3.1001 \(\int \frac{\cos ^4(c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^4} \, dx\)

Optimal. Leaf size=649 \[ \frac{\sin (c+d x) \left (-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-68 a^4 b^3 B+65 a^2 b^5 B+24 a^6 b B-60 a^7 C-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{6 b^5 d \left (a^2-b^2\right )^3}+\frac{a \left (-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-28 a^5 b^3 B+35 a^3 b^5 B+8 a^7 b B-20 a^8 C-20 a b^7 B+8 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}-\frac{\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^3(c+d x) \left (a^2 b^2 (A+10 C)+2 a^3 b B-5 a^4 C-7 a b^3 B+4 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \cos ^2(c+d x) \left (a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^3 b^3 B-8 a^5 b B+20 a^6 C-27 a b^5 B+12 A b^6\right )}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left (-a^4 b^2 (A-27 C)+a^2 b^4 (2 A-23 C)-11 a^3 b^3 B+4 a^5 b B-10 a^6 C+12 a b^5 B-b^6 (6 A-C)\right )}{2 b^4 d \left (a^2-b^2\right )^3}+\frac{x \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{2 b^6} \]

[Out]

((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/(2*b^6) + (a*(8*A*b^8 + 8*a^7*b*B - 28*a^5*b^3*B + 35*a^3*b^5*B - 2
0*a*b^7*B - a^6*b^2*(2*A - 69*C) + 7*a^4*b^4*(A - 12*C) - 8*a^2*b^6*(A - 5*C) - 20*a^8*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) + ((24*a^6*b*B - 68*a^4*b^3*B +
65*a^2*b^5*B - 6*b^7*B - a^5*b^2*(6*A - 167*C) + a^3*b^4*(17*A - 146*C) - 2*a*b^6*(13*A - 12*C) - 60*a^7*C)*Si
n[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) - ((4*a^5*b*B - 11*a^3*b^3*B + 12*a*b^5*B - 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*(b*B
- a*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 + 2*a^3*b*B - 7*a*
b^3*B - 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 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^5*B + 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]))

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Rubi [A]  time = 12.1992, antiderivative size = 649, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.146, Rules used = {3047, 3049, 3023, 2735, 2659, 205} \[ \frac{\sin (c+d x) \left (-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-68 a^4 b^3 B+65 a^2 b^5 B+24 a^6 b B-60 a^7 C-2 a b^6 (13 A-12 C)-6 b^7 B\right )}{6 b^5 d \left (a^2-b^2\right )^3}+\frac{a \left (-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-28 a^5 b^3 B+35 a^3 b^5 B+8 a^7 b B-20 a^8 C-20 a b^7 B+8 A b^8\right ) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^6 d \sqrt{a-b} \sqrt{a+b} \left (a^2-b^2\right )^3}-\frac{\sin (c+d x) \cos ^4(c+d x) \left (A b^2-a (b B-a C)\right )}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^3}+\frac{\sin (c+d x) \cos ^3(c+d x) \left (a^2 b^2 (A+10 C)+2 a^3 b B-5 a^4 C-7 a b^3 B+4 A b^4\right )}{6 b^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^2}-\frac{\sin (c+d x) \cos ^2(c+d x) \left (a^4 b^2 (2 A-53 C)+a^2 b^4 (A+48 C)+20 a^3 b^3 B-8 a^5 b B+20 a^6 C-27 a b^5 B+12 A b^6\right )}{6 b^3 d \left (a^2-b^2\right )^3 (a+b \cos (c+d x))}-\frac{\sin (c+d x) \cos (c+d x) \left (-a^4 b^2 (A-27 C)+a^2 b^4 (2 A-23 C)-11 a^3 b^3 B+4 a^5 b B-10 a^6 C+12 a b^5 B-b^6 (6 A-C)\right )}{2 b^4 d \left (a^2-b^2\right )^3}+\frac{x \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{2 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*x)/(2*b^6) + (a*(8*A*b^8 + 8*a^7*b*B - 28*a^5*b^3*B + 35*a^3*b^5*B - 2
0*a*b^7*B - a^6*b^2*(2*A - 69*C) + 7*a^4*b^4*(A - 12*C) - 8*a^2*b^6*(A - 5*C) - 20*a^8*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) + ((24*a^6*b*B - 68*a^4*b^3*B +
