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

Optimal. Leaf size=206 \[ \frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d \sqrt {a-b} \sqrt {a+b}}+\frac {\sin (c+d x) \left (3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right )}{3 b^3 d}+\frac {x \left (-2 a^3 C+2 a^2 b B-a b^2 (2 A+C)+b^3 B\right )}{2 b^4}+\frac {(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac {C \sin (c+d x) \cos ^2(c+d x)}{3 b d} \]

[Out]

1/2*(2*a^2*b*B+b^3*B-2*a^3*C-a*b^2*(2*A+C))*x/b^4+1/3*(3*A*b^2-3*B*a*b+3*C*a^2+2*C*b^2)*sin(d*x+c)/b^3/d+1/2*(
B*b-C*a)*cos(d*x+c)*sin(d*x+c)/b^2/d+1/3*C*cos(d*x+c)^2*sin(d*x+c)/b/d+2*a^2*(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^4/d/(a-b)^(1/2)/(a+b)^(1/2)

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Rubi [A]  time = 0.57, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ \frac {\sin (c+d x) \left (3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right )}{3 b^3 d}+\frac {2 a^2 \left (A b^2-a (b B-a C)\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d \sqrt {a-b} \sqrt {a+b}}+\frac {x \left (2 a^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right )}{2 b^4}+\frac {(b B-a C) \sin (c+d x) \cos (c+d x)}{2 b^2 d}+\frac {C \sin (c+d x) \cos ^2(c+d x)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*(2*A + C))*x)/(2*b^4) + (2*a^2*(A*b^2 - a*(b*B - a*C))*ArcTan[(Sqrt[a -
b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*
Sin[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*b^2*d) + (C*Cos[c + d*x]^2*Sin[c + d*x])/
(3*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 ^2(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 ^2(c+d x) \sin (c+d x)}{3 b d}+\frac {\int \frac {\cos (c+d x) \left (2 a C+b (3 A+2 C) \cos (c+d x)+3 (b B-a C) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b}\\ &=\frac {(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac {\int \frac {3 a (b B-a C)+b (3 b B+a C) \cos (c+d x)+2 \left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^2}\\ &=\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac {(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac {\int \frac {3 a b (b B-a C)+3 \left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3}\\ &=\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) x}{2 b^4}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac {(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac {\left (a^2 \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^4}\\ &=\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) x}{2 b^4}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac {(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {C \cos ^2(c+d x) \sin (c+d x)}{3 b d}+\frac {\left (2 a^2 \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^4 d}\\ &=\frac {\left (2 a^2 b B+b^3 B-2 a^3 C-a b^2 (2 A+C)\right ) x}{2 b^4}+\frac {2 a^2 \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^4 \sqrt {a+b} d}+\frac {\left (3 A b^2-3 a b B+3 a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b^3 d}+\frac {(b B-a C) \cos (c+d x) \sin (c+d x)}{2 b^2 d}+\frac {C \cos ^2(c+d x) \sin (c+d x)}{3 b d}\\ \end {align*}

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Mathematica [A]  time = 0.68, size = 179, normalized size = 0.87 \[ \frac {3 b \sin (c+d x) \left (4 a^2 C-4 a b B+4 A b^2+3 b^2 C\right )-\frac {24 a^2 \left (a (a C-b B)+A b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}-6 (c+d x) \left (2 a^3 C-2 a^2 b B+a b^2 (2 A+C)-b^3 B\right )+3 b^2 (b B-a C) \sin (2 (c+d x))+b^3 C \sin (3 (c+d x))}{12 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*(-2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*(2*A + C))*(c + d*x) - (24*a^2*(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] + 3*b*(4*A*b^2 - 4*a*b*B + 4*a^2*C + 3*b^2*C)*Sin
[c + d*x] + 3*b^2*(b*B - a*C)*Sin[2*(c + d*x)] + b^3*C*Sin[3*(c + d*x)])/(12*b^4*d)

