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

Optimal. Leaf size=409 \[ -\frac{\left (3 a^3 A b+23 a^2 b^2 B-12 a^4 B-8 a A b^3-6 b^4 B\right ) \sin (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{a \left (-7 a^4 A b^3+8 a^2 A b^5+2 a^6 A b+28 a^5 b^2 B-35 a^3 b^4 B-8 a^7 B+20 a b^6 B-8 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{a \left (a^2 A b-4 a^3 B+9 a b^2 B-6 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{a^2 \left (-2 a^2 A b^3+a^4 A b+11 a^3 b^2 B-4 a^5 B-12 a b^4 B+6 A b^5\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{x (A b-4 a B)}{b^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 5.1748, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2989, 3047, 3031, 3023, 2735, 2659, 205} \[ -\frac{\left (3 a^3 A b+23 a^2 b^2 B-12 a^4 B-8 a A b^3-6 b^4 B\right ) \sin (c+d x)}{6 b^4 d \left (a^2-b^2\right )^2}-\frac{a \left (-7 a^4 A b^3+8 a^2 A b^5+2 a^6 A b+28 a^5 b^2 B-35 a^3 b^4 B-8 a^7 B+20 a b^6 B-8 A b^7\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a (A b-a B) \sin (c+d x) \cos ^3(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{a \left (a^2 A b-4 a^3 B+9 a b^2 B-6 A b^3\right ) \sin (c+d x) \cos ^2(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{a^2 \left (-2 a^2 A b^3+a^4 A b+11 a^3 b^2 B-4 a^5 B-12 a b^4 B+6 A b^5\right ) \sin (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{x (A b-4 a B)}{b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2989

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[((b*c - a*d)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)
*(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 - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (
A*b + a*B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)
*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2,
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] && GtQ[m, 1] && LtQ[n, -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]

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 ^4(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^4} \, dx &=\frac{a (A b-a B) \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 a (A b-a B)+3 b (A b-a B) \cos (c+d x)+\left (a A b-4 a^2 B+3 b^2 B\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^3} \, dx}{3 b \left (a^2-b^2\right )}\\ &=\frac{a (A b-a B) \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{a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\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 a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\right )+2 b \left (2 a^2 A b+3 A b^3+a^3 B-6 a b^2 B\right ) \cos (c+d x)-\left (3 a^3 A b-8 a A b^3-12 a^4 B+23 a^2 b^2 B-6 b^4 B\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{a (A b-a B) \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{a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\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^2 \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\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 a b \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\right )+\left (a^2-b^2\right ) \left (3 a^4 A b-4 a^2 A b^3+6 A b^5-12 a^5 B+25 a^3 b^2 B-18 a b^4 B\right ) \cos (c+d x)-b \left (a^2-b^2\right ) \left (3 a^3 A b-8 a A b^3-12 a^4 B+23 a^2 b^2 B-6 b^4 B\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 (3 a^3 A b-8 a A b^3-12 a^4 B+23 a^2 b^2 B-6 b^4 B\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \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{a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\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^2 \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\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 a b^2 \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\right )+6 b \left (a^2-b^2\right )^3 (A b-4 a B) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^5 \left (a^2-b^2\right )^3}\\ &=\frac{(A b-4 a B) x}{b^5}-\frac{\left (3 a^3 A b-8 a A b^3-12 a^4 B+23 a^2 b^2 B-6 b^4 B\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \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{a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\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^2 \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (a \left (2 a^6 A b-7 a^4 A b^3+8 a^2 A b^5-8 A b^7-8 a^7 B+28 a^5 b^2 B-35 a^3 b^4 B+20 a b^6 B\right )\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )^3}\\ &=\frac{(A b-4 a B) x}{b^5}-\frac{\left (3 a^3 A b-8 a A b^3-12 a^4 B+23 a^2 b^2 B-6 b^4 B\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \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{a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\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^2 \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\right ) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\left (a \left (2 a^6 A b-7 a^4 A b^3+8 a^2 A b^5-8 A b^7-8 a^7 B+28 a^5 b^2 B-35 a^3 b^4 B+20 a b^6 B\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 )}{b^5 \left (a^2-b^2\right )^3 d}\\ &=\frac{(A b-4 a B) x}{b^5}-\frac{a \left (2 a^6 A b-7 a^4 A b^3+8 a^2 A b^5-8 A b^7-8 a^7 B+28 a^5 b^2 B-35 a^3 b^4 B+20 a b^6 B\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 (3 a^3 A b-8 a A b^3-12 a^4 B+23 a^2 b^2 B-6 b^4 B\right ) \sin (c+d x)}{6 b^4 \left (a^2-b^2\right )^2 d}+\frac{a (A b-a B) \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{a \left (a^2 A b-6 A b^3-4 a^3 B+9 a b^2 B\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^2 \left (a^4 A b-2 a^2 A b^3+6 A b^5-4 a^5 B+11 a^3 b^2 B-12 a b^4 B\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 [B]  time = 6.61236, size = 1278, normalized size = 3.12 \[ \frac{96 B (c+d x) a^{10}-24 A b (c+d x) a^9+288 b B (c+d x) \cos (c+d x) a^9-96 b B \sin (c+d x) a^9-144 b^2 B (c+d x) a^8-72 A b^2 (c+d x) \cos (c+d x) a^8+144 b^2 B (c+d x) \cos (2 (c+d x)) a^8+24 A b^2 \sin (c+d x) a^8-120 b^2 B \sin (2 (c+d x)) a^8+36 A b^3 (c+d x) a^7-792 b^3 B (c+d x) \cos (c+d x) a^7-36 A b^3 (c+d x) \cos (2 (c+d x)) a^7+24 b^3 B (c+d x) \cos (3 (c+d x)) a^7+228 b^3 B \sin (c+d x) a^7+30 A b^3 \sin (2 (c+d x)) a^7-44 b^3 B \sin (3 (c+d x)) a^7-144 b^4 B (c+d x) a^6+198 A b^4 (c+d x) \cos (c+d x) a^6-432 b^4 B (c+d x) \cos (2 (c+d x)) a^6-6 A b^4 (c+d x) \cos (3 (c+d x)) a^6-57 A b^4 \sin (c+d x) a^6+336 b^4 B \sin (2 (c+d x)) a^6+11 A b^4 \sin (3 (c+d x)) a^6-3 b^4 B \sin (4 (c+d x)) a^6+36 A b^5 (c+d x) a^5+648 b^5 B (c+d x) \cos (c+d x) a^5+108 A b^5 (c+d x) \cos (2 (c+d x)) a^5-72 b^5 B (c+d x) \cos (3 (c+d x)) a^5-135 b^5 B \sin (c+d x) a^5-90 A b^5 \sin (2 (c+d x)) a^5+125 b^5 B \sin (3 (c+d x)) a^5+336 b^6 B (c+d x) a^4-162 A b^6 (c+d x) \cos (c+d x) a^4+432 b^6 B (c+d x) \cos (2 (c+d x)) a^4+18 A b^6 (c+d x) \cos (3 (c+d x)) a^4+72 A b^6 \sin (c+d x) a^4-300 b^6 B \sin (2 (c+d x)) a^4-32 A b^6 \sin (3 (c+d x)) a^4+9 b^6 B \sin (4 (c+d x)) a^4-84 A b^7 (c+d x) a^3-72 b^7 B (c+d x) \cos (c+d x) a^3-108 A b^7 (c+d x) \cos (2 (c+d x)) a^3+72 b^7 B (c+d x) \cos (3 (c+d x)) a^3-90 b^7 B \sin (c+d x) a^3+120 A b^7 \sin (2 (c+d x)) a^3-114 b^7 B \sin (3 (c+d x)) a^3-144 b^8 B (c+d x) a^2+18 A b^8 (c+d x) \cos (c+d x) a^2-144 b^8 B (c+d x) \cos (2 (c+d x)) a^2-18 A b^8 (c+d x) \cos (3 (c+d x)) a^2+36 A b^8 \sin (c+d x) a^2+18 b^8 B \sin (2 (c+d x)) a^2+36 A