65*a^2*b^5*B - 6*b^7*B - a^5*b^2*(6*A - 167*C) + a^3*b^4*(17*A - 146*C) - 2*a*b^6*(13*A - 12*C) - 60*a^7*C)*Si
n[c + d*x])/(6*b^5*(a^2 - b^2)^3*d) - ((4*a^5*b*B - 11*a^3*b^3*B + 12*a*b^5*B - 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*(b*B
- a*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 + 2*a^3*b*B - 7*a*
b^3*B - 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 - 8*a^5*b*B + 20*a^3*b^3*B - 27*a*b^5*B + 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 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+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 ^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 (b B-a C)\right )+3 b (b B-a (A+C)) \cos (c+d x)-\left (2 A b^2-2 a b B+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 (b B-a 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+2 a^3 b B-7 a b^3 B-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+2 a^3 b B-7 a b^3 B-5 a^4 C+a^2 b^2 (A+10 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)-2 \left (4 a^3 b B-9 a b^3 B-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 (b B-a 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+2 a^3 b B-7 a b^3 B-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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+a^4 b^2 (2 A-53 C)+20 a^6 C+a^2 b^4 (A+48 C)\right )+b \left (2 a^4 b B+7 a^2 b^3 B+6 b^5 B-a^3 b^2 (5 A-8 C)-5 a^5 C-2 a b^4 (5 A+9 C)\right ) \cos (c+d x)+6 \left (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (b B-a 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+2 a^3 b B-7 a b^3 B-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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+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 (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (4 a^5 b B-7 a^3 b^3 B+18 a b^5 B-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 \left (24 a^6 b B-68 a^4 b^3 B+65 a^2 b^5 B-6 b^7 B-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-2 a b^6 (13 A-12 C)-60 a^7 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{\left (24 a^6 b B-68 a^4 b^3 B+65 a^2 b^5 B-6 b^7 B-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-2 a b^6 (13 A-12 C)-60 a^7 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}-\frac{\left (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (b B-a 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+2 a^3 b B-7 a b^3 B-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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+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 (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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^2-8 a b B+20 a^2 C+b^2 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^2-8 a b B+20 a^2 C+b^2 C\right ) x}{2 b^6}+\frac{\left (24 a^6 b B-68 a^4 b^3 B+65 a^2 b^5 B-6 b^7 B-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-2 a b^6 (13 A-12 C)-60 a^7 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}-\frac{\left (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (b B-a 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+2 a^3 b B-7 a b^3 B-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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+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 (a \left (8 A b^8+8 a^7 b B-28 a^5 b^3 B+35 a^3 b^5 B-20 a b^7 B-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-20 a^8 C\right )\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^2-8 a b B+20 a^2 C+b^2 C\right ) x}{2 b^6}+\frac{\left (24 a^6 b B-68 a^4 b^3 B+65 a^2 b^5 B-6 b^7 