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

Optimal. Leaf size=514 \[ -\frac{a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+60 a^6 C+2 b^6 (13 A-12 C)\right ) \sin (c+d x)}{6 b^5 d \left (a^2-b^2\right )^3}+\frac{\left (-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 C+8 a A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\left (a^2 b^2 (A+10 C)-5 a^4 C+4 A b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{\left (a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^6 C+12 A b^6\right ) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+10 a^6 C+b^6 (6 A-C)\right ) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3}+\frac{x \left (C \left (20 a^2+b^2\right )+2 A b^2\right )}{2 b^6} \]

[Out]

((2*A*b^2 + (20*a^2 + b^2)*C)*x)/(2*b^6) + ((8*a*A*b^8 - a^7*b^2*(2*A - 69*C) + 7*a^5*b^4*(A - 12*C) - 8*a^3*b
^6*(A - 5*C) - 20*a^9*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*(a^2
- b^2)^3*d) - (a*(a^4*b^2*(6*A - 167*C) - a^2*b^4*(17*A - 146*C) + 2*b^6*(13*A - 12*C) + 60*a^6*C)*Sin[c + d*
x])/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*b^2*(A - 27*C) - a^2*b^4*(2*A - 23*C) + b^6*(6*A - C) + 10*a^6*C)*Cos[c +
d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*
(a + b*Cos[c + d*x])^3) + ((4*A*b^4 - 5*a^4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 -
b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((12*A*b^6 + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c +
d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 2.21043, antiderivative size = 514, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {3048, 3047, 3049, 3023, 2735, 2659, 205} \[ -\frac{a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+60 a^6 C+2 b^6 (13 A-12 C)\right ) \sin (c+d x)}{6 b^5 d \left (a^2-b^2\right )^3}+\frac{\left (-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 C+8 a A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^4(c+d x)}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\left (a^2 b^2 (A+10 C)-5 a^4 C+4 A b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{\left (a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^6 C+12 A b^6\right ) \sin (c+d x) \cos ^2(c+d x)}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}+\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+10 a^6 C+b^6 (6 A-C)\right ) \sin (c+d x) \cos (c+d x)}{2 b^4 d \left (a^2-b^2\right )^3}+\frac{x \left (C \left (20 a^2+b^2\right )+2 A b^2\right )}{2 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*A*b^2 + (20*a^2 + b^2)*C)*x)/(2*b^6) + ((8*a*A*b^8 - a^7*b^2*(2*A - 69*C) + 7*a^5*b^4*(A - 12*C) - 8*a^3*b
^6*(A - 5*C) - 20*a^9*C)*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^6*Sqrt[a + b]*(a^2
- b^2)^3*d) - (a*(a^4*b^2*(6*A - 167*C) - a^2*b^4*(17*A - 146*C) + 2*b^6*(13*A - 12*C) + 60*a^6*C)*Sin[c + d*
x])/(6*b^5*(a^2 - b^2)^3*d) + ((a^4*b^2*(A - 27*C) - a^2*b^4*(2*A - 23*C) + b^6*(6*A - C) + 10*a^6*C)*Cos[c +
