3.1002 \(\int \frac{\cos ^3(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=461 \[ -\frac{\sin (c+d x) \left (23 a^2 b^2 C+3 a^3 b B-12 a^4 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 C-8 a b^7 B+2 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left (3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 C-6 a b^5 B+2 A b^6\right )}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{x (b B-4 a C)}{b^5} \]

[Out]

((b*B - 4*a*C)*x)/b^5 - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C
- 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5
*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Sin[c + d*x])/(6*b^
4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d
*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2
*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*
b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

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Rubi [A]  time = 9.76056, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{\sin (c+d x) \left (23 a^2 b^2 C+3 a^3 b B-12 a^4 C-8 a b^3 B+5 A b^4-6 b^4 C\right )}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{\left (a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 C-8 a b^7 B+2 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\sin (c+d x) \cos ^3(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^2(c+d x) \left (a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 C-6 a b^3 B+3 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}+\frac{a \sin (c+d x) \left (3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 C-6 a b^5 B+2 A b^6\right )}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{x (b B-4 a C)}{b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*B - 4*a*C)*x)/b^5 - ((2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C
- 35*a^4*b^4*C + a^2*b^6*(3*A + 20*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5
*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Sin[c + d*x])/(6*b^
4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^3*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Cos[c + d
*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Cos[c + d*x]^2*Sin[c + d*x])/(6*b^2
*(a^2 - b^2)^2*d*(a + b*Cos[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*
b^2*C + 3*a^2*b^4*(A + 4*C))*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

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]

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 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 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 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{\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx &=-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}-\frac{\int \frac{\cos ^2(c+d x) \left (3 \left (A b^2-a (b B-a C)\right )+3 b (b B-a (A+C)) \cos (c+d x)-\left (A b^2-a b B+4 a^2 C-3 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{\int \frac{\cos (c+d x) \left (2 \left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right )+2 b \left (2 a^2 b B+3 b^3 B+a^3 C-a b^2 (5 A+6 C)\right ) \cos (c+d x)-\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{6 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{-3 b \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+\left (a^2-b^2\right ) \left (3 a^4 b B-4 a^2 b^3 B+6 b^5 B-12 a^5 C+25 a^3 b^2 C-a b^4 (5 A+18 C)\right ) \cos (c+d x)-b \left (a^2-b^2\right ) \left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )^3}\\ &=-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\int \frac{-3 b^2 \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right )+6 b \left (a^2-b^2\right )^3 (b B-4 a C) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-4 a C) x}{b^5}-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=\frac{(b B-4 a C) x}{b^5}-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 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^5 \left (a^2-b^2\right )^3 d}\\ &=\frac{(b B-4 a C) x}{b^5}-\frac{\left (3 a^2 A b^6+2 A b^8+2 a^7 b B-7 a^5 b^3 B+8 a^3 b^5 B-8 a b^7 B-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+20 a^2 b^6 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{(a-b)^{7/2} b^5 (a+b)^{7/2} d}-\frac{\left (5 A b^4+3 a^3 b B-8 a b^3 B-12 a^4 C+23 a^2 b^2 C-6 b^4 C\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}-\frac{\left (A b^2-a (b B-a C)\right ) \cos ^3(c+d x) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^3}+\frac{\left (3 A b^4+a^3 b B-6 a b^3 B-4 a^4 C+a^2 b^2 (2 A+9 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^2}+\frac{a \left (2 A b^6-a^5 b B+2 a^3 b^3 B-6 a b^5 B+4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [A]  time = 6.83562, size = 532, normalized size = 1.15 \[ \frac{-a^3 A b^2 \sin (c+d x)+a^4 b B \sin (c+d x)+a^5 (-C) \sin (c+d x)}{3 b^4 d \left (b^2-a^2\right ) (a+b \cos (c+d x))^3}+\frac{-4 a^4 A b^2 \sin (c+d x)+9 a^2 A b^4 \sin (c+d x)-12 a^3 b^3 B \sin (c+d x)+15 a^4 b^2 C \sin (c+d x)+7 a^5 b B \sin (c+d x)-10 a^6 C \sin (c+d x)}{6 b^4 d \left (b^2-a^2\right )^2 (a+b \cos (c+d x))^2}+\frac{-2 a^5 A b^2 \sin (c+d x)+5 a^3 A b^4 \sin (c+d x)-32 a^4 b^3 B \sin (c+d x)+36 a^2 b^5 B \sin (c+d x)+71 a^5 b^2 C \sin (c+d x)-60 a^3 b^4 C \sin (c+d x)+11 a^6 b B \sin (c+d x)-26 a^7 C \sin (c+d x)-18 a A b^6 \sin (c+d x)}{6 b^4 d \left (b^2-a^2\right )^3 (a+b \cos (c+d x))}-\frac{\left (-3 a^2 A b^6+7 a^5 b^3 B-8 a^3 b^5 B-28 a^6 b^2 C+35 a^4 b^4 C-20 a^2 b^6 C-2 a^7 b B+8 a^8 C+8 a b^7 B-2 A b^8\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{b^5 d \left (a^2-b^2\right )^3 \sqrt{b^2-a^2}}+\frac{(c+d x) (b B-4 a C)}{b^5 d}+\frac{C \sin (c+d x)}{b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*B - 4*a*C)*(c + d*x))/(b^5*d) - ((-3*a^2*A*b^6 - 2*A*b^8 - 2*a^7*b*B + 7*a^5*b^3*B - 8*a^3*b^5*B + 8*a*b^7
*B + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - 20*a^2*b^6*C)*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]
])/(b^5*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) + (C*Sin[c + d*x])/(b^4*d) + (-(a^3*A*b^2*Sin[c + d*x]) + a^4*b*B*Si
n[c + d*x] - a^5*C*Sin[c + d*x])/(3*b^4*(-a^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (-4*a^4*A*b^2*Sin[c + d*x] +
9*a^2*A*b^4*Sin[c + d*x] + 7*a^5*b*B*Sin[c + d*x] - 12*a^3*b^3*B*Sin[c + d*x] - 10*a^6*C*Sin[c + d*x] + 15*a^4
*b^2*C*Sin[c + d*x])/(6*b^4*(-a^2 + b^2)^2*d*(a + b*Cos[c + d*x])^2) + (-2*a^5*A*b^2*Sin[c + d*x] + 5*a^3*A*b^
4*Sin[c + d*x] - 18*a*A*b^6*Sin[c + d*x] + 11*a^6*b*B*Sin[c + d*x] - 32*a^4*b^3*B*Sin[c + d*x] + 36*a^2*b^5*B*
Sin[c + d*x] - 26*a^7*C*Sin[c + d*x] + 71*a^5*b^2*C*Sin[c + d*x] - 60*a^3*b^4*C*Sin[c + d*x])/(6*b^4*(-a^2 + b
^2)^3*d*(a + b*Cos[c + d*x]))