B-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-2 a b^6 (13 A-12 C)-60 a^7 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}-\frac{\left (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (b B-a 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+2 a^3 b B-7 a b^3 B-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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+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 (a \left (8 A b^8+8 a^7 b B-28 a^5 b^3 B+35 a^3 b^5 B-20 a b^7 B-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-20 a^8 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b^6 \left (a^2-b^2\right )^3 d}\\ &=\frac{\left (2 A b^2-8 a b B+20 a^2 C+b^2 C\right ) x}{2 b^6}+\frac{a \left (8 A b^8+8 a^7 b B-28 a^5 b^3 B+35 a^3 b^5 B-20 a b^7 B-a^6 b^2 (2 A-69 C)+7 a^4 b^4 (A-12 C)-8 a^2 b^6 (A-5 C)-20 a^8 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{\left (24 a^6 b B-68 a^4 b^3 B+65 a^2 b^5 B-6 b^7 B-a^5 b^2 (6 A-167 C)+a^3 b^4 (17 A-146 C)-2 a b^6 (13 A-12 C)-60 a^7 C\right ) \sin (c+d x)}{6 b^5 \left (a^2-b^2\right )^3 d}-\frac{\left (4 a^5 b B-11 a^3 b^3 B+12 a b^5 B-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 (b B-a 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+2 a^3 b B-7 a b^3 B-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^6-8 a^5 b B+20 a^3 b^3 B-27 a b^5 B+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 [C]  time = 6.90157, size = 658, normalized size = 1.01 \[ \frac{(c+d x) \left (20 a^2 C-8 a b B+2 A b^2+b^2 C\right )}{2 b^6 d}+\frac{a^4 A b^2 \sin (c+d x)-a^5 b B \sin (c+d x)+a^6 C \sin (c+d x)}{3 b^5 d \left (b^2-a^2\right ) (a+b \cos (c+d x))^3}+\frac{7 a^5 A b^2 \sin (c+d x)-12 a^3 A b^4 \sin (c+d x)+15 a^4 b^3 B \sin (c+d x)-18 a^5 b^2 C \sin (c+d x)-10 a^6 b B \sin (c+d x)+13 a^7 C \sin (c+d x)}{6 b^5 d \left (b^2-a^2\right )^2 (a+b \cos (c+d x))^2}+\frac{11 a^6 A b^2 \sin (c+d x)-32 a^4 A b^4 \sin (c+d x)+36 a^2 A b^6 \sin (c+d x)+71 a^5 b^3 B \sin (c+d x)-60 a^3 b^5 B \sin (c+d x)-122 a^6 b^2 C \sin (c+d x)+90 a^4 b^4 C \sin (c+d x)-26 a^7 b B \sin (c+d x)+47 a^8 C \sin (c+d x)}{6 b^5 d \left (b^2-a^2\right )^3 (a+b \cos (c+d x))}+\frac{a \left (2 a^6 A b^2-7 a^4 A b^4+8 a^2 A b^6+28 a^5 b^3 B-35 a^3 b^5 B-69 a^6 b^2 C+84 a^4 b^4 C-40 a^2 b^6 C-8 a^7 b B+20 a^8 C+20 a b^7 B-8 A b^8\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{b^6 d \left (a^2-b^2\right )^3 \sqrt{b^2-a^2}}+\frac{(4 a C-b B) \left (-\frac{\sin (c+d x)}{2 b^5}-\frac{i \cos (c+d x)}{2 b^5}\right )}{d}+\frac{(4 a C-b B) \left (-\frac{\sin (c+d x)}{2 b^5}+\frac{i \cos (c+d x)}{2 b^5}\right )}{d}+\frac{C \sin (2 (c+d x))}{4 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*A*b^2 - 8*a*b*B + 20*a^2*C + b^2*C)*(c + d*x))/(2*b^6*d) + (a*(2*a^6*A*b^2 - 7*a^4*A*b^4 + 8*a^2*A*b^6 - 8
*A*b^8 - 8*a^7*b*B + 28*a^5*b^3*B - 35*a^3*b^5*B + 20*a*b^7*B + 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) + ((-(b*
B) + 4*a*C)*(((-I/2)*Cos[c + d*x])/b^5 - Sin[c + d*x]/(2*b^5)))/d + ((-(b*B) + 4*a*C)*(((I/2)*Cos[c + d*x])/b^
5 - Sin[c + d*x]/(2*b^5)))/d + (a^4*A*b^2*Sin[c + d*x] - a^5*b*B*Sin[c + d*x] + a^6*C*Sin[c + d*x])/(3*b^5*(-a
^2 + b^2)*d*(a + b*Cos[c + d*x])^3) + (7*a^5*A*b^2*Sin[c + d*x] - 12*a^3*A*b^4*Sin[c + d*x] - 10*a^6*b*B*Sin[c
+ d*x] + 15*a^4*b^3*B*Sin[c + d*x] + 13*a^7*C*Sin[c + d*x] - 18*a^5*b^2*C*Sin[c + d*x])/(6*b^5*(-a^2 + b^2)^2
*d*(a + b*Cos[c + d*x])^2) + (11*a^6*A*b^2*Sin[c + d*x] - 32*a^4*A*b^4*Sin[c + d*x] + 36*a^2*A*b^6*Sin[c + d*x
] - 26*a^7*b*B*Sin[c + d*x] + 71*a^5*b^3*B*Sin[c + d*x] - 60*a^3*b^5*B*Sin[c + d*x] + 47*a^8*C*Sin[c + d*x] -
