 + 3*a^5*b^8 -
a^6*b^7 - a^7*b^6)))*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 -
2*B*a^5*b))/(2*(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4)))*(-(a + b)^5*(a - b)^5)^
(1/2)*(2*A*b^6 + 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b))/(2*
(b^14 - 5*a^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))))*(-(a + b)^5*(a - b)^5)^(1/2)*(2*A*b^6
+ 6*C*a^6 + A*a^2*b^4 + 5*B*a^3*b^3 + 12*C*a^2*b^4 - 15*C*a^4*b^2 - 6*B*a*b^5 - 2*B*a^5*b)*1i)/(d*(b^14 - 5*a
^2*b^12 + 10*a^4*b^10 - 10*a^6*b^8 + 5*a^8*b^6 - a^10*b^4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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