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

Optimal. Leaf size=279 \[ -\frac {2 a^3 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {\sin (c+d x) \cos (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )}{8 b^3 d}+\frac {\sin (c+d x) \left (-3 a^3 C+3 a^2 b B-a b^2 (3 A+2 C)+2 b^3 B\right )}{3 b^4 d}-\frac {x \left (-8 a^4 C+8 a^3 b B-4 a^2 b^2 (2 A+C)+4 a b^3 B-b^4 (4 A+3 C)\right )}{8 b^5}+\frac {(b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 b d} \]

[Out]

-1/8*(8*a^3*b*B+4*a*b^3*B-8*a^4*C-4*a^2*b^2*(2*A+C)-b^4*(4*A+3*C))*x/b^5+1/3*(3*a^2*b*B+2*b^3*B-3*a^3*C-a*b^2*
(3*A+2*C))*sin(d*x+c)/b^4/d+1/8*(4*A*b^2-4*B*a*b+4*C*a^2+3*C*b^2)*cos(d*x+c)*sin(d*x+c)/b^3/d+1/3*(B*b-C*a)*co
s(d*x+c)^2*sin(d*x+c)/b^2/d+1/4*C*cos(d*x+c)^3*sin(d*x+c)/b/d-2*a^3*(A*b^2-a*(B*b-C*a))*arctan((a-b)^(1/2)*tan
(1/2*d*x+1/2*c)/(a+b)^(1/2))/b^5/d/(a-b)^(1/2)/(a+b)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.94, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3049, 3023, 2735, 2659, 205} \[ \frac {\sin (c+d x) \left (3 a^2 b B-3 a^3 C-a b^2 (3 A+2 C)+2 b^3 B\right )}{3 b^4 d}-\frac {2 a^3 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {\sin (c+d x) \cos (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )}{8 b^3 d}-\frac {x \left (-4 a^2 b^2 (2 A+C)+8 a^3 b B-8 a^4 C+4 a b^3 B-b^4 (4 A+3 C)\right )}{8 b^5}+\frac {(b B-a C) \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}+\frac {C \sin (c+d x) \cos ^3(c+d x)}{4 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((8*a^3*b*B + 4*a*b^3*B - 8*a^4*C - 4*a^2*b^2*(2*A + C) - b^4*(4*A + 3*C))*x)/(8*b^5) - (2*a^3*(A*b^2 - a*(b*
B - a*C))*ArcTan[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^5*Sqrt[a + b]*d) + ((3*a^2*b*B +
2*b^3*B - 3*a^3*C - a*b^2*(3*A + 2*C))*Sin[c + d*x])/(3*b^4*d) + ((4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Cos[
c + d*x]*Sin[c + d*x])/(8*b^3*d) + ((b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*b^2*d) + (C*Cos[c + d*x]^3*Sin
[c + d*x])/(4*b*d)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2735

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

Rule 3023

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

Rule 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])))