b^8 \sin (3 (c+d x)) a^2-9 b^8 B \sin (4 (c+d x)) a^2+36 A b^9 (c+d x) a-72 b^9 B (c+d x) \cos (c+d x) a+36 A b^9 (c+d x) \cos (2 (c+d x)) a-24 b^9 B (c+d x) \cos (3 (c+d x)) a+18 b^9 B \sin (c+d x) a+18 b^9 B \sin (3 (c+d x)) a+18 A b^{10} (c+d x) \cos (c+d x)+6 A b^{10} (c+d x) \cos (3 (c+d x))+6 b^{10} B \sin (2 (c+d x))+3 b^{10} B \sin (4 (c+d x))}{24 b^5 \left (b^2-a^2\right )^3 d (a+b \cos (c+d x))^3}-\frac{a \left (8 B a^7-2 A b a^6-28 b^2 B a^5+7 A b^3 a^4+35 b^4 B a^3-8 A b^5 a^2-20 b^6 B a+8 A b^7\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{b^5 \left (a^2-b^2\right )^3 \sqrt{b^2-a^2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(-2*a^6*A*b + 7*a^4*A*b^3 - 8*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 28*a^5*b^2*B + 35*a^3*b^4*B - 20*a*b^6*B)*A
rcTanh[((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)) + (-24*a^9*A*b*(c
+ d*x) + 36*a^7*A*b^3*(c + d*x) + 36*a^5*A*b^5*(c + d*x) - 84*a^3*A*b^7*(c + d*x) + 36*a*A*b^9*(c + d*x) + 96
*a^10*B*(c + d*x) - 144*a^8*b^2*B*(c + d*x) - 144*a^6*b^4*B*(c + d*x) + 336*a^4*b^6*B*(c + d*x) - 144*a^2*b^8*
B*(c + d*x) - 72*a^8*A*b^2*(c + d*x)*Cos[c + d*x] + 198*a^6*A*b^4*(c + d*x)*Cos[c + d*x] - 162*a^4*A*b^6*(c +
d*x)*Cos[c + d*x] + 18*a^2*A*b^8*(c + d*x)*Cos[c + d*x] + 18*A*b^10*(c + d*x)*Cos[c + d*x] + 288*a^9*b*B*(c +
d*x)*Cos[c + d*x] - 792*a^7*b^3*B*(c + d*x)*Cos[c + d*x] + 648*a^5*b^5*B*(c + d*x)*Cos[c + d*x] - 72*a^3*b^7*B
*(c + d*x)*Cos[c + d*x] - 72*a*b^9*B*(c + d*x)*Cos[c + d*x] - 36*a^7*A*b^3*(c + d*x)*Cos[2*(c + d*x)] + 108*a^
5*A*b^5*(c + d*x)*Cos[2*(c + d*x)] - 108*a^3*A*b^7*(c + d*x)*Cos[2*(c + d*x)] + 36*a*A*b^9*(c + d*x)*Cos[2*(c
+ d*x)] + 144*a^8*b^2*B*(c + d*x)*Cos[2*(c + d*x)] - 432*a^6*b^4*B*(c + d*x)*Cos[2*(c + d*x)] + 432*a^4*b^6*B*
(c + d*x)*Cos[2*(c + d*x)] - 144*a^2*b^8*B*(c + d*x)*Cos[2*(c + d*x)] - 6*a^6*A*b^4*(c + d*x)*Cos[3*(c + d*x)]
+ 18*a^4*A*b^6*(c + d*x)*Cos[3*(c + d*x)] - 18*a^2*A*b^8*(c + d*x)*Cos[3*(c + d*x)] + 6*A*b^10*(c + d*x)*Cos[
3*(c + d*x)] + 24*a^7*b^3*B*(c + d*x)*Cos[3*(c + d*x)] - 72*a^5*b^5*B*(c + d*x)*Cos[3*(c + d*x)] + 72*a^3*b^7*
B*(c + d*x)*Cos[3*(c + d*x)] - 24*a*b^9*B*(c + d*x)*Cos[3*(c + d*x)] + 24*a^8*A*b^2*Sin[c + d*x] - 57*a^6*A*b^
4*Sin[c + d*x] + 72*a^4*A*b^6*Sin[c + d*x] + 36*a^2*A*b^8*Sin[c + d*x] - 96*a^9*b*B*Sin[c + d*x] + 228*a^7*b^3
*B*Sin[c + d*x] - 135*a^5*b^5*B*Sin[c + d*x] - 90*a^3*b^7*B*Sin[c + d*x] + 18*a*b^9*B*Sin[c + d*x] + 30*a^7*A*
b^3*Sin[2*(c + d*x)] - 90*a^5*A*b^5*Sin[2*(c + d*x)] + 120*a^3*A*b^7*Sin[2*(c + d*x)] - 120*a^8*b^2*B*Sin[2*(c
+ d*x)] + 336*a^6*b^4*B*Sin[2*(c + d*x)] - 300*a^4*b^6*B*Sin[2*(c + d*x)] + 18*a^2*b^8*B*Sin[2*(c + d*x)] + 6
*b^10*B*Sin[2*(c + d*x)] + 11*a^6*A*b^4*Sin[3*(c + d*x)] - 32*a^4*A*b^6*Sin[3*(c + d*x)] + 36*a^2*A*b^8*Sin[3*
(c + d*x)] - 44*a^7*b^3*B*Sin[3*(c + d*x)] + 125*a^5*b^5*B*Sin[3*(c + d*x)] - 114*a^3*b^7*B*Sin[3*(c + d*x)] +
18*a*b^9*B*Sin[3*(c + d*x)] - 3*a^6*b^4*B*Sin[4*(c + d*x)] + 9*a^4*b^6*B*Sin[4*(c + d*x)] - 9*a^2*b^8*B*Sin[4
*(c + d*x)] + 3*b^10*B*Sin[4*(c + d*x)])/(24*b^5*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.141, size = 2787, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/d*a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*
c)*A+40/d*a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*
x+1/2*c)^3*B-4/d*a^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan
(1/2*d*x+1/2*c)^5*A+35/d*a^4/b/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-
b)/((a-b)*(a+b))^(1/2))*B+8/d*a^8/b^5/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2
*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+8/d*a*b^2/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*
