\(\int \frac {\cos (c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{a+b \cos (c+d x)} \, dx\) [979]

Optimal result

Integrand size = 39, antiderivative size = 144 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\left (b^2 (2 A+C)-2 a (b B-a C)\right ) x}{2 b^3}-\frac {2 a \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} d}+\frac {(b B-a C) \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3128, 3102, 2814, 2738, 211} \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=-\frac {2 a \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d \sqrt {a-b} \sqrt {a+b}}+\frac {x \left (-\frac {2 a (b B-a C)}{b^2}+2 A+C\right )}{2 b}+\frac {(b B-a C) \sin (c+d x)}{b^2 d}+\frac {C \sin (c+d x) \cos (c+d x)}{2 b d} \]

[In]

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

[Out]

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

Rule 211

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

Rule 2738

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 2814

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 3102

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 3128

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*} \text {integral}& = \frac {C \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int \frac {a C+b (2 A+C) \cos (c+d x)+2 (b B-a C) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b} \\ & = \frac {(b B-a C) \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int \frac {a b C+\left (b^2 (2 A+C)-2 a (b B-a C)\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^2} \\ & = \frac {\left (b^2 (2 A+C)-2 a (b B-a C)\right ) x}{2 b^3}+\frac {(b B-a C) \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (a \left (A b^2-a (b B-a C)\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^3} \\ & = \frac {\left (b^2 (2 A+C)-2 a (b B-a C)\right ) x}{2 b^3}+\frac {(b B-a C) \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (2 a \left (A b^2-a (b B-a C)\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^3 d} \\ & = \frac {\left (b^2 (2 A+C)-2 a (b B-a C)\right ) x}{2 b^3}-\frac {2 a \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} d}+\frac {(b B-a C) \sin (c+d x)}{b^2 d}+\frac {C \cos (c+d x) \sin (c+d x)}{2 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {2 \left (2 A b^2-2 a b B+2 a^2 C+b^2 C\right ) (c+d x)+\frac {8 a \left (A b^2+a (-b B+a C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+4 b (b B-a C) \sin (c+d x)+b^2 C \sin (2 (c+d x))}{4 b^3 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (\left (B \,b^{2}-a b C -\frac {1}{2} C \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B \,b^{2}-a b C +\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}-2 B a b +2 a^{2} C +C \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 a \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(180\)
default	\(\frac {\frac {\frac {2 \left (\left (B \,b^{2}-a b C -\frac {1}{2} C \,b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (B \,b^{2}-a b C +\frac {1}{2} C \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\left (2 A \,b^{2}-2 B a b +2 a^{2} C +C \,b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 a \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{d}\)	\(180\)
risch	\(\frac {x A}{b}-\frac {x B a}{b^{2}}+\frac {x C \,a^{2}}{b^{3}}+\frac {C x}{2 b}-\frac {i {\mathrm e}^{i \left (d x +c \right )} B}{2 b d}+\frac {i {\mathrm e}^{i \left (d x +c \right )} C a}{2 b^{2} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} B}{2 b d}-\frac {i {\mathrm e}^{-i \left (d x +c \right )} C a}{2 b^{2} d}-\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d b}+\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {-i a^{2}+i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d b}-\frac {a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) B}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{3}}+\frac {\sin \left (2 d x +2 c \right ) C}{4 b d}\)	\(571\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.17 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\left [\frac {{\left (2 \, C a^{4} - 2 \, B a^{3} b + {\left (2 \, A - C\right )} a^{2} b^{2} + 2 \, B a b^{3} - {\left (2 \, A + C\right )} b^{4}\right )} d x - {\left (C a^{3} - B a^{2} b + A a b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - {\left (2 \, C a^{3} b - 2 \, B a^{2} b^{2} - 2 \, C a b^{3} + 2 \, B b^{4} - {\left (C a^{2} b^{2} - C b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} d}, \frac {{\left (2 \, C a^{4} - 2 \, B a^{3} b + {\left (2 \, A - C\right )} a^{2} b^{2} + 2 \, B a b^{3} - {\left (2 \, A + C\right )} b^{4}\right )} d x - 2 \, {\left (C a^{3} - B a^{2} b + A a b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (2 \, C a^{3} b - 2 \, B a^{2} b^{2} - 2 \, C a b^{3} + 2 \, B b^{4} - {\left (C a^{2} b^{2} - C b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} - b^{5}\right )} d}\right ] \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(cos(d*x+c)*(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 de

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.66 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\frac {\frac {{\left (2 \, C a^{2} - 2 \, B a b + 2 \, A b^{2} + C b^{2}\right )} {\left (d x + c\right )}}{b^{3}} + \frac {4 \, {\left (C a^{3} - B a^{2} b + A a b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{3}} - \frac {2 \, {\left (2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2} b^{2}}}{2 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 5594, normalized size of antiderivative = 38.85 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

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