Rubi steps

\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx &=\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {\cos ^2(c+d x) \left (3 a C+b (4 A+3 C) \cos (c+d x)+4 (b B-a C) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{4 b}\\ &=\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {\cos (c+d x) \left (8 a (b B-a C)+b (8 b B+a C) \cos (c+d x)+3 \left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{12 b^2}\\ &=\frac {\left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {3 a \left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right )+b \left (12 A b^2+4 a b B-4 a^2 C+9 b^2 C\right ) \cos (c+d x)+8 \left (3 a^2 b B+2 b^3 B-3 a^3 C-a b^2 (3 A+2 C)\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^3}\\ &=\frac {\left (3 a^2 b B+2 b^3 B-3 a^3 C-a b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d}+\frac {\left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {3 a b \left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right )-3 \left (8 a^3 b B+4 a b^3 B-8 a^4 C-4 a^2 b^2 (2 A+C)-b^4 (4 A+3 C)\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^4}\\ &=-\frac {\left (8 a^3 b B+4 a b^3 B-8 a^4 C-4 a^2 b^2 (2 A+C)-b^4 (4 A+3 C)\right ) x}{8 b^5}+\frac {\left (3 a^2 b B+2 b^3 B-3 a^3 C-a b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d}+\frac {\left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac {\left (a^3 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^5}\\ &=-\frac {\left (8 a^3 b B+4 a b^3 B-8 a^4 C-4 a^2 b^2 (2 A+C)-b^4 (4 A+3 C)\right ) x}{8 b^5}+\frac {\left (3 a^2 b B+2 b^3 B-3 a^3 C-a b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d}+\frac {\left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}-\frac {\left (2 a^3 \left (A b^2-a (b B-a 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^5 d}\\ &=-\frac {\left (8 a^3 b B+4 a b^3 B-8 a^4 C-4 a^2 b^2 (2 A+C)-b^4 (4 A+3 C)\right ) x}{8 b^5}-\frac {2 a^3 \left (A b^2-a (b B-a 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^5 \sqrt {a+b} d}+\frac {\left (3 a^2 b B+2 b^3 B-3 a^3 C-a b^2 (3 A+2 C)\right ) \sin (c+d x)}{3 b^4 d}+\frac {\left (4 A b^2-4 a b B+4 a^2 C+3 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {(b B-a C) \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {C \cos ^3(c+d x) \sin (c+d x)}{4 b d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.90, size = 238, normalized size = 0.85 \[ \frac {24 b^2 \sin (2 (c+d x)) \left (a^2 C-a b B+A b^2+b^2 C\right )+\frac {192 a^3 \left (a (a C-b B)+A b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+24 b \sin (c+d x) \left (-4 a^3 C+4 a^2 b B-a b^2 (4 A+3 C)+3 b^3 B\right )+12 (c+d x) \left (8 a^4 C-8 a^3 b B+4 a^2 b^2 (2 A+C)-4 a b^3 B+b^4 (4 A+3 C)\right )+8 b^3 (b B-a C) \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*(-8*a^3*b*B - 4*a*b^3*B + 8*a^4*C + 4*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*(c + d*x) + (192*a^3*(A*b^2 + a
*(-(b*B) + a*C))*ArcTanh[((a - b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + 24*b*(4*a^2*b*B + 3*
b^3*B - 4*a^3*C - a*b^2*(4*A + 3*C))*Sin[c + d*x] + 24*b^2*(A*b^2 - a*b*B + a^2*C + b^2*C)*Sin[2*(c + d*x)] +
8*b^3*(b*B - a*C)*Sin[3*(c + d*x)] + 3*b^4*C*Sin[4*(c + d*x)])/(96*b^5*d)

________________________________________________________________________________________

fricas [A]  time = 0.55, size = 777, normalized size = 2.78 \[ \left [\frac {3 \, {\left (8 \, C a^{6} - 8 \, B a^{5} b + 4 \, {\left (2 \, A - C\right )} a^{4} b^{2} + 4 \, B a^{3} b^{3} - {\left (4 \, A + C\right )} a^{2} b^{4} + 4 \, B a b^{5} - {\left (4 \, A + 3 \, C\right )} b^{6}\right )} d x - 12 \, {\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (24 \, C a^{5} b - 24 \, B a^{4} b^{2} + 8 \, {\left (3 \, A - C\right )} a^{3} b^{3} + 8 \, B a^{2} b^{4} - 8 \, {\left (3 \, A + 2 \, C\right )} a b^{5} + 16 \, B b^{6} - 6 \, {\left (C a^{2} b^{4} - C b^{6}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (C a^{3} b^{3} - B a^{2} b^{4} - C a b^{5} + B b^{6}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (4 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + {\left (4 \, A - C\right )} a^{2} b^{4} + 4 \, B a b^{5} - {\left (4 \, A + 3 \, C\right )} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{2} b^{5} - b^{7}\right )} d}, \frac {3 \, {\left (8 \, C a^{6} - 8 \, B a^{5} b + 4 \, {\left (2 \, A - C\right )} a^{4} b^{2} + 4 \, B a^{3} b^{3} - {\left (4 \, A + C\right )} a^{2} b^{4} + 4 \, B a b^{5} - {\left (4 \, A + 3 \, C\right )} b^{6}\right )} d x - 24 \, {\left (C a^{5} - B a^{4} b + A a^{3} b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (24 \, C a^{5} b - 24 \, B a^{4} b^{2} + 8 \, {\left (3 \, A - C\right )} a^{3} b^{3} + 8 \, B a^{2} b^{4} - 8 \, {\left (3 \, A + 2 \, C\right )} a b^{5} + 16 \, B b^{6} - 6 \, {\left (C a^{2} b^{4} - C b^{6}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (C a^{3} b^{3} - B a^{2} b^{4} - C a b^{5} + B b^{6}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (4 \, C a^{4} b^{2} - 4 \, B a^{3} b^{3} + {\left (4 \, A - C\right )} a^{2} b^{4} + 4 \, B a b^{5} - {\left (4 \, A + 3 \, C\right )} b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, {\left (a^{2} b^{5} - b^{7}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/24*(3*(8*C*a^6 - 8*B*a^5*b + 4*(2*A - C)*a^4*b^2 + 4*B*a^3*b^3 - (4*A + C)*a^2*b^4 + 4*B*a*b^5 - (4*A + 3*C
)*b^6)*d*x - 12*(C*a^5 - B*a^4*b + A*a^3*b^2)*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)) - (24*C*a^5*b - 24*B*a^4*b^2 + 8*(3*A - C)*a^3*b^3 + 8*B*a^2*b^4 - 8*(3*A + 2*C)*a*b^5 + 16*B
*b^6 - 6*(C*a^2*b^4 - C*b^6)*cos(d*x + c)^3 + 8*(C*a^3*b^3 - B*a^2*b^4 - C*a*b^5 + B*b^6)*cos(d*x + c)^2 - 3*(
4*C*a^4*b^2 - 4*B*a^3*b^3 + (4*A - C)*a^2*b^4 + 4*B*a*b^5 - (4*A + 3*C)*b^6)*cos(d*x + c))*sin(d*x + c))/((a^2
*b^5 - b^7)*d), 1/24*(3*(8*C*a^6 - 8*B*a^5*b + 4*(2*A - C)*a^4*b^2 + 4*B*a^3*b^3 - (4*A + C)*a^2*b^4 + 4*B*a*b
^5 - (4*A + 3*C)*b^6)*d*x - 24*(C*a^5 - B*a^4*b + A*a^3*b^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqr
t(a^2 - b^2)*sin(d*x + c))) - (24*C*a^5*b - 24*B*a^4*b^2 + 8*(3*A - C)*a^3*b^3 + 8*B*a^2*b^4 - 8*(3*A + 2*C)*a
*b^5 + 16*B*b^6 - 6*(C*a^2*b^4 - C*b^6)*cos(d*x + c)^3 + 8*(C*a^3*b^3 - B*a^2*b^4 - C*a*b^5 + B*b^6)*cos(d*x +
c)^2 - 3*(4*C*a^4*b^2 - 4*B*a^3*b^3 + (4*A - C)*a^2*b^4 + 4*B*a*b^5 - (4*A + 3*C)*b^6)*cos(d*x + c))*sin(d*x
+ c))/((a^2*b^5 - b^7)*d)]