d*x]*Sin[c + d*x])/(2*b^4*(a^2 - b^2)^3*d) - ((A*b^2 + a^2*C)*Cos[c + d*x]^4*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*
(a + b*Cos[c + d*x])^3) + ((4*A*b^4 - 5*a^4*C + a^2*b^2*(A + 10*C))*Cos[c + d*x]^3*Sin[c + d*x])/(6*b^2*(a^2 -
b^2)^2*d*(a + b*Cos[c + d*x])^2) - ((12*A*b^6 + a^4*b^2*(2*A - 53*C) + 20*a^6*C + a^2*b^4*(A + 48*C))*Cos[c +
d*x]^2*Sin[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Cos[c + d*x]))

Rule 3048

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[((c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(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, 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 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 3049

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((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*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(m + n + 2)), x] + Dist[1/(d*(m + n + 2)), Int[(a + b*Sin[e + f*x]
)^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2)
- C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*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] && GtQ[m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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 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) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^4} \, dx &=-\frac{\left (A b^2+a^2 C\right ) \cos ^4(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 ^3(c+d x) \left (4 \left (A b^2+a^2 C\right )-3 a b (A+C) \cos (c+d x)-\left (2 A b^2+5 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^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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 ^2(c+d x) \left (3 \left (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right )-2 a b \left (5 A b^2-\left (a^2-6 b^2\right ) C\right ) \cos (c+d x)+2 \left (a^2 b^2 (A-18 C)-3 b^4 (2 A-C)+10 a^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^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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{\left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\int \frac{\cos (c+d x) \left (2 \left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right )-a b \left (a^2 b^2 (5 A-8 C)+5 a^4 C+2 b^4 (5 A+9 C)\right ) \cos (c+d x)-6 \left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )^3}\\ &=\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d}-\frac{\left (A b^2+a^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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{\left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\int \frac{-6 a \left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right )+2 b \left (a^4 b^2 (A-25 C)+10 a^6 C+3 b^6 (2 A+C)+a^2 b^4 (8 A+27 C)\right ) \cos (c+d x)+2 a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+2 b^6 (13 A-12 C)+60 a^6 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{12 b^4 \left (a^2-b^2\right )^3}\\ &=-\frac{a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+2 b^6 (13 A-12 C)+60 a^6 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d}-\frac{\left (A b^2+a^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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{\left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}-\frac{\int \frac{-6 a b \left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right )-6 \left (a^2-b^2\right )^3 \left (2 A b^2+\left (20 a^2+b^2\right ) C\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{12 b^5 \left (a^2-b^2\right )^3}\\ &=\frac{\left (2 A b^2+\left (20 a^2+b^2\right ) C\right ) x}{2 b^6}-\frac{a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+2 b^6 (13 A-12 C)+60 a^6 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d}-\frac{\left (A b^2+a^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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{\left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (8 a A b^8-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 C\right ) \int \frac{1}{a+b \cos (c+d x)} \, dx}{2 b^6 \left (a^2-b^2\right )^3}\\ &=\frac{\left (2 A b^2+\left (20 a^2+b^2\right ) C\right ) x}{2 b^6}-\frac{a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+2 b^6 (13 A-12 C)+60 a^6 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d}-\frac{\left (A b^2+a^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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{\left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}+\frac{\left (8 a A b^8-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 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^6 \left (a^2-b^2\right )^3 d}\\ &=\frac{\left (2 A b^2+\left (20 a^2+b^2\right ) C\right ) x}{2 b^6}+\frac{\left (8 a A b^8-a^7 b^2 (2 A-69 C)+7 a^5 b^4 (A-12 C)-8 a^3 b^6 (A-5 C)-20 a^9 C\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^6 \sqrt{a+b} \left (a^2-b^2\right )^3 d}-\frac{a \left (a^4 b^2 (6 A-167 C)-a^2 b^4 (17 A-146 C)+2 b^6 (13 A-12 C)+60 a^6 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}+\frac{\left (a^4 b^2 (A-27 C)-a^2 b^4 (2 A-23 C)+b^6 (6 A-C)+10 a^6 C\right ) \cos (c+d x) \sin (c+d x)}{2 b^4 \left (a^2-b^2\right )^3 d}-\frac{\left (A b^2+a^2 C\right ) \cos ^4(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 (4 A b^4-5 a^4 C+a^2 b^2 (A+10 C)\right ) \cos ^3(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{\left (12 A b^6+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{6 b^3 \left (a^2-b^2\right )^3 d (a+b \cos (c+d x))}\\ \end{align*}