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fricas [A]  time = 0.52, size = 599, normalized size = 2.91 \[ \left [-\frac {3 \, {\left (2 \, C a^{5} - 2 \, B a^{4} b + {\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} - {\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} d x + 3 \, {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (6 \, C a^{4} b - 6 \, B a^{3} b^{2} + 2 \, {\left (3 \, A - C\right )} a^{2} b^{3} + 6 \, B a b^{4} - 2 \, {\left (3 \, A + 2 \, C\right )} b^{5} + 2 \, {\left (C a^{2} b^{3} - C b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} - C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )} d}, -\frac {3 \, {\left (2 \, C a^{5} - 2 \, B a^{4} b + {\left (2 \, A - C\right )} a^{3} b^{2} + B a^{2} b^{3} - {\left (2 \, A + C\right )} a b^{4} + B b^{5}\right )} d x - 6 \, {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (6 \, C a^{4} b - 6 \, B a^{3} b^{2} + 2 \, {\left (3 \, A - C\right )} a^{2} b^{3} + 6 \, B a b^{4} - 2 \, {\left (3 \, A + 2 \, C\right )} b^{5} + 2 \, {\left (C a^{2} b^{3} - C b^{5}\right )} \cos \left (d x + c\right )^{2} - 3 \, {\left (C a^{3} b^{2} - B a^{2} b^{3} - C a b^{4} + B b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (a^{2} b^{4} - b^{6}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(3*(2*C*a^5 - 2*B*a^4*b + (2*A - C)*a^3*b^2 + B*a^2*b^3 - (2*A + C)*a*b^4 + B*b^5)*d*x + 3*(C*a^4 - B*a^
3*b + A*a^2*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)) - (6*C*a^4*b
- 6*B*a^3*b^2 + 2*(3*A - C)*a^2*b^3 + 6*B*a*b^4 - 2*(3*A + 2*C)*b^5 + 2*(C*a^2*b^3 - C*b^5)*cos(d*x + c)^2 -
3*(C*a^3*b^2 - B*a^2*b^3 - C*a*b^4 + B*b^5)*cos(d*x + c))*sin(d*x + c))/((a^2*b^4 - b^6)*d), -1/6*(3*(2*C*a^5
- 2*B*a^4*b + (2*A - C)*a^3*b^2 + B*a^2*b^3 - (2*A + C)*a*b^4 + B*b^5)*d*x - 6*(C*a^4 - B*a^3*b + A*a^2*b^2)*s
qrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqrt(a^2 - b^2)*sin(d*x + c))) - (6*C*a^4*b - 6*B*a^3*b^2 + 2*(3*
A - C)*a^2*b^3 + 6*B*a*b^4 - 2*(3*A + 2*C)*b^5 + 2*(C*a^2*b^3 - C*b^5)*cos(d*x + c)^2 - 3*(C*a^3*b^2 - B*a^2*b
^3 - C*a*b^4 + B*b^5)*cos(d*x + c))*sin(d*x + c))/((a^2*b^4 - b^6)*d)]

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giac [B]  time = 0.22, size = 424, normalized size = 2.06 \[ -\frac {\frac {3 \, {\left (2 \, C a^{3} - 2 \, B a^{2} b + 2 \, A a b^{2} + C a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{b^{4}} + \frac {12 \, {\left (C a^{4} - B a^{3} b + A a^{2} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{4}} - \frac {2 \, {\left (6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, B a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, C a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{3}}}{6 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*(2*C*a^3 - 2*B*a^2*b + 2*A*a*b^2 + C*a*b^2 - B*b^3)*(d*x + c)/b^4 + 12*(C*a^4 - B*a^3*b + A*a^2*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))/s
qrt(a^2 - b^2)))/(sqrt(a^2 - b^2)*b^4) - 2*(6*C*a^2*tan(1/2*d*x + 1/2*c)^5 - 6*B*a*b*tan(1/2*d*x + 1/2*c)^5 +
3*C*a*b*tan(1/2*d*x + 1/2*c)^5 + 6*A*b^2*tan(1/2*d*x + 1/2*c)^5 - 3*B*b^2*tan(1/2*d*x + 1/2*c)^5 + 6*C*b^2*tan
(1/2*d*x + 1/2*c)^5 + 12*C*a^2*tan(1/2*d*x + 1/2*c)^3 - 12*B*a*b*tan(1/2*d*x + 1/2*c)^3 + 12*A*b^2*tan(1/2*d*x
+ 1/2*c)^3 + 4*C*b^2*tan(1/2*d*x + 1/2*c)^3 + 6*C*a^2*tan(1/2*d*x + 1/2*c) - 6*B*a*b*tan(1/2*d*x + 1/2*c) - 3
*C*a*b*tan(1/2*d*x + 1/2*c) + 6*A*b^2*tan(1/2*d*x + 1/2*c) + 3*B*b^2*tan(1/2*d*x + 1/2*c) + 6*C*b^2*tan(1/2*d*
x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 + 1)^3*b^3))/d