________________________________________________________________________________________

giac [B]  time = 0.25, size = 801, normalized size = 2.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(8*C*a^4 - 8*B*a^3*b + 8*A*a^2*b^2 + 4*C*a^2*b^2 - 4*B*a*b^3 + 4*A*b^4 + 3*C*b^4)*(d*x + c)/b^5 + 48*(
C*a^5 - B*a^4*b + A*a^3*b^2)*(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)))/(sqrt(a^2 - b^2)*b^5) - 2*(24*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 24
*B*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 12*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 24*A*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 12*B
*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 24*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 12*A*b^3*tan(1/2*d*x + 1/2*c)^7 - 24*B*b^3
*tan(1/2*d*x + 1/2*c)^7 + 15*C*b^3*tan(1/2*d*x + 1/2*c)^7 + 72*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 72*B*a^2*b*tan(1
/2*d*x + 1/2*c)^5 + 12*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 72*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 12*B*a*b^2*tan(1/2
*d*x + 1/2*c)^5 + 40*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 + 12*A*b^3*tan(1/2*d*x + 1/2*c)^5 - 40*B*b^3*tan(1/2*d*x +
1/2*c)^5 - 9*C*b^3*tan(1/2*d*x + 1/2*c)^5 + 72*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 72*B*a^2*b*tan(1/2*d*x + 1/2*c)
^3 - 12*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 72*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 12*B*a*b^2*tan(1/2*d*x + 1/2*c)^3
+ 40*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 12*A*b^3*tan(1/2*d*x + 1/2*c)^3 - 40*B*b^3*tan(1/2*d*x + 1/2*c)^3 + 9*C
*b^3*tan(1/2*d*x + 1/2*c)^3 + 24*C*a^3*tan(1/2*d*x + 1/2*c) - 24*B*a^2*b*tan(1/2*d*x + 1/2*c) - 12*C*a^2*b*tan
(1/2*d*x + 1/2*c) + 24*A*a*b^2*tan(1/2*d*x + 1/2*c) + 12*B*a*b^2*tan(1/2*d*x + 1/2*c) + 24*C*a*b^2*tan(1/2*d*x
+ 1/2*c) - 12*A*b^3*tan(1/2*d*x + 1/2*c) - 24*B*b^3*tan(1/2*d*x + 1/2*c) - 15*C*b^3*tan(1/2*d*x + 1/2*c))/((t
an(1/2*d*x + 1/2*c)^2 + 1)^4*b^4))/d