Mathematica [B]  time = 6.6876, size = 1452, normalized size = 2.82 \[ \frac{a \left (20 C a^8+2 A b^2 a^6-69 b^2 C a^6-7 A b^4 a^4+84 b^4 C a^4+8 A b^6 a^2-40 b^6 C a^2-8 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^6 \left (a^2-b^2\right )^3 \sqrt{b^2-a^2} d}-\frac{960 C (c+d x) a^{11}+2880 b C (c+d x) \cos (c+d x) a^{10}-960 b C \sin (c+d x) a^{10}+96 A b^2 (c+d x) a^9-1392 b^2 C (c+d x) a^9+1440 b^2 C (c+d x) \cos (2 (c+d x)) a^9-1200 b^2 C \sin (2 (c+d x)) a^9+288 A b^3 (c+d x) \cos (c+d x) a^8-7776 b^3 C (c+d x) \cos (c+d x) a^8+240 b^3 C (c+d x) \cos (3 (c+d x)) a^8-96 A b^3 \sin (c+d x) a^8+2232 b^3 C \sin (c+d x) a^8-440 b^3 C \sin (3 (c+d x)) a^8-144 A b^4 (c+d x) a^7-1512 b^4 C (c+d x) a^7+144 A b^4 (c+d x) \cos (2 (c+d x)) a^7-4248 b^4 C (c+d x) \cos (2 (c+d x)) a^7-120 A b^4 \sin (2 (c+d x)) a^7+3300 b^4 C \sin (2 (c+d x)) a^7-30 b^4 C \sin (4 (c+d x)) a^7-792 A b^5 (c+d x) \cos (c+d x) a^6+6084 b^5 C (c+d x) \cos (c+d x) a^6+24 A b^5 (c+d x) \cos (3 (c+d x)) a^6-708 b^5 C (c+d x) \cos (3 (c+d x)) a^6+228 A b^5 \sin (c+d x) a^6-1086 b^5 C \sin (c+d x) a^6-44 A b^5 \sin (3 (c+d x)) a^6+1253 b^5 C \sin (3 (c+d x)) a^6+3 b^5 C \sin (5 (c+d x)) a^6-144 A b^6 (c+d x) a^5+3288 b^6 C (c+d x) a^5-432 A b^6 (c+d x) \cos (2 (c+d x)) a^5+4104 b^6 C (c+d x) \cos (2 (c+d x)) a^5+360 A b^6 \sin (2 (c+d x)) a^5-2772 b^6 C \sin (2 (c+d x)) a^5+90 b^6 C \sin (4 (c+d x)) a^5+648 A b^7 (c+d x) \cos (c+d x) a^4-396 b^7 C (c+d x) \cos (c+d x) a^4-72 A b^7 (c+d x) \cos (3 (c+d x)) a^4+684 b^7 C (c+d x) \cos (3 (c+d x)) a^4-288 A b^7 \sin (c+d x) a^4-750 b^7 C \sin (c+d x) a^4+128 A b^7 \sin (3 (c+d x)) a^4-1143 b^7 C \sin (3 (c+d x)) a^4-9 b^7 C \sin (5 (c+d x)) a^4+336 A b^8 (c+d x) a^3-1272 b^8 C (c+d x) a^3+432 A b^8 (c+d x) \cos (2 (c+d x)) a^3-1224 b^8 C (c+d x) \cos (2 (c+d x)) a^3-480 A b^8 \sin (2 (c+d x)) a^3+372 b^8 C \sin (2 (c+d x)) a^3-90 b^8 C \sin (4 (c+d x)) a^3-72 A b^9 (c+d x) \cos (c+d x) a^2-756 b^9 C (c+d x) \cos (c+d x) a^2+72 A b^9 (c+d x) \cos (3 (c+d x)) a^2-204 b^9 C (c+d x) \cos (3 (c+d x)) a^2-144 A b^9 \sin (c+d x) a^2+270 b^9 C \sin (c+d x) a^2-144 A b^9 \sin (3 (c+d x)) a^2+279 b^9 C \sin (3 (c+d x)) a^2+9 b^9 C \sin (5 (c+d x)) a^2-144 A b^{10} (c+d x) a-72 b^{10} C (c+d x) a-144 A b^{10} (c+d x) \cos (2 (c+d x)) a-72 b^{10} C (c+d x) \cos (2 (c+d x)) a+60 b^{10} C \sin (2 (c+d x)) a+30 b^{10} C \sin (4 (c+d x)) a-72 A b^{11} (c+d x) \cos (c+d x)-36 b^{11} C (c+d x) \cos (c+d x)-24 A b^{11} (c+d x) \cos (3 (c+d x))-12 b^{11} C (c+d x) \cos (3 (c+d x))-6 b^{11} C \sin (c+d x)-9 b^{11} C \sin (3 (c+d x))-3 b^{11} C \sin (5 (c+d x))}{96 b^6 \left (b^2-a^2\right )^3 d (a+b \cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8*A*b^8 + 20*a^8*C - 69*a^6*b^2*C + 84*a^4*b^4*C - 40*a^2*b^6*C)
*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/(b^6*(a^2 - b^2)^3*Sqrt[-a^2 + b^2]*d) - (96*a^9*A*b^2*
(c + d*x) - 144*a^7*A*b^4*(c + d*x) - 144*a^5*A*b^6*(c + d*x) + 336*a^3*A*b^8*(c + d*x) - 144*a*A*b^10*(c + d*
x) + 960*a^11*C*(c + d*x) - 1392*a^9*b^2*C*(c + d*x) - 1512*a^7*b^4*C*(c + d*x) + 3288*a^5*b^6*C*(c + d*x) - 1
272*a^3*b^8*C*(c + d*x) - 72*a*b^10*C*(c + d*x) + 288*a^8*A*b^3*(c + d*x)*Cos[c + d*x] - 792*a^6*A*b^5*(c + d*
x)*Cos[c + d*x] + 648*a^4*A*b^7*(c + d*x)*Cos[c + d*x] - 72*a^2*A*b^9*(c + d*x)*Cos[c + d*x] - 72*A*b^11*(c +