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maple [B]  time = 0.13, size = 814, normalized size = 3.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*a^2/b^2/((a-b)*(a+b))^(1/2)*arctan(tan(1/2*d*x+1/2*c)*(a-b)/((a-b)*(a+b))^(1/2))*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))*B+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))*C+2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*A-2/d/b^2/(1+ta
n(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*B*a-1/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*B+2/d/b^3
/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*C*a^2+1/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5
*C*a+2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^5*C+4/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*
c)^3*A-4/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*B*a+4/d/b^3/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*
d*x+1/2*c)^3*C*a^2+4/3/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)^3*C+2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*
tan(1/2*d*x+1/2*c)*A-2/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*B*a+2/d/b^3/(1+tan(1/2*d*x+1/2*c)^2
)^3*tan(1/2*d*x+1/2*c)*C*a^2+2/d/b/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*C+1/d/b/(1+tan(1/2*d*x+1/2*c)
^2)^3*tan(1/2*d*x+1/2*c)*B-1/d/b^2/(1+tan(1/2*d*x+1/2*c)^2)^3*tan(1/2*d*x+1/2*c)*C*a-2/d/b^2*arctan(tan(1/2*d*
x+1/2*c))*A*a+2/d/b^3*arctan(tan(1/2*d*x+1/2*c))*a^2*B+1/d/b*arctan(tan(1/2*d*x+1/2*c))*B-2/d/b^4*arctan(tan(1
/2*d*x+1/2*c))*C*a^3-1/d/b^2*arctan(tan(1/2*d*x+1/2*c))*C*a