________________________________________________________________________________________

maple [B]  time = 0.13, size = 1580, normalized size = 5.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*a*A+10/3/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1
/2*c)^3*B-6/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*a*A-1/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1
/2*d*x+1/2*c)^7*C*a^2+1/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*C*a^2-6/d/b^2/(1+tan(1/2*d*x+1/2
*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a*A-5/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*C-1/d/b/(1+tan(1/2*d
*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*A+2/d/b^3*arctan(tan(1/2*d*x+1/2*c))*A*a^2+2/d/b^5*arctan(tan(1/2*d*x+1/2*
c))*a^4*C+3/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*C+5/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2
*d*x+1/2*c)*C-6/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a^3*C+3/4/d/b*arctan(tan(1/2*d*x+1/2*c))
*C+1/d/b*arctan(tan(1/2*d*x+1/2*c))*A+1/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*A+1/d/b^3*arctan(t
an(1/2*d*x+1/2*c))*C*a^2+1/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*A-1/d/b/(1+tan(1/2*d*x+1/2*c)^2)^
4*tan(1/2*d*x+1/2*c)^5*A+1/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*B*a+2/d*a^4/b^4/((a-b)*(a+b))
^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*B+2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2
*c)^7*B-2/d/b^4*arctan(tan(1/2*d*x+1/2*c))*B*a^3-1/d/b^2*arctan(tan(1/2*d*x+1/2*c))*B*a+1/d/b^2/(1+tan(1/2*d*x
+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*B*a-3/4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*C+2/d/b/(1+tan(1
/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*B-1/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*B*a+6/d/b^3/(1+t
an(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*a^2*B+1/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*C*a^2-
2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*C*a-2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c
)^7*a^3*C+10/3/d/b/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*B-2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/
2*d*x+1/2*c)*a*A+6/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*a^2*B-2/d/b^4/(1+tan(1/2*d*x+1/2*c)^2
)^4*tan(1/2*d*x+1/2*c)*a^3*C-10/3/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^5*C*a-2/d/b^2/(1+tan(1/2
*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*C*a-10/3/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*C*a-2/d*a
^3/b^3/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*A-1/d/b^3/(1+tan(1/2*d*x+1/2*c
)^2)^4*tan(1/2*d*x+1/2*c)^5*C*a^2+2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)*a^2*B-6/d/b^4/(1+tan(1
/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*a^3*C-2/d*a^5/b^5/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/
((a-b)*(a+b))^(1/2))*C+2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^4*tan(1/2*d*x+1/2*c)^7*a^2*B-1/d/b^2/(1+tan(1/2*d*x+1/
2*c)^2)^4*tan(1/2*d*x+1/2*c)^3*B*a