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Maple [B]  time = 0.054, size = 3571, normalized size = 7.8 \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)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x)

[Out]

6/d*b^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/(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^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/(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^4*B*arctan(tan(1/2*d*x+1/2*c))+2/d/b^4*C*tan(1/2*d*x+1/2*c)/(tan(1/2*d*x+1/2*c)^2+1)-8/d
/b^5*C*arctan(tan(1/2*d*x+1/2*c))*a-12/d*b/(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/(a-b)/(a^
3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*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^3/(
a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*C+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^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^6/b^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*B+6/d*a^4/b/(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+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/(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*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+5/d*a^4/b/(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-18/d*a^5/b^
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
+3/d*a^2*b/(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-3/d*a^2*b/(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+2/d*a^6/b^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-18/d*a^5/b^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*a^6/b^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-5/d*a^4/b/(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+6/d*a^7/b^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)*C-1/d*a^5/b^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)*B-2/d*a^6/b^3/(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)*B+6/d*a^7/b^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)^5*C+6/d*a^4/b/(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+1/d*a^5/b
^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*
B-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))*a
^3*B-2/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+12/d/b^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^7/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan
(1/2*d*x+1/2*c)^3*C-24/d*b/(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/(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^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/(a^2+2*a*b+b^2)
/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-4/d/b^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^6/(a^2
+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+44/3/d/b/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b
)^3*a^4/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-116/3/d/b^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^5/(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/(a*tan(1/2*d*x+1/2*c)^2-t
an(1/2*d*x+1/2*c)^2*b+a+b)^3*a^2/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-4/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*B+4/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)*B+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^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+4/3/
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^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^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^7*B-3/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))*a^2*A+8/d/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^8*C-20/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^2-28/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^6*C+8/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)*t
an(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a*B+35/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))*a^4*C+7/d/b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arc
tan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*a^5*B