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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)^2*(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 = 9.56, size = 7119, normalized size = 34.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^2*(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)^5*(2*A*b^2 - B*b^2 + 2*C*a^2 + 2*C*b^2 - 2*B*a*b + C*a*b))/b^3 + (4*tan(c/2 + (d*x)/2)^3*
(3*A*b^2 + 3*C*a^2 + C*b^2 - 3*B*a*b))/(3*b^3) + (tan(c/2 + (d*x)/2)*(2*A*b^2 + B*b^2 + 2*C*a^2 + 2*C*b^2 - 2*
B*a*b - C*a*b))/b^3)/(d*(3*tan(c/2 + (d*x)/2)^2 + 3*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^6 + 1)) + (atan(
((((8*tan(c/2 + (d*x)/2)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a^8*b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 +
16*A^2*a^4*b^5 - 8*A^2*a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B^2*a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6
*b^3 - 8*B^2*a^7*b^2 + C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 13*C^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*
a^7*b^2 - 4*A*B*a*b^8 - 2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B*a^2*b^7 - 20*A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B
*a^5*b^4 + 16*A*B*a^6*b^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^4*b^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3
- 16*A*C*a^7*b^2 + 6*B*C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 32*B*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*
a^7*b^2))/b^6 + (((8*(4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^13 - 2*B*a^2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C
*a^2*b^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*a*b^12 + 2*B*a*b^12 + 2*C*a*b^12))/b^9 - (8*tan(c/2
+ (d*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b
*1i))/b^10)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i))/b^4)*((B*b^3*1i)/2 - C*a^3*1i
- b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i)*1i)/b^4 + (((8*tan(c/2 + (d*x)/2)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8
+ 16*C^2*a^8*b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2*a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3
*b^6 + 16*B^2*a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 + C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*
a^4*b^5 - 13*C^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8 - 2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B
*a^2*b^7 - 20*A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*b^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6
+ 20*A*C*a^4*b^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B*C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*
a^4*b^5 - 32*B*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*a^7*b^2))/b^6 - (((8*(4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^1
3 - 2*B*a^2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C*a^2*b^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*
a*b^12 + 2*B*a*b^12 + 2*C*a*b^12))/b^9 + (8*tan(c/2 + (d*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*((B*b^3*1i)
/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i))/b^10)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1
i)/2) + B*a^2*b*1i))/b^4)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i)*1i)/b^4)/((16*(4*
C^3*a^11 - 6*C^3*a^10*b - 4*A^3*a^4*b^7 + 4*A^3*a^5*b^6 + B^3*a^3*b^8 - 2*B^3*a^4*b^7 + 5*B^3*a^5*b^6 - 6*B^3*
a^6*b^5 + 6*B^3*a^7*b^4 - 4*B^3*a^8*b^3 - C^3*a^6*b^5 + 2*C^3*a^7*b^4 - 5*C^3*a^8*b^3 + 6*C^3*a^9*b^2 - 12*B*C
^2*a^10*b - A*B^2*a^2*b^9 + 2*A*B^2*a^3*b^8 - 9*A*B^2*a^4*b^7 + 12*A*B^2*a^5*b^6 - 16*A*B^2*a^6*b^5 + 12*A*B^2
*a^7*b^4 + 4*A^2*B*a^3*b^8 - 6*A^2*B*a^4*b^7 + 14*A^2*B*a^5*b^6 - 12*A^2*B*a^6*b^5 - A*C^2*a^4*b^7 + 2*A*C^2*a
^5*b^6 - 9*A*C^2*a^6*b^5 + 12*A*C^2*a^7*b^4 - 16*A*C^2*a^8*b^3 + 12*A*C^2*a^9*b^2 - 4*A^2*C*a^4*b^7 + 6*A^2*C*
a^5*b^6 - 14*A^2*C*a^6*b^5 + 12*A^2*C*a^7*b^4 + 3*B*C^2*a^5*b^6 - 6*B*C^2*a^6*b^5 + 15*B*C^2*a^7*b^4 - 18*B*C^
2*a^8*b^3 + 18*B*C^2*a^9*b^2 - 3*B^2*C*a^4*b^7 + 6*B^2*C*a^5*b^6 - 15*B^2*C*a^6*b^5 + 18*B^2*C*a^7*b^4 - 18*B^
2*C*a^8*b^3 + 12*B^2*C*a^9*b^2 + 2*A*B*C*a^3*b^8 - 4*A*B*C*a^4*b^7 + 18*A*B*C*a^5*b^6 - 24*A*B*C*a^6*b^5 + 32*
A*B*C*a^7*b^4 - 24*A*B*C*a^8*b^3))/b^9 + (((8*tan(c/2 + (d*x)/2)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a