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 10.81, size = 9661, normalized size = 34.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((tan(c/2 + (d*x)/2)^7*(4*A*b^3 - 8*B*b^3 + 8*C*a^3 + 5*C*b^3 + 8*A*a*b^2 - 4*B*a*b^2 - 8*B*a^2*b + 8*C*a*b^
2 + 4*C*a^2*b))/(4*b^4) + (tan(c/2 + (d*x)/2)^3*(72*C*a^3 - 40*B*b^3 - 12*A*b^3 + 9*C*b^3 + 72*A*a*b^2 + 12*B*
a*b^2 - 72*B*a^2*b + 40*C*a*b^2 - 12*C*a^2*b))/(12*b^4) + (tan(c/2 + (d*x)/2)^5*(12*A*b^3 - 40*B*b^3 + 72*C*a^
3 - 9*C*b^3 + 72*A*a*b^2 - 12*B*a*b^2 - 72*B*a^2*b + 40*C*a*b^2 + 12*C*a^2*b))/(12*b^4) - (tan(c/2 + (d*x)/2)*
(4*A*b^3 + 8*B*b^3 - 8*C*a^3 + 5*C*b^3 - 8*A*a*b^2 - 4*B*a*b^2 + 8*B*a^2*b - 8*C*a*b^2 + 4*C*a^2*b))/(4*b^4))/
(d*(4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) - (a
tan(((((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 256*C^2*
a^10*b + 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8 + 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7
*b^4 + 16*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a^7*b^4
+ 256*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 2
56*C^2*a^6*b^5 - 256*C^2*a^7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*A*C*a*
b^10 - 24*B*C*a*b^10 + 256*B*C*a^10*b + 96*A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 +
512*A*B*a^6*b^5 - 512*A*B*a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b^7 - 4
64*A*C*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9 - 152*
B*C*a^3*b^8 + 264*B*C*a^4*b^7 - 368*B*C*a^5*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 - 512*B*
C*a^9*b^2))/(2*b^8) + (((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^2*b^14
- 16*B*a^3*b^13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^5*b^11
+ 32*C*a^6*b^10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 - (tan(c/2 + (d*x)/2)*(128*a*b^12 - 256*a^2*b
^11 + 128*a^3*b^10)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2 - B
*a^3*b*1i))/(2*b^13))*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2 -
B*a^3*b*1i))/b^5)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2 - B*
a^3*b*1i)*1i)/b^5 + (((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*
b^10 + 256*C^2*a^10*b + 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8 + 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^
5 - 128*A^2*a^7*b^4 + 16*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 -
256*B^2*a^7*b^4 + 256*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*
C^2*a^5*b^6 + 256*C^2*a^6*b^5 - 256*C^2*a^7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b
^10 - 72*A*C*a*b^10 - 24*B*C*a*b^10 + 256*B*C*a^10*b + 96*A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 51
2*A*B*a^5*b^6 + 512*A*B*a^6*b^5 - 512*A*B*a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*
A*C*a^4*b^7 - 464*A*C*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C
*a^2*b^9 - 152*B*C*a^3*b^8 + 264*B*C*a^4*b^7 - 368*B*C*a^5*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a
^8*b^3 - 512*B*C*a^9*b^2))/(2*b^8) - (((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12
+ 16*B*a^2*b^14 - 16*B*a^3*b^13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12
- 48*C*a^5*b^11 + 32*C*a^6*b^10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 + (tan(c/2 + (d*x)/2)*(128*a*b
^12 - 256*a^2*b^11 + 128*a^3*b^10)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*
a*b^3*1i)/2 - B*a^3*b*1i))/(2*b^13))*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (
B*a*b^3*1i)/2 - B*a^3*b*1i))/b^5)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a
*b^3*1i)/2 - B*a^3*b*1i)*1i)/b^5)/((64*C^3*a^14 - 96*C^3*a^13*b - 16*A^3*a^3*b^11 + 32*A^3*a^4*b^10 - 80*A^3*a
^5*b^9 + 96*A^3*a^6*b^8 - 96*A^3*a^7*b^7 + 64*A^3*a^8*b^6 + 16*B^3*a^6*b^8 - 32*B^3*a^7*b^7 + 80*B^3*a^8*b^6 -
96*B^3*a^9*b^5 + 96*B^3*a^10*b^4 - 64*B^3*a^11*b^3 - 9*C^3*a^5*b^9 + 18*C^3*a^6*b^8 - 33*C^3*a^7*b^7 + 48*C^3