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 4.346, size = 6165, normalized size = 13.37 \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)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

[-1/12*(12*(4*C*a^9*b^3 - B*a^8*b^4 - 16*C*a^7*b^5 + 4*B*a^6*b^6 + 24*C*a^5*b^7 - 6*B*a^4*b^8 - 16*C*a^3*b^9 +
4*B*a^2*b^10 + 4*C*a*b^11 - B*b^12)*d*x*cos(d*x + c)^3 + 36*(4*C*a^10*b^2 - B*a^9*b^3 - 16*C*a^8*b^4 + 4*B*a^
7*b^5 + 24*C*a^6*b^6 - 6*B*a^5*b^7 - 16*C*a^4*b^8 + 4*B*a^3*b^9 + 4*C*a^2*b^10 - B*a*b^11)*d*x*cos(d*x + c)^2
+ 36*(4*C*a^11*b - B*a^10*b^2 - 16*C*a^9*b^3 + 4*B*a^8*b^4 + 24*C*a^7*b^5 - 6*B*a^6*b^6 - 16*C*a^5*b^7 + 4*B*a
^4*b^8 + 4*C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c) + 12*(4*C*a^12 - B*a^11*b - 16*C*a^10*b^2 + 4*B*a^9*b^3 +
24*C*a^8*b^4 - 6*B*a^7*b^5 - 16*C*a^6*b^6 + 4*B*a^5*b^7 + 4*C*a^4*b^8 - B*a^3*b^9)*d*x + 3*(8*C*a^11 - 2*B*a^1
0*b - 28*C*a^9*b^2 + 7*B*a^8*b^3 + 35*C*a^7*b^4 - 8*B*a^6*b^5 - (3*A + 20*C)*a^5*b^6 + 8*B*a^4*b^7 - 2*A*a^3*b
^8 + (8*C*a^8*b^3 - 2*B*a^7*b^4 - 28*C*a^6*b^5 + 7*B*a^5*b^6 + 35*C*a^4*b^7 - 8*B*a^3*b^8 - (3*A + 20*C)*a^2*b
^9 + 8*B*a*b^10 - 2*A*b^11)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 2*B*a^8*b^3 - 28*C*a^7*b^4 + 7*B*a^6*b^5 + 35*C*
a^5*b^6 - 8*B*a^4*b^7 - (3*A + 20*C)*a^3*b^8 + 8*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x + c)^2 + 3*(8*C*a^10*b - 2*B*
a^9*b^2 - 28*C*a^8*b^3 + 7*B*a^7*b^4 + 35*C*a^6*b^5 - 8*B*a^5*b^6 - (3*A + 20*C)*a^4*b^7 + 8*B*a^3*b^8 - 2*A*a
^2*b^9)*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*(24
*C*a^11*b - 6*B*a^10*b^2 - 92*C*a^9*b^3 + 23*B*a^8*b^4 + (4*A + 133*C)*a^7*b^5 - 43*B*a^6*b^6 + (7*A - 71*C)*a
^5*b^7 + 26*B*a^4*b^8 - (11*A - 6*C)*a^3*b^9 + 6*(C*a^8*b^4 - 4*C*a^6*b^6 + 6*C*a^4*b^8 - 4*C*a^2*b^10 + C*b^1
2)*cos(d*x + c)^3 + (44*C*a^9*b^3 - 11*B*a^8*b^4 + (2*A - 169*C)*a^7*b^5 + 43*B*a^6*b^6 - (7*A - 239*C)*a^5*b^
7 - 68*B*a^4*b^8 + (23*A - 132*C)*a^3*b^9 + 36*B*a^2*b^10 - 18*(A - C)*a*b^11)*cos(d*x + c)^2 + 3*(20*C*a^10*b
^2 - 5*B*a^9*b^3 - 77*C*a^8*b^4 + 20*B*a^7*b^5 + (A + 110*C)*a^6*b^6 - 35*B*a^5*b^7 + (8*A - 59*C)*a^4*b^8 + 2
0*B*a^3*b^9 - 3*(3*A - 2*C)*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^10 + 6*a^4*b^12 - 4*a^2*
b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^15)*d*cos(d*x + c)^2 +
3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4*a^4*b^12 + a^2*b^14)*d*cos(d*x + c) + (a^11*b^5 - 4*a^9*b^7 + 6*a^7*b