d*x)*Cos[c + d*x] + 2880*a^10*b*C*(c + d*x)*Cos[c + d*x] - 7776*a^8*b^3*C*(c + d*x)*Cos[c + d*x] + 6084*a^6*b^
5*C*(c + d*x)*Cos[c + d*x] - 396*a^4*b^7*C*(c + d*x)*Cos[c + d*x] - 756*a^2*b^9*C*(c + d*x)*Cos[c + d*x] - 36*
b^11*C*(c + d*x)*Cos[c + d*x] + 144*a^7*A*b^4*(c + d*x)*Cos[2*(c + d*x)] - 432*a^5*A*b^6*(c + d*x)*Cos[2*(c +
d*x)] + 432*a^3*A*b^8*(c + d*x)*Cos[2*(c + d*x)] - 144*a*A*b^10*(c + d*x)*Cos[2*(c + d*x)] + 1440*a^9*b^2*C*(c
+ d*x)*Cos[2*(c + d*x)] - 4248*a^7*b^4*C*(c + d*x)*Cos[2*(c + d*x)] + 4104*a^5*b^6*C*(c + d*x)*Cos[2*(c + d*x
)] - 1224*a^3*b^8*C*(c + d*x)*Cos[2*(c + d*x)] - 72*a*b^10*C*(c + d*x)*Cos[2*(c + d*x)] + 24*a^6*A*b^5*(c + d*
x)*Cos[3*(c + d*x)] - 72*a^4*A*b^7*(c + d*x)*Cos[3*(c + d*x)] + 72*a^2*A*b^9*(c + d*x)*Cos[3*(c + d*x)] - 24*A
*b^11*(c + d*x)*Cos[3*(c + d*x)] + 240*a^8*b^3*C*(c + d*x)*Cos[3*(c + d*x)] - 708*a^6*b^5*C*(c + d*x)*Cos[3*(c
+ d*x)] + 684*a^4*b^7*C*(c + d*x)*Cos[3*(c + d*x)] - 204*a^2*b^9*C*(c + d*x)*Cos[3*(c + d*x)] - 12*b^11*C*(c
+ d*x)*Cos[3*(c + d*x)] - 96*a^8*A*b^3*Sin[c + d*x] + 228*a^6*A*b^5*Sin[c + d*x] - 288*a^4*A*b^7*Sin[c + d*x]
- 144*a^2*A*b^9*Sin[c + d*x] - 960*a^10*b*C*Sin[c + d*x] + 2232*a^8*b^3*C*Sin[c + d*x] - 1086*a^6*b^5*C*Sin[c
+ d*x] - 750*a^4*b^7*C*Sin[c + d*x] + 270*a^2*b^9*C*Sin[c + d*x] - 6*b^11*C*Sin[c + d*x] - 120*a^7*A*b^4*Sin[2
*(c + d*x)] + 360*a^5*A*b^6*Sin[2*(c + d*x)] - 480*a^3*A*b^8*Sin[2*(c + d*x)] - 1200*a^9*b^2*C*Sin[2*(c + d*x)
] + 3300*a^7*b^4*C*Sin[2*(c + d*x)] - 2772*a^5*b^6*C*Sin[2*(c + d*x)] + 372*a^3*b^8*C*Sin[2*(c + d*x)] + 60*a*
b^10*C*Sin[2*(c + d*x)] - 44*a^6*A*b^5*Sin[3*(c + d*x)] + 128*a^4*A*b^7*Sin[3*(c + d*x)] - 144*a^2*A*b^9*Sin[3
*(c + d*x)] - 440*a^8*b^3*C*Sin[3*(c + d*x)] + 1253*a^6*b^5*C*Sin[3*(c + d*x)] - 1143*a^4*b^7*C*Sin[3*(c + d*x
)] + 279*a^2*b^9*C*Sin[3*(c + d*x)] - 9*b^11*C*Sin[3*(c + d*x)] - 30*a^7*b^4*C*Sin[4*(c + d*x)] + 90*a^5*b^6*C
*Sin[4*(c + d*x)] - 90*a^3*b^8*C*Sin[4*(c + d*x)] + 30*a*b^10*C*Sin[4*(c + d*x)] + 3*a^6*b^5*C*Sin[5*(c + d*x)
] - 9*a^4*b^7*C*Sin[5*(c + d*x)] + 9*a^2*b^9*C*Sin[5*(c + d*x)] - 3*b^11*C*Sin[5*(c + d*x)])/(96*b^6*(-a^2 + b
^2)^3*d*(a + b*Cos[c + d*x])^3)

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 2919, normalized size = 5.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/d/b^5/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)^3*a*C-8/d/b^5/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/
2*c)*a*C-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((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b
))^(1/2))*A+40/d*a^3/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*
(a-b))^(1/2))*C+2/d/b^4*arctan(tan(1/2*d*x+1/2*c))*A+1/d/b^4*arctan(tan(1/2*d*x+1/2*c))*C+1/d/b^4/(tan(1/2*d*x
+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)*C-1/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)^3*C+20/d/b^6*arctan(
tan(1/2*d*x+1/2*c))*a^2*C-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*A-20/d*a^9/b^6/(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))*C+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((a