*a^8*b^6 - 88*C^3*a^9*b^5 + 104*C^3*a^10*b^4 - 104*C^3*a^11*b^3 + 96*C^3*a^12*b^2 - 192*B*C^2*a^13*b - 48*A*B^
2*a^5*b^9 + 96*A*B^2*a^6*b^8 - 240*A*B^2*a^7*b^7 + 288*A*B^2*a^8*b^6 - 288*A*B^2*a^9*b^5 + 192*A*B^2*a^10*b^4
+ 48*A^2*B*a^4*b^10 - 96*A^2*B*a^5*b^9 + 240*A^2*B*a^6*b^8 - 288*A^2*B*a^7*b^7 + 288*A^2*B*a^8*b^6 - 192*A^2*B
*a^9*b^5 - 9*A*C^2*a^3*b^11 + 18*A*C^2*a^4*b^10 - 57*A*C^2*a^5*b^9 + 96*A*C^2*a^6*b^8 - 192*A*C^2*a^7*b^7 + 24
0*A*C^2*a^8*b^6 - 288*A*C^2*a^9*b^5 + 288*A*C^2*a^10*b^4 - 288*A*C^2*a^11*b^3 + 192*A*C^2*a^12*b^2 - 24*A^2*C*
a^3*b^11 + 48*A^2*C*a^4*b^10 - 120*A^2*C*a^5*b^9 + 168*A^2*C*a^6*b^8 - 264*A^2*C*a^7*b^7 + 288*A^2*C*a^8*b^6 -
288*A^2*C*a^9*b^5 + 192*A^2*C*a^10*b^4 + 9*B*C^2*a^4*b^10 - 18*B*C^2*a^5*b^9 + 57*B*C^2*a^6*b^8 - 96*B*C^2*a^
7*b^7 + 192*B*C^2*a^8*b^6 - 240*B*C^2*a^9*b^5 + 288*B*C^2*a^10*b^4 - 288*B*C^2*a^11*b^3 + 288*B*C^2*a^12*b^2 -
24*B^2*C*a^5*b^9 + 48*B^2*C*a^6*b^8 - 120*B^2*C*a^7*b^7 + 168*B^2*C*a^8*b^6 - 264*B^2*C*a^9*b^5 + 288*B^2*C*a
^10*b^4 - 288*B^2*C*a^11*b^3 + 192*B^2*C*a^12*b^2 + 48*A*B*C*a^4*b^10 - 96*A*B*C*a^5*b^9 + 240*A*B*C*a^6*b^8 -
336*A*B*C*a^7*b^7 + 528*A*B*C*a^8*b^6 - 576*A*B*C*a^9*b^5 + 576*A*B*C*a^10*b^4 - 384*A*B*C*a^11*b^3)/b^12 + (
((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 256*C^2*a^10*b
+ 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8 + 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7*b^4 +
16*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a^7*b^4 + 256
*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 256*C^2
*a^6*b^5 - 256*C^2*a^7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*A*C*a*b^10 -
24*B*C*a*b^10 + 256*B*C*a^10*b + 96*A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 + 512*A
*B*a^6*b^5 - 512*A*B*a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b^7 - 464*A*C
*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9 - 152*B*C*a^
3*b^8 + 264*B*C*a^4*b^7 - 368*B*C*a^5*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 - 512*B*C*a^9*
b^2))/(2*b^8) + (((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^2*b^14 - 16*
B*a^3*b^13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^5*b^11 + 32*
C*a^6*b^10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 - (tan(c/2 + (d*x)/2)*(128*a*b^12 - 256*a^2*b^11 +
128*a^3*b^10)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2 - B*a^3*b
*1i))/(2*b^13))*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2 - B*a^3
*b*1i))/b^5)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2 - B*a^3*b*
1i))/b^5 - (((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 25
6*C^2*a^10*b + 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8 + 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A
^2*a^7*b^4 + 16*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a
^7*b^4 + 256*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b
^6 + 256*C^2*a^6*b^5 - 256*C^2*a^7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*
A*C*a*b^10 - 24*B*C*a*b^10 + 256*B*C*a^10*b + 96*A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5
*b^6 + 512*A*B*a^6*b^5 - 512*A*B*a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b
^7 - 464*A*C*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9
- 152*B*C*a^3*b^8 + 264*B*C*a^4*b^7 - 368*B*C*a^5*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 -
512*B*C*a^9*b^2))/(2*b^8) - (((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^
2*b^14 - 16*B*a^3*b^13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^
5*b^11 + 32*C*a^6*b^10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 + (tan(c/2 + (d*x)/2)*(128*a*b^12 - 256
*a^2*b^11 + 128*a^3*b^10)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)
/2 - B*a^3*b*1i))/(2*b^13))*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1
i)/2 - B*a^3*b*1i))/b^5)*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/
2 - B*a^3*b*1i))/b^5))*(b^2*(A*a^2*1i + (C*a^2*1i)/2) + C*a^4*1i + b^4*((A*1i)/2 + (C*3i)/8) - (B*a*b^3*1i)/2