^8*b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2*a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B
^2*a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 + C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 1
3*C^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8 - 2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B*a^2*b^7 -
20*A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*b^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^
4*b^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B*C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 3
2*B*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*a^7*b^2))/b^6 + (((8*(4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^13 - 2*B*a^2
*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C*a^2*b^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*a*b^12 + 2*
B*a*b^12 + 2*C*a*b^12))/b^9 - (8*tan(c/2 + (d*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*((B*b^3*1i)/2 - C*a^3*
1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i))/b^10)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a
^2*b*1i))/b^4)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i))/b^4 - (((8*tan(c/2 + (d*x)/
2)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a^8*b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2
*a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B^2*a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 +
C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 13*C^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8
- 2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B*a^2*b^7 - 20*A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*
b^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^4*b^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B
*C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 32*B*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*a^7*b^2))/b^6 - (((8*(
4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^13 - 2*B*a^2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C*a^2*b^11 + 2*C*a^3*b^
10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*a*b^12 + 2*B*a*b^12 + 2*C*a*b^12))/b^9 + (8*tan(c/2 + (d*x)/2)*(8*a*b^10
- 16*a^2*b^9 + 8*a^3*b^8)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i))/b^10)*((B*b^3*1i
)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i))/b^4)*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1
i)/2) + B*a^2*b*1i))/b^4))*((B*b^3*1i)/2 - C*a^3*1i - b^2*(A*a*1i + (C*a*1i)/2) + B*a^2*b*1i)*2i)/(b^4*d) + (a
^2*atan(((a^2*(-(a + b)*(a - b))^(1/2)*((8*tan(c/2 + (d*x)/2)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a^8*
b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2*a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B^2*
a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 + C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 13*C
^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8 - 2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B*a^2*b^7 - 20*
A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*b^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^4*b
^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B*C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 32*B
*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*a^7*b^2))/b^6 + (a^2*(-(a + b)*(a - b))^(1/2)*((8*(4*A*a^3*b^10 - 8*A*a^2
*b^11 - 2*B*b^13 - 2*B*a^2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C*a^2*b^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C
*a^5*b^8 + 4*A*a*b^12 + 2*B*a*b^12 + 2*C*a*b^12))/b^9 - (8*a^2*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*
b^2 + C*a^2 - B*a*b)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/(b^6*(b^6 - a^2*b^4)))*(A*b^2 + C*a^2 - B*a*b))/(b^6
- a^2*b^4))*(A*b^2 + C*a^2 - B*a*b)*1i)/(b^6 - a^2*b^4) + (a^2*(-(a + b)*(a - b))^(1/2)*((8*tan(c/2 + (d*x)/2
)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a^8*b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2*
a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B^2*a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 +
C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 13*C^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8 -
2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B*a^2*b^7 - 20*A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*b