-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+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)*ar
ctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+69/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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-84/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((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*C-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)*arctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+4/d*a^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)*A-30/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-4/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^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*A+44/3/d*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^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^
3*A-24/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^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^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+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*A+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*A-12/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-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*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)*A-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)*A+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)*A-12/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-12/d*a^8/b^
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)*C-3
/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+34/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+6/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-24/d*a^8/b^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*C+212/3/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^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*a^8/b^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*C+3/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+34/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-30/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-60/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^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*C

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 5.45533, size = 5520, normalized size = 10.74 \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+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

[1/12*(6*(20*C*a^10*b^3 + (2*A - 79*C)*a^8*b^5 - 4*(2*A - 29*C)*a^6*b^7 + 2*(6*A - 37*C)*a^4*b^9 - 8*(A - 2*C)
*a^2*b^11 + (2*A + C)*b^13)*d*x*cos(d*x + c)^3 + 18*(20*C*a^11*b^2 + (2*A - 79*C)*a^9*b^4 - 4*(2*A - 29*C)*a^7
*b^6 + 2*(6*A - 37*C)*a^5*b^8 - 8*(A - 2*C)*a^3*b^10 + (2*A + C)*a*b^12)*d*x*cos(d*x + c)^2 + 18*(20*C*a^12*b
+ (2*A - 79*C)*a^10*b^3 - 4*(2*A - 29*C)*a^8*b^5 + 2*(6*A - 37*C)*a^6*b^7 - 8*(A - 2*C)*a^4*b^9 + (2*A + C)*a^