122*a^6*b^2*C*Sin[c + d*x] + 90*a^4*b^4*C*Sin[c + d*x])/(6*b^5*(-a^2 + b^2)^3*d*(a + b*Cos[c + d*x])) + (C*Sin
[2*(c + d*x)])/(4*b^4*d)

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 4367, normalized size = 6.7 \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+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x)

[Out]

-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)^3*C+2/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)*B+1/d/b^4/(tan(1/2*d*x+1/2
*c)^2+1)^2*tan(1/2*d*x+1/2*c)*C+2/d/b^4/(tan(1/2*d*x+1/2*c)^2+1)^2*tan(1/2*d*x+1/2*c)^3*B-8/d/b^5*arctan(tan(1
/2*d*x+1/2*c))*a*B+20/d/b^6*arctan(tan(1/2*d*x+1/2*c))*a^2*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*A+1/d*a^5/b^2/(a*tan(1/2*d*x+1/2*c)^2-ta
n(1/2*d*x+1/2*c)^2*b+a+b)^3/(a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tan(1/2*d*x+1/2*c)^5*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-24/d*a^8/b^5/(a*ta
n(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2*c)^2*b+a+b)^3/(a^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
)*C+12/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^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^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+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*B-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-116/3/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^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+6/d*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)*B-1/d*a^5/b^2/(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+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-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+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-
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)*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*A+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+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+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)*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-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-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-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+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+21
2/3/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+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-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-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)*B+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)*B+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-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+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*B-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*B-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-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((a-b)*tan(1/2*d*x+1/2*c)/((a+
b)*(a-b))^(1/2))*B-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((a-b)*tan(1/2*d*x+1/2
*c)/((a+b)*(a-b))^(1/2))*B+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((a-b)*tan(1/2*d
*x+1/2*c)/((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((a-b)*ta
n(1/2*d*x+1/2*c)/((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)*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)*a
rctan((a-b)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*A+20/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+40/d*a^3/(a*tan(1/2*d*x+1/2*c)^2-tan(1/2*d*x+1/2