x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-28/d*a^6/b^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(ta
n(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B-2/d*a^7/b^4/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*ar
ctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A+2/d/b^4*A*arctan(tan(1/2*d*x+1/2*c))-24/d*a^2*b/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+6/d*a^4/b/
(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-18/
d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/
2*c)^5*B-12/d*a^2*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(
1/2*d*x+1/2*c)^5*A+5/d*a^4/b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+
b^3)*tan(1/2*d*x+1/2*c)^5*B-1/d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2
*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-4/d*a^6/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*
a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-2/d*a^6/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b
)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*A-2/d*a^6/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c
)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+1/d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/
2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+6/d*a^4/b/(tan(1/2*d*x+1/2*c)^2
*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+44/3/d*a^4/b/(tan(1/2*
d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A-5/d*a^4/b/
(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+12/
d*a^7/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/
2*c)^3*B-18/d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*ta
n(1/2*d*x+1/2*c)*B-12/d*a^2*b/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2
-b^3)*tan(1/2*d*x+1/2*c)*A+6/d*a^7/b^4/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*
b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+2/d*a^6/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/
(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-116/3/d*a^5/b^2/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+
a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B-2/d*a^6/b^3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+
1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*A+6/d*a^7/b^4/(tan(1/2*d*x+1/2*c)^2*a-t
an(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B+7/d*a^5/b^2/(a^6-3*a^4*b^2+3
*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-20/d*a^2*b/(a^6-3*a^4
*b^2+3*a^2*b^4-b^6)/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+20/d*a^3/(tan(1
/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*B+20/d*a^
3/(tan(1/2*d*x+1/2*c)^2*a-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a+b)/(a^3-3*a^2*b+3*a*b^2-b^3)*tan(1/2*d*x+1/2*c)*B-8
/d/b^5*B*arctan(tan(1/2*d*x+1/2*c))*a+2/d/b^4*B*tan(1/2*d*x+1/2*c)/(1+tan(1/2*d*x+1/2*c)^2)-8/d*a^3/(a^6-3*a^4
*b^2+3*a^2*b^4-b^6)/((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., 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)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.16999, size = 5723, normalized size = 13.99 \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)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

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

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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)**4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.49078, size = 1304, normalized size = 3.19 \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)^4*(A+B*cos(d*x+c))/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

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