^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^4*b^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B*
C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 32*B*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*a^7*b^2))/b^6 - (a^2*(-
(a + b)*(a - b))^(1/2)*((8*(4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^13 - 2*B*a^2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9
- 2*C*a^2*b^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*a*b^12 + 2*B*a*b^12 + 2*C*a*b^12))/b^9 + (8*a
^2*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/(b
^6*(b^6 - a^2*b^4)))*(A*b^2 + C*a^2 - B*a*b))/(b^6 - a^2*b^4))*(A*b^2 + C*a^2 - B*a*b)*1i)/(b^6 - a^2*b^4))/((
16*(4*C^3*a^11 - 6*C^3*a^10*b - 4*A^3*a^4*b^7 + 4*A^3*a^5*b^6 + B^3*a^3*b^8 - 2*B^3*a^4*b^7 + 5*B^3*a^5*b^6 -
6*B^3*a^6*b^5 + 6*B^3*a^7*b^4 - 4*B^3*a^8*b^3 - C^3*a^6*b^5 + 2*C^3*a^7*b^4 - 5*C^3*a^8*b^3 + 6*C^3*a^9*b^2 -
12*B*C^2*a^10*b - A*B^2*a^2*b^9 + 2*A*B^2*a^3*b^8 - 9*A*B^2*a^4*b^7 + 12*A*B^2*a^5*b^6 - 16*A*B^2*a^6*b^5 + 12
*A*B^2*a^7*b^4 + 4*A^2*B*a^3*b^8 - 6*A^2*B*a^4*b^7 + 14*A^2*B*a^5*b^6 - 12*A^2*B*a^6*b^5 - A*C^2*a^4*b^7 + 2*A
*C^2*a^5*b^6 - 9*A*C^2*a^6*b^5 + 12*A*C^2*a^7*b^4 - 16*A*C^2*a^8*b^3 + 12*A*C^2*a^9*b^2 - 4*A^2*C*a^4*b^7 + 6*
A^2*C*a^5*b^6 - 14*A^2*C*a^6*b^5 + 12*A^2*C*a^7*b^4 + 3*B*C^2*a^5*b^6 - 6*B*C^2*a^6*b^5 + 15*B*C^2*a^7*b^4 - 1
8*B*C^2*a^8*b^3 + 18*B*C^2*a^9*b^2 - 3*B^2*C*a^4*b^7 + 6*B^2*C*a^5*b^6 - 15*B^2*C*a^6*b^5 + 18*B^2*C*a^7*b^4 -
18*B^2*C*a^8*b^3 + 12*B^2*C*a^9*b^2 + 2*A*B*C*a^3*b^8 - 4*A*B*C*a^4*b^7 + 18*A*B*C*a^5*b^6 - 24*A*B*C*a^6*b^5
+ 32*A*B*C*a^7*b^4 - 24*A*B*C*a^8*b^3))/b^9 + (a^2*(-(a + b)*(a - b))^(1/2)*((8*tan(c/2 + (d*x)/2)*(B^2*b^9 -
8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a^8*b + 4*A^2*a^2*b^7 - 12*A^2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2*a^5*b^4 + 7*
B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B^2*a^4*b^5 - 16*B^2*a^5*b^4 + 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 + C^2*a^2*b^7
- 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 13*C^2*a^5*b^4 + 16*C^2*a^6*b^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8 - 2*B*C*a*b^8
+ 16*B*C*a^8*b + 12*A*B*a^2*b^7 - 20*A*B*a^3*b^6 + 28*A*B*a^4*b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*b^3 + 4*A*C*a
^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^4*b^5 - 28*A*C*a^5*b^4 + 32*A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B*C*a^2*b^7 -
14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 32*B*C*a^5*b^4 + 32*B*C*a^6*b^3 - 32*B*C*a^7*b^2))/b^6 + (a^2*(-(a + b)*(a -
b))^(1/2)*((8*(4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^13 - 2*B*a^2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C*a^2*b
^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*a*b^12 + 2*B*a*b^12 + 2*C*a*b^12))/b^9 - (8*a^2*tan(c/2 +
(d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/(b^6*(b^6 - a^
2*b^4)))*(A*b^2 + C*a^2 - B*a*b))/(b^6 - a^2*b^4))*(A*b^2 + C*a^2 - B*a*b))/(b^6 - a^2*b^4) - (a^2*(-(a + b)*(
a - b))^(1/2)*((8*tan(c/2 + (d*x)/2)*(B^2*b^9 - 8*C^2*a^9 - 3*B^2*a*b^8 + 16*C^2*a^8*b + 4*A^2*a^2*b^7 - 12*A^
2*a^3*b^6 + 16*A^2*a^4*b^5 - 8*A^2*a^5*b^4 + 7*B^2*a^2*b^7 - 13*B^2*a^3*b^6 + 16*B^2*a^4*b^5 - 16*B^2*a^5*b^4
+ 16*B^2*a^6*b^3 - 8*B^2*a^7*b^2 + C^2*a^2*b^7 - 3*C^2*a^3*b^6 + 7*C^2*a^4*b^5 - 13*C^2*a^5*b^4 + 16*C^2*a^6*b
^3 - 16*C^2*a^7*b^2 - 4*A*B*a*b^8 - 2*B*C*a*b^8 + 16*B*C*a^8*b + 12*A*B*a^2*b^7 - 20*A*B*a^3*b^6 + 28*A*B*a^4*
b^5 - 32*A*B*a^5*b^4 + 16*A*B*a^6*b^3 + 4*A*C*a^2*b^7 - 12*A*C*a^3*b^6 + 20*A*C*a^4*b^5 - 28*A*C*a^5*b^4 + 32*
A*C*a^6*b^3 - 16*A*C*a^7*b^2 + 6*B*C*a^2*b^7 - 14*B*C*a^3*b^6 + 26*B*C*a^4*b^5 - 32*B*C*a^5*b^4 + 32*B*C*a^6*b
^3 - 32*B*C*a^7*b^2))/b^6 - (a^2*(-(a + b)*(a - b))^(1/2)*((8*(4*A*a^3*b^10 - 8*A*a^2*b^11 - 2*B*b^13 - 2*B*a^
2*b^11 + 6*B*a^3*b^10 - 4*B*a^4*b^9 - 2*C*a^2*b^11 + 2*C*a^3*b^10 - 6*C*a^4*b^9 + 4*C*a^5*b^8 + 4*A*a*b^12 + 2
*B*a*b^12 + 2*C*a*b^12))/b^9 + (8*a^2*tan(c/2 + (d*x)/2)*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*(8*a
*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/(b^6*(b^6 - a^2*b^4)))*(A*b^2 + C*a^2 - B*a*b))/(b^6 - a^2*b^4))*(A*b^2 + C*a
^2 - B*a*b))/(b^6 - a^2*b^4)))*(-(a + b)*(a - b))^(1/2)*(A*b^2 + C*a^2 - B*a*b)*2i)/(d*(b^6 - a^2*b^4))

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

[Out]

Timed out

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