*c)^2*b+a+b)^3/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tan(1/2*d*x+1/2*c)^3*B+20/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-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)*tan(1/2*d*x+1/2*c)/((a+b)*(a-b))^(1/2))*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+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

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Fricas [B]  time = 5.50197, size = 7788, normalized size = 12. \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)+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 - 8*B*a^9*b^4 + (2*A - 79*C)*a^8*b^5 + 32*B*a^7*b^6 - 4*(2*A - 29*C)*a^6*b^7 - 48*B*a^
5*b^8 + 2*(6*A - 37*C)*a^4*b^9 + 32*B*a^3*b^10 - 8*(A - 2*C)*a^2*b^11 - 8*B*a*b^12 + (2*A + C)*b^13)*d*x*cos(d
*x + c)^3 + 18*(20*C*a^11*b^2 - 8*B*a^10*b^3 + (2*A - 79*C)*a^9*b^4 + 32*B*a^8*b^5 - 4*(2*A - 29*C)*a^7*b^6 -
48*B*a^6*b^7 + 2*(6*A - 37*C)*a^5*b^8 + 32*B*a^4*b^9 - 8*(A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + (2*A + C)*a*b^12)
*d*x*cos(d*x + c)^2 + 18*(20*C*a^12*b - 8*B*a^11*b^2 + (2*A - 79*C)*a^10*b^3 + 32*B*a^9*b^4 - 4*(2*A - 29*C)*a
^8*b^5 - 48*B*a^7*b^6 + 2*(6*A - 37*C)*a^6*b^7 + 32*B*a^5*b^8 - 8*(A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + (2*A + C)
*a^2*b^11)*d*x*cos(d*x + c) + 6*(20*C*a^13 - 8*B*a^12*b + (2*A - 79*C)*a^11*b^2 + 32*B*a^10*b^3 - 4*(2*A - 29*
C)*a^9*b^4 - 48*B*a^8*b^5 + 2*(6*A - 37*C)*a^7*b^6 + 32*B*a^6*b^7 - 8*(A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + (2*A +
C)*a^3*b^10)*d*x - 3*(20*C*a^12 - 8*B*a^11*b + (2*A - 69*C)*a^10*b^2 + 28*B*a^9*b^3 - 7*(A - 12*C)*a^8*b^4 -
35*B*a^7*b^5 + 8*(A - 5*C)*a^6*b^6 + 20*B*a^5*b^7 - 8*A*a^4*b^8 + (20*C*a^9*b^3 - 8*B*a^8*b^4 + (2*A - 69*C)*a
^7*b^5 + 28*B*a^6*b^6 - 7*(A - 12*C)*a^5*b^7 - 35*B*a^4*b^8 + 8*(A - 5*C)*a^3*b^9 + 20*B*a^2*b^10 - 8*A*a*b^11
)*cos(d*x + c)^3 + 3*(20*C*a^10*b^2 - 8*B*a^9*b^3 + (2*A - 69*C)*a^8*b^4 + 28*B*a^7*b^5 - 7*(A - 12*C)*a^6*b^6
- 35*B*a^5*b^7 + 8*(A - 5*C)*a^4*b^8 + 20*B*a^3*b^9 - 8*A*a^2*b^10)*cos(d*x + c)^2 + 3*(20*C*a^11*b - 8*B*a^1
0*b^2 + (2*A - 69*C)*a^9*b^3 + 28*B*a^8*b^4 - 7*(A - 12*C)*a^7*b^5 - 35*B*a^6*b^6 + 8*(A - 5*C)*a^5*b^7 + 20*B
*a^4*b^8 - 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 - 24*B*a^11*b^2 + (6*A - 227*C)*a^10*b^3 + 92*B*a^9*b^4 - (23*A - 313*C)*a^8*b^5 - 1
33*B*a^7*b^6 + (43*A - 170*C)*a^6*b^7 + 71*B*a^5*b^8 - 2*(13*A - 12*C)*a^4*b^9 - 6*B*a^3*b^10 - 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 + 3*(5*C*a^9*b^4 - 2*B*a^8*b^5 - 20*C*a^7*b
^6 + 8*B*a^6*b^7 + 30*C*a^5*b^8 - 12*B*a^4*b^9 - 20*C*a^3*b^10 + 8*B*a^2*b^11 + 5*C*a*b^12 - 2*B*b^13)*cos(d*x
+ c)^3 + (110*C*a^10*b^3 - 44*B*a^9*b^4 + (11*A - 421*C)*a^8*b^5 + 169*B*a^7*b^6 - (43*A - 590*C)*a^6*b^7 - 2
39*B*a^5*b^8 + 2*(34*A - 171*C)*a^4*b^9 + 132*B*a^3*b^10 - 9*(4*A - 7*C)*a^2*b^11 - 18*B*a*b^12)*cos(d*x + c)^
2 + 3*(50*C*a^11*b^2 - 20*B*a^10*b^3 + 5*(A - 38*C)*a^9*b^4 + 77*B*a^8*b^5 - (20*A - 263*C)*a^7*b^6 - 110*B*a^
6*b^7 + (35*A - 146*C)*a^5*b^8 + 59*B*a^4*b^9 - (20*A - 23*C)*a^3*b^10 - 6*B*a^2*b^11)*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 - 8*B
*a^9*b^4 + (2*A - 79*C)*a^8*b^5 + 32*B*a^7*b^6 - 4*(2*A - 29*C)*a^6*b^7 - 48*B*a^5*b^8 + 2*(6*A - 37*C)*a^4*b^
9 + 32*B*a^3*b^10 - 8*(A - 2*C)*a^2*b^11 - 8*B*a*b^12 + (2*A + C)*b^13)*d*x*cos(d*x + c)^3 + 9*(20*C*a^11*b^2
- 8*B*a^10*b^3 + (2*A - 79*C)*a^9*b^4 + 32*B*a^8*b^5 - 4*(2*A - 29*C)*a^7*b^6 - 48*B*a^6*b^7 + 2*(6*A - 37*C)*
a^5*b^8 + 32*B*a^4*b^9 - 8*(A - 2*C)*a^3*b^10 - 8*B*a^2*b^11 + (2*A + C)*a*b^12)*d*x*cos(d*x + c)^2 + 9*(20*C*
a^12*b - 8*B*a^11*b^2 + (2*A - 79*C)*a^10*b^3 + 32*B*a^9*b^4 - 4*(2*A - 29*C)*a^8*b^5 - 48*B*a^7*b^6 + 2*(6*A
- 37*C)*a^6*b^7 + 32*B*a^5*b^8 - 8*(A - 2*C)*a^4*b^9 - 8*B*a^3*b^10 + (2*A + C)*a^2*b^11)*d*x*cos(d*x + c) + 3