^9 - 4*a^5*b^11 + a^3*b^13)*d), -1/6*(6*(4*C*a^9*b^3 - B*a^8*b^4 - 16*C*a^7*b^5 + 4*B*a^6*b^6 + 24*C*a^5*b^7 -
6*B*a^4*b^8 - 16*C*a^3*b^9 + 4*B*a^2*b^10 + 4*C*a*b^11 - B*b^12)*d*x*cos(d*x + c)^3 + 18*(4*C*a^10*b^2 - B*a^
9*b^3 - 16*C*a^8*b^4 + 4*B*a^7*b^5 + 24*C*a^6*b^6 - 6*B*a^5*b^7 - 16*C*a^4*b^8 + 4*B*a^3*b^9 + 4*C*a^2*b^10 -
B*a*b^11)*d*x*cos(d*x + c)^2 + 18*(4*C*a^11*b - B*a^10*b^2 - 16*C*a^9*b^3 + 4*B*a^8*b^4 + 24*C*a^7*b^5 - 6*B*a
^6*b^6 - 16*C*a^5*b^7 + 4*B*a^4*b^8 + 4*C*a^3*b^9 - B*a^2*b^10)*d*x*cos(d*x + c) + 6*(4*C*a^12 - B*a^11*b - 16
*C*a^10*b^2 + 4*B*a^9*b^3 + 24*C*a^8*b^4 - 6*B*a^7*b^5 - 16*C*a^6*b^6 + 4*B*a^5*b^7 + 4*C*a^4*b^8 - B*a^3*b^9)
*d*x - 3*(8*C*a^11 - 2*B*a^10*b - 28*C*a^9*b^2 + 7*B*a^8*b^3 + 35*C*a^7*b^4 - 8*B*a^6*b^5 - (3*A + 20*C)*a^5*b
^6 + 8*B*a^4*b^7 - 2*A*a^3*b^8 + (8*C*a^8*b^3 - 2*B*a^7*b^4 - 28*C*a^6*b^5 + 7*B*a^5*b^6 + 35*C*a^4*b^7 - 8*B*
a^3*b^8 - (3*A + 20*C)*a^2*b^9 + 8*B*a*b^10 - 2*A*b^11)*cos(d*x + c)^3 + 3*(8*C*a^9*b^2 - 2*B*a^8*b^3 - 28*C*a
^7*b^4 + 7*B*a^6*b^5 + 35*C*a^5*b^6 - 8*B*a^4*b^7 - (3*A + 20*C)*a^3*b^8 + 8*B*a^2*b^9 - 2*A*a*b^10)*cos(d*x +
c)^2 + 3*(8*C*a^10*b - 2*B*a^9*b^2 - 28*C*a^8*b^3 + 7*B*a^7*b^4 + 35*C*a^6*b^5 - 8*B*a^5*b^6 - (3*A + 20*C)*a
^4*b^7 + 8*B*a^3*b^8 - 2*A*a^2*b^9)*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))) - (24*C*a^11*b - 6*B*a^10*b^2 - 92*C*a^9*b^3 + 23*B*a^8*b^4 + (4*A + 133*C)*a^7*b^5 - 43*B*a^
6*b^6 + (7*A - 71*C)*a^5*b^7 + 26*B*a^4*b^8 - (11*A - 6*C)*a^3*b^9 + 6*(C*a^8*b^4 - 4*C*a^6*b^6 + 6*C*a^4*b^8
- 4*C*a^2*b^10 + C*b^12)*cos(d*x + c)^3 + (44*C*a^9*b^3 - 11*B*a^8*b^4 + (2*A - 169*C)*a^7*b^5 + 43*B*a^6*b^6
- (7*A - 239*C)*a^5*b^7 - 68*B*a^4*b^8 + (23*A - 132*C)*a^3*b^9 + 36*B*a^2*b^10 - 18*(A - C)*a*b^11)*cos(d*x +
c)^2 + 3*(20*C*a^10*b^2 - 5*B*a^9*b^3 - 77*C*a^8*b^4 + 20*B*a^7*b^5 + (A + 110*C)*a^6*b^6 - 35*B*a^5*b^7 + (8
*A - 59*C)*a^4*b^8 + 20*B*a^3*b^9 - 3*(3*A - 2*C)*a^2*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^8 - 4*a^6*b^10
+ 6*a^4*b^12 - 4*a^2*b^14 + b^16)*d*cos(d*x + c)^3 + 3*(a^9*b^7 - 4*a^7*b^9 + 6*a^5*b^11 - 4*a^3*b^13 + a*b^1
5)*d*cos(d*x + c)^2 + 3*(a^10*b^6 - 4*a^8*b^8 + 6*a^6*b^10 - 4*a^4*b^12 + a^2*b^14)*d*cos(d*x + c) + (a^11*b^5
- 4*a^9*b^7 + 6*a^7*b^9 - 4*a^5*b^11 + a^3*b^13)*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)**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.34957, size = 1654, normalized size = 3.59 \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)^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