2*b^11)*d*x*cos(d*x + c) + 6*(20*C*a^13 + (2*A - 79*C)*a^11*b^2 - 4*(2*A - 29*C)*a^9*b^4 + 2*(6*A - 37*C)*a^7*
b^6 - 8*(A - 2*C)*a^5*b^8 + (2*A + C)*a^3*b^10)*d*x - 3*(20*C*a^12 + (2*A - 69*C)*a^10*b^2 - 7*(A - 12*C)*a^8*
b^4 + 8*(A - 5*C)*a^6*b^6 - 8*A*a^4*b^8 + (20*C*a^9*b^3 + (2*A - 69*C)*a^7*b^5 - 7*(A - 12*C)*a^5*b^7 + 8*(A -
5*C)*a^3*b^9 - 8*A*a*b^11)*cos(d*x + c)^3 + 3*(20*C*a^10*b^2 + (2*A - 69*C)*a^8*b^4 - 7*(A - 12*C)*a^6*b^6 +
8*(A - 5*C)*a^4*b^8 - 8*A*a^2*b^10)*cos(d*x + c)^2 + 3*(20*C*a^11*b + (2*A - 69*C)*a^9*b^3 - 7*(A - 12*C)*a^7*
b^5 + 8*(A - 5*C)*a^5*b^7 - 8*A*a^3*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*(60*C*a^12*b + (6*A - 227*C)*a^10*b^3 - (23*A - 313*C)*a^8*b^5 + (43*A - 170*C)*
a^6*b^7 - 2*(13*A - 12*C)*a^4*b^9 - 3*(C*a^8*b^5 - 4*C*a^6*b^7 + 6*C*a^4*b^9 - 4*C*a^2*b^11 + C*b^13)*cos(d*x
+ c)^4 + 15*(C*a^9*b^4 - 4*C*a^7*b^6 + 6*C*a^5*b^8 - 4*C*a^3*b^10 + C*a*b^12)*cos(d*x + c)^3 + (110*C*a^10*b^3
+ (11*A - 421*C)*a^8*b^5 - (43*A - 590*C)*a^6*b^7 + 2*(34*A - 171*C)*a^4*b^9 - 9*(4*A - 7*C)*a^2*b^11)*cos(d*
x + c)^2 + 3*(50*C*a^11*b^2 + 5*(A - 38*C)*a^9*b^4 - (20*A - 263*C)*a^7*b^6 + (35*A - 146*C)*a^5*b^8 - (20*A -
23*C)*a^3*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^9 - 4*a^6*b^11 + 6*a^4*b^13 - 4*a^2*b^15 + b^17)*d*cos(d*
x + c)^3 + 3*(a^9*b^8 - 4*a^7*b^10 + 6*a^5*b^12 - 4*a^3*b^14 + a*b^16)*d*cos(d*x + c)^2 + 3*(a^10*b^7 - 4*a^8*
b^9 + 6*a^6*b^11 - 4*a^4*b^13 + a^2*b^15)*d*cos(d*x + c) + (a^11*b^6 - 4*a^9*b^8 + 6*a^7*b^10 - 4*a^5*b^12 + a
^3*b^14)*d), 1/6*(3*(20*C*a^10*b^3 + (2*A - 79*C)*a^8*b^5 - 4*(2*A - 29*C)*a^6*b^7 + 2*(6*A - 37*C)*a^4*b^9 -
8*(A - 2*C)*a^2*b^11 + (2*A + C)*b^13)*d*x*cos(d*x + c)^3 + 9*(20*C*a^11*b^2 + (2*A - 79*C)*a^9*b^4 - 4*(2*A -
29*C)*a^7*b^6 + 2*(6*A - 37*C)*a^5*b^8 - 8*(A - 2*C)*a^3*b^10 + (2*A + C)*a*b^12)*d*x*cos(d*x + c)^2 + 9*(20*
C*a^12*b + (2*A - 79*C)*a^10*b^3 - 4*(2*A - 29*C)*a^8*b^5 + 2*(6*A - 37*C)*a^6*b^7 - 8*(A - 2*C)*a^4*b^9 + (2*
A + C)*a^2*b^11)*d*x*cos(d*x + c) + 3*(20*C*a^13 + (2*A - 79*C)*a^11*b^2 - 4*(2*A - 29*C)*a^9*b^4 + 2*(6*A - 3
7*C)*a^7*b^6 - 8*(A - 2*C)*a^5*b^8 + (2*A + C)*a^3*b^10)*d*x - 3*(20*C*a^12 + (2*A - 69*C)*a^10*b^2 - 7*(A - 1
2*C)*a^8*b^4 + 8*(A - 5*C)*a^6*b^6 - 8*A*a^4*b^8 + (20*C*a^9*b^3 + (2*A - 69*C)*a^7*b^5 - 7*(A - 12*C)*a^5*b^7
+ 8*(A - 5*C)*a^3*b^9 - 8*A*a*b^11)*cos(d*x + c)^3 + 3*(20*C*a^10*b^2 + (2*A - 69*C)*a^8*b^4 - 7*(A - 12*C)*a
^6*b^6 + 8*(A - 5*C)*a^4*b^8 - 8*A*a^2*b^10)*cos(d*x + c)^2 + 3*(20*C*a^11*b + (2*A - 69*C)*a^9*b^3 - 7*(A - 1
2*C)*a^7*b^5 + 8*(A - 5*C)*a^5*b^7 - 8*A*a^3*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))) - (60*C*a^12*b + (6*A - 227*C)*a^10*b^3 - (23*A - 313*C)*a^8*b^5 + (43*A - 170*
C)*a^6*b^7 - 2*(13*A - 12*C)*a^4*b^9 - 3*(C*a^8*b^5 - 4*C*a^6*b^7 + 6*C*a^4*b^9 - 4*C*a^2*b^11 + C*b^13)*cos(d
*x + c)^4 + 15*(C*a^9*b^4 - 4*C*a^7*b^6 + 6*C*a^5*b^8 - 4*C*a^3*b^10 + C*a*b^12)*cos(d*x + c)^3 + (110*C*a^10*