- B*a^3*b*1i)*2i)/(b^5*d) - (a^3*atan(((a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 128*C
^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 256*C^2*a^10*b + 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8 + 25
6*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7*b^4 + 16*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 112*B^
2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a^7*b^4 + 256*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C^2*a
^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 256*C^2*a^6*b^5 - 256*C^2*a^7*b^4 + 256*C^2*a^8*
b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*A*C*a*b^10 - 24*B*C*a*b^10 + 256*B*C*a^10*b + 96*A*B*
a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 + 512*A*B*a^6*b^5 - 512*A*B*a^7*b^4 + 256*A*B*a^
8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b^7 - 464*A*C*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*a^7*
b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9 - 152*B*C*a^3*b^8 + 264*B*C*a^4*b^7 - 368*B*C*a^5*b^6
+ 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 - 512*B*C*a^9*b^2))/(2*b^8) + (a^3*(-(a + b)*(a - b))^(
1/2)*((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^2*b^14 - 16*B*a^3*b^13 +
48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^5*b^11 + 32*C*a^6*b^10 -
16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 - (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a
^2 - B*a*b)*(128*a*b^12 - 256*a^2*b^11 + 128*a^3*b^10))/(2*b^8*(b^7 - a^2*b^5)))*(A*b^2 + C*a^2 - B*a*b))/(b^7
- a^2*b^5))*(A*b^2 + C*a^2 - B*a*b)*1i)/(b^7 - a^2*b^5) + (a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*
(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 256*C^2*a^10*b + 112*A^2*a^2*b^9 -
208*A^2*a^3*b^8 + 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7*b^4 + 16*B^2*a^2*b^9 - 48*
B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a^7*b^4 + 256*B^2*a^8*b^3 - 128*B^
2*a^9*b^2 + 51*C^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 256*C^2*a^6*b^5 - 256*C^2*a^
7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*A*C*a*b^10 - 24*B*C*a*b^10 + 256*
B*C*a^10*b + 96*A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 + 512*A*B*a^6*b^5 - 512*A*B*
a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b^7 - 464*A*C*a^5*b^6 + 512*A*C*a^
6*b^5 - 512*A*C*a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9 - 152*B*C*a^3*b^8 + 264*B*C*a^4*b
^7 - 368*B*C*a^5*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 - 512*B*C*a^9*b^2))/(2*b^8) - (a^3*
(-(a + b)*(a - b))^(1/2)*((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^2*b^
14 - 16*B*a^3*b^13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^5*b^
11 + 32*C*a^6*b^10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 + (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b)
)^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(128*a*b^12 - 256*a^2*b^11 + 128*a^3*b^10))/(2*b^8*(b^7 - a^2*b^5)))*(A*b^2 +
C*a^2 - B*a*b))/(b^7 - a^2*b^5))*(A*b^2 + C*a^2 - B*a*b)*1i)/(b^7 - a^2*b^5))/((64*C^3*a^14 - 96*C^3*a^13*b -
16*A^3*a^3*b^11 + 32*A^3*a^4*b^10 - 80*A^3*a^5*b^9 + 96*A^3*a^6*b^8 - 96*A^3*a^7*b^7 + 64*A^3*a^8*b^6 + 16*B^3
*a^6*b^8 - 32*B^3*a^7*b^7 + 80*B^3*a^8*b^6 - 96*B^3*a^9*b^5 + 96*B^3*a^10*b^4 - 64*B^3*a^11*b^3 - 9*C^3*a^5*b^
9 + 18*C^3*a^6*b^8 - 33*C^3*a^7*b^7 + 48*C^3*a^8*b^6 - 88*C^3*a^9*b^5 + 104*C^3*a^10*b^4 - 104*C^3*a^11*b^3 +
96*C^3*a^12*b^2 - 192*B*C^2*a^13*b - 48*A*B^2*a^5*b^9 + 96*A*B^2*a^6*b^8 - 240*A*B^2*a^7*b^7 + 288*A*B^2*a^8*b
^6 - 288*A*B^2*a^9*b^5 + 192*A*B^2*a^10*b^4 + 48*A^2*B*a^4*b^10 - 96*A^2*B*a^5*b^9 + 240*A^2*B*a^6*b^8 - 288*A
^2*B*a^7*b^7 + 288*A^2*B*a^8*b^6 - 192*A^2*B*a^9*b^5 - 9*A*C^2*a^3*b^11 + 18*A*C^2*a^4*b^10 - 57*A*C^2*a^5*b^9
+ 96*A*C^2*a^6*b^8 - 192*A*C^2*a^7*b^7 + 240*A*C^2*a^8*b^6 - 288*A*C^2*a^9*b^5 + 288*A*C^2*a^10*b^4 - 288*A*C