*(20*C*a^13 - 8*B*a^12*b + (2*A - 79*C)*a^11*b^2 + 32*B*a^10*b^3 - 4*(2*A - 29*C)*a^9*b^4 - 48*B*a^8*b^5 + 2*(
6*A - 37*C)*a^7*b^6 + 32*B*a^6*b^7 - 8*(A - 2*C)*a^5*b^8 - 8*B*a^4*b^9 + (2*A + C)*a^3*b^10)*d*x - 3*(20*C*a^1
2 - 8*B*a^11*b + (2*A - 69*C)*a^10*b^2 + 28*B*a^9*b^3 - 7*(A - 12*C)*a^8*b^4 - 35*B*a^7*b^5 + 8*(A - 5*C)*a^6*
b^6 + 20*B*a^5*b^7 - 8*A*a^4*b^8 + (20*C*a^9*b^3 - 8*B*a^8*b^4 + (2*A - 69*C)*a^7*b^5 + 28*B*a^6*b^6 - 7*(A -
12*C)*a^5*b^7 - 35*B*a^4*b^8 + 8*(A - 5*C)*a^3*b^9 + 20*B*a^2*b^10 - 8*A*a*b^11)*cos(d*x + c)^3 + 3*(20*C*a^10
*b^2 - 8*B*a^9*b^3 + (2*A - 69*C)*a^8*b^4 + 28*B*a^7*b^5 - 7*(A - 12*C)*a^6*b^6 - 35*B*a^5*b^7 + 8*(A - 5*C)*a
^4*b^8 + 20*B*a^3*b^9 - 8*A*a^2*b^10)*cos(d*x + c)^2 + 3*(20*C*a^11*b - 8*B*a^10*b^2 + (2*A - 69*C)*a^9*b^3 +
28*B*a^8*b^4 - 7*(A - 12*C)*a^7*b^5 - 35*B*a^6*b^6 + 8*(A - 5*C)*a^5*b^7 + 20*B*a^4*b^8 - 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 - 24*B*a^11
*b^2 + (6*A - 227*C)*a^10*b^3 + 92*B*a^9*b^4 - (23*A - 313*C)*a^8*b^5 - 133*B*a^7*b^6 + (43*A - 170*C)*a^6*b^7
+ 71*B*a^5*b^8 - 2*(13*A - 12*C)*a^4*b^9 - 6*B*a^3*b^10 - 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 + 3*(5*C*a^9*b^4 - 2*B*a^8*b^5 - 20*C*a^7*b^6 + 8*B*a^6*b^7 + 30*C*a^5*b^8 - 12*
B*a^4*b^9 - 20*C*a^3*b^10 + 8*B*a^2*b^11 + 5*C*a*b^12 - 2*B*b^13)*cos(d*x + c)^3 + (110*C*a^10*b^3 - 44*B*a^9*
b^4 + (11*A - 421*C)*a^8*b^5 + 169*B*a^7*b^6 - (43*A - 590*C)*a^6*b^7 - 239*B*a^5*b^8 + 2*(34*A - 171*C)*a^4*b
^9 + 132*B*a^3*b^10 - 9*(4*A - 7*C)*a^2*b^11 - 18*B*a*b^12)*cos(d*x + c)^2 + 3*(50*C*a^11*b^2 - 20*B*a^10*b^3
+ 5*(A - 38*C)*a^9*b^4 + 77*B*a^8*b^5 - (20*A - 263*C)*a^7*b^6 - 110*B*a^6*b^7 + (35*A - 146*C)*a^5*b^8 + 59*B
*a^4*b^9 - (20*A - 23*C)*a^3*b^10 - 6*B*a^2*b^11)*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+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 = 1.35508, size = 1939, normalized size = 2.99 \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)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/6*(6*(20*C*a^9 - 8*B*a^8*b + 2*A*a^7*b^2 - 69*C*a^7*b^2 + 28*B*a^6*b^3 - 7*A*a^5*b^4 + 84*C*a^5*b^4 - 35*B*a
^4*b^5 + 8*A*a^3*b^6 - 40*C*a^3*b^6 + 20*B*a^2*b^7 - 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 - 18*B*a^9*b*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 + 42*B*a^8*b^2*tan(1/2*d*x + 1/2*c)^5 - 4
8*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 + 24*B*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 - 117*B*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 + 24*B*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 + 105*B*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 - 60*B*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 - 36*B*a^9*b*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 + 152*B*a^7*b^3
*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 - 236*B*a
^5*b^5*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 +
120*B*a^3*b^7*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) -
18*B*a^9*b*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) - 42*B*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) + 24*B*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) + 117*B*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) + 24*B*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) - 105*B*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) - 60*B*a^3*b^7*ta
n(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 - 8*B*a*b + 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 - 2*B*b*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) - 2*B*b*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