b^3 + (11*A - 421*C)*a^8*b^5 - (43*A - 590*C)*a^6*b^7 + 2*(34*A - 171*C)*a^4*b^9 - 9*(4*A - 7*C)*a^2*b^11)*cos
(d*x + c)^2 + 3*(50*C*a^11*b^2 + 5*(A - 38*C)*a^9*b^4 - (20*A - 263*C)*a^7*b^6 + (35*A - 146*C)*a^5*b^8 - (20*
A - 23*C)*a^3*b^10)*cos(d*x + c))*sin(d*x + c))/((a^8*b^9 - 4*a^6*b^11 + 6*a^4*b^13 - 4*a^2*b^15 + b^17)*d*cos
(d*x + c)^3 + 3*(a^9*b^8 - 4*a^7*b^10 + 6*a^5*b^12 - 4*a^3*b^14 + a*b^16)*d*cos(d*x + c)^2 + 3*(a^10*b^7 - 4*a
^8*b^9 + 6*a^6*b^11 - 4*a^4*b^13 + a^2*b^15)*d*cos(d*x + c) + (a^11*b^6 - 4*a^9*b^8 + 6*a^7*b^10 - 4*a^5*b^12
+ a^3*b^14)*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+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**4,x)

[Out]

Timed out

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Giac [B]  time = 1.95398, size = 1389, normalized size = 2.7 \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+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(6*(20*C*a^9 + 2*A*a^7*b^2 - 69*C*a^7*b^2 - 7*A*a^5*b^4 + 84*C*a^5*b^4 + 8*A*a^3*b^6 - 40*C*a^3*b^6 - 8*A*
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^6 - 3*a^4*b^8 + 3*a^2*b^10 - b^12)*sqrt(a^2 - b^2)) - 2*(36*C*a^10*tan(1/2*d*
x + 1/2*c)^5 - 81*C*a^9*b*tan(1/2*d*x + 1/2*c)^5 + 6*A*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 48*C*a^8*b^2*tan(1/2*d
*x + 1/2*c)^5 - 15*A*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 + 213*C*a^7*b^3*tan(1/2*d*x + 1/2*c)^5 - 6*A*a^6*b^4*tan(1
/2*d*x + 1/2*c)^5 - 48*C*a^6*b^4*tan(1/2*d*x + 1/2*c)^5 + 45*A*a^5*b^5*tan(1/2*d*x + 1/2*c)^5 - 162*C*a^5*b^5*
tan(1/2*d*x + 1/2*c)^5 - 6*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 + 90*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^5 - 60*A*a^3*b
^7*tan(1/2*d*x + 1/2*c)^5 + 36*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^10*tan(1/2*d*x + 1/2*c)^3 + 12*A*a^8*
b^2*tan(1/2*d*x + 1/2*c)^3 - 284*C*a^8*b^2*tan(1/2*d*x + 1/2*c)^3 - 56*A*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 392*
C*a^6*b^4*tan(1/2*d*x + 1/2*c)^3 + 116*A*a^4*b^6*tan(1/2*d*x + 1/2*c)^3 - 180*C*a^4*b^6*tan(1/2*d*x + 1/2*c)^3
- 72*A*a^2*b^8*tan(1/2*d*x + 1/2*c)^3 + 36*C*a^10*tan(1/2*d*x + 1/2*c) + 81*C*a^9*b*tan(1/2*d*x + 1/2*c) + 6*
A*a^8*b^2*tan(1/2*d*x + 1/2*c) - 48*C*a^8*b^2*tan(1/2*d*x + 1/2*c) + 15*A*a^7*b^3*tan(1/2*d*x + 1/2*c) - 213*C
*a^7*b^3*tan(1/2*d*x + 1/2*c) - 6*A*a^6*b^4*tan(1/2*d*x + 1/2*c) - 48*C*a^6*b^4*tan(1/2*d*x + 1/2*c) - 45*A*a^
5*b^5*tan(1/2*d*x + 1/2*c) + 162*C*a^5*b^5*tan(1/2*d*x + 1/2*c) - 6*A*a^4*b^6*tan(1/2*d*x + 1/2*c) + 90*C*a^4*
b^6*tan(1/2*d*x + 1/2*c) + 60*A*a^3*b^7*tan(1/2*d*x + 1/2*c) + 36*A*a^2*b^8*tan(1/2*d*x + 1/2*c))/((a^6*b^5 -
3*a^4*b^7 + 3*a^2*b^9 - b^11)*(a*tan(1/2*d*x + 1/2*c)^2 - b*tan(1/2*d*x + 1/2*c)^2 + a + b)^3) + 3*(20*C*a^2 +
2*A*b^2 + C*b^2)*(d*x + c)/b^6 - 6*(8*C*a*tan(1/2*d*x + 1/2*c)^3 + C*b*tan(1/2*d*x + 1/2*c)^3 + 8*C*a*tan(1/2
*d*x + 1/2*c) - C*b*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^2*b^5))/d