-1/3*(3*(8*C*a^8 - 2*B*a^7*b - 28*C*a^6*b^2 + 7*B*a^5*b^3 + 35*C*a^4*b^4 - 8*B*a^3*b^5 - 3*A*a^2*b^6 - 20*C*a^
2*b^6 + 8*B*a*b^7 - 2*A*b^8)*(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^6*b^5 - 3*a^4*b^7 + 3*a^2*b^9 - b^11)*sqrt(a^2 - b^2)) - (1
8*C*a^9*tan(1/2*d*x + 1/2*c)^5 - 6*B*a^8*b*tan(1/2*d*x + 1/2*c)^5 - 42*C*a^8*b*tan(1/2*d*x + 1/2*c)^5 + 15*B*a
^7*b^2*tan(1/2*d*x + 1/2*c)^5 - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 117
*C*a^6*b^3*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 45*B*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 -
24*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^5 - 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^5
- 105*C*a^4*b^5*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 + 60*B*a^3*b^6*tan(1/2*d*x + 1/2*
c)^5 + 60*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^5 - 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c)^5 - 36*B*a^2*b^7*tan(1/2*d*x +
1/2*c)^5 + 18*A*a*b^8*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^9*tan(1/2*d*x + 1/2*c)^3 - 12*B*a^8*b*tan(1/2*d*x + 1/2*
c)^3 - 152*C*a^7*b^2*tan(1/2*d*x + 1/2*c)^3 + 56*B*a^6*b^3*tan(1/2*d*x + 1/2*c)^3 + 4*A*a^5*b^4*tan(1/2*d*x +
1/2*c)^3 + 236*C*a^5*b^4*tan(1/2*d*x + 1/2*c)^3 - 116*B*a^4*b^5*tan(1/2*d*x + 1/2*c)^3 + 32*A*a^3*b^6*tan(1/2*
d*x + 1/2*c)^3 - 120*C*a^3*b^6*tan(1/2*d*x + 1/2*c)^3 + 72*B*a^2*b^7*tan(1/2*d*x + 1/2*c)^3 - 36*A*a*b^8*tan(1
/2*d*x + 1/2*c)^3 + 18*C*a^9*tan(1/2*d*x + 1/2*c) - 6*B*a^8*b*tan(1/2*d*x + 1/2*c) + 42*C*a^8*b*tan(1/2*d*x +
1/2*c) - 15*B*a^7*b^2*tan(1/2*d*x + 1/2*c) - 24*C*a^7*b^2*tan(1/2*d*x + 1/2*c) + 6*B*a^6*b^3*tan(1/2*d*x + 1/2
*c) - 117*C*a^6*b^3*tan(1/2*d*x + 1/2*c) + 6*A*a^5*b^4*tan(1/2*d*x + 1/2*c) + 45*B*a^5*b^4*tan(1/2*d*x + 1/2*c
) - 24*C*a^5*b^4*tan(1/2*d*x + 1/2*c) + 3*A*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*B*a^4*b^5*tan(1/2*d*x + 1/2*c) +
105*C*a^4*b^5*tan(1/2*d*x + 1/2*c) + 6*A*a^3*b^6*tan(1/2*d*x + 1/2*c) - 60*B*a^3*b^6*tan(1/2*d*x + 1/2*c) + 60
*C*a^3*b^6*tan(1/2*d*x + 1/2*c) + 27*A*a^2*b^7*tan(1/2*d*x + 1/2*c) - 36*B*a^2*b^7*tan(1/2*d*x + 1/2*c) + 18*A
*a*b^8*tan(1/2*d*x + 1/2*c))/((a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d
*x + 1/2*c)^2 + a + b)^3) + 3*(4*C*a - B*b)*(d*x + c)/b^5 - 6*C*tan(1/2*d*x + 1/2*c)/((tan(1/2*d*x + 1/2*c)^2
+ 1)*b^4))/d