^2*a^11*b^3 + 192*A*C^2*a^12*b^2 - 24*A^2*C*a^3*b^11 + 48*A^2*C*a^4*b^10 - 120*A^2*C*a^5*b^9 + 168*A^2*C*a^6*b
^8 - 264*A^2*C*a^7*b^7 + 288*A^2*C*a^8*b^6 - 288*A^2*C*a^9*b^5 + 192*A^2*C*a^10*b^4 + 9*B*C^2*a^4*b^10 - 18*B*
C^2*a^5*b^9 + 57*B*C^2*a^6*b^8 - 96*B*C^2*a^7*b^7 + 192*B*C^2*a^8*b^6 - 240*B*C^2*a^9*b^5 + 288*B*C^2*a^10*b^4
- 288*B*C^2*a^11*b^3 + 288*B*C^2*a^12*b^2 - 24*B^2*C*a^5*b^9 + 48*B^2*C*a^6*b^8 - 120*B^2*C*a^7*b^7 + 168*B^2
*C*a^8*b^6 - 264*B^2*C*a^9*b^5 + 288*B^2*C*a^10*b^4 - 288*B^2*C*a^11*b^3 + 192*B^2*C*a^12*b^2 + 48*A*B*C*a^4*b
^10 - 96*A*B*C*a^5*b^9 + 240*A*B*C*a^6*b^8 - 336*A*B*C*a^7*b^7 + 528*A*B*C*a^8*b^6 - 576*A*B*C*a^9*b^5 + 576*A
*B*C*a^10*b^4 - 384*A*B*C*a^11*b^3)/b^12 + (a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)*(16*A^2*b^11 - 1
28*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 256*C^2*a^10*b + 112*A^2*a^2*b^9 - 208*A^2*a^3*b^8
+ 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7*b^4 + 16*B^2*a^2*b^9 - 48*B^2*a^3*b^8 + 11
2*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a^7*b^4 + 256*B^2*a^8*b^3 - 128*B^2*a^9*b^2 + 51*C
^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 256*C^2*a^6*b^5 - 256*C^2*a^7*b^4 + 256*C^2*
a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*A*C*a*b^10 - 24*B*C*a*b^10 + 256*B*C*a^10*b + 96*
A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 + 512*A*B*a^6*b^5 - 512*A*B*a^7*b^4 + 256*A*
B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b^7 - 464*A*C*a^5*b^6 + 512*A*C*a^6*b^5 - 512*A*C*
a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9 - 152*B*C*a^3*b^8 + 264*B*C*a^4*b^7 - 368*B*C*a^5
*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 - 512*B*C*a^9*b^2))/(2*b^8) + (a^3*(-(a + b)*(a - b
))^(1/2)*((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^2*b^14 - 16*B*a^3*b^
13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^5*b^11 + 32*C*a^6*b^
10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 - (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 +
C*a^2 - B*a*b)*(128*a*b^12 - 256*a^2*b^11 + 128*a^3*b^10))/(2*b^8*(b^7 - a^2*b^5)))*(A*b^2 + C*a^2 - B*a*b))/
(b^7 - a^2*b^5))*(A*b^2 + C*a^2 - B*a*b))/(b^7 - a^2*b^5) - (a^3*(-(a + b)*(a - b))^(1/2)*((tan(c/2 + (d*x)/2)
*(16*A^2*b^11 - 128*C^2*a^11 + 9*C^2*b^11 - 48*A^2*a*b^10 - 27*C^2*a*b^10 + 256*C^2*a^10*b + 112*A^2*a^2*b^9 -
208*A^2*a^3*b^8 + 256*A^2*a^4*b^7 - 256*A^2*a^5*b^6 + 256*A^2*a^6*b^5 - 128*A^2*a^7*b^4 + 16*B^2*a^2*b^9 - 48
*B^2*a^3*b^8 + 112*B^2*a^4*b^7 - 208*B^2*a^5*b^6 + 256*B^2*a^6*b^5 - 256*B^2*a^7*b^4 + 256*B^2*a^8*b^3 - 128*B
^2*a^9*b^2 + 51*C^2*a^2*b^9 - 81*C^2*a^3*b^8 + 136*C^2*a^4*b^7 - 216*C^2*a^5*b^6 + 256*C^2*a^6*b^5 - 256*C^2*a
^7*b^4 + 256*C^2*a^8*b^3 - 256*C^2*a^9*b^2 + 24*A*C*b^11 - 32*A*B*a*b^10 - 72*A*C*a*b^10 - 24*B*C*a*b^10 + 256
*B*C*a^10*b + 96*A*B*a^2*b^9 - 224*A*B*a^3*b^8 + 416*A*B*a^4*b^7 - 512*A*B*a^5*b^6 + 512*A*B*a^6*b^5 - 512*A*B
*a^7*b^4 + 256*A*B*a^8*b^3 + 152*A*C*a^2*b^9 - 264*A*C*a^3*b^8 + 368*A*C*a^4*b^7 - 464*A*C*a^5*b^6 + 512*A*C*a
^6*b^5 - 512*A*C*a^7*b^4 + 512*A*C*a^8*b^3 - 256*A*C*a^9*b^2 + 72*B*C*a^2*b^9 - 152*B*C*a^3*b^8 + 264*B*C*a^4*
b^7 - 368*B*C*a^5*b^6 + 464*B*C*a^6*b^5 - 512*B*C*a^7*b^4 + 512*B*C*a^8*b^3 - 512*B*C*a^9*b^2))/(2*b^8) - (a^3
*(-(a + b)*(a - b))^(1/2)*((16*A*b^16 + 12*C*b^16 + 16*A*a^2*b^14 - 48*A*a^3*b^13 + 32*A*a^4*b^12 + 16*B*a^2*b
^14 - 16*B*a^3*b^13 + 48*B*a^4*b^12 - 32*B*a^5*b^11 + 4*C*a^2*b^14 - 4*C*a^3*b^13 + 16*C*a^4*b^12 - 48*C*a^5*b
^11 + 32*C*a^6*b^10 - 16*A*a*b^15 - 16*B*a*b^15 - 12*C*a*b^15)/b^12 + (a^3*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b
))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(128*a*b^12 - 256*a^2*b^11 + 128*a^3*b^10))/(2*b^8*(b^7 - a^2*b^5)))*(A*b^2 +
C*a^2 - B*a*b))/(b^7 - a^2*b^5))*(A*b^2 + C*a^2 - B*a*b))/(b^7 - a^2*b^5)))*(-(a + b)*(a - b))^(1/2)*(A*b^2 +
C*a^2 - B*a*b)*2i)/(d*(b^7 - a^2*b^5))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________