\(\int \frac {\cos (c+d x) (A+B \cos (c+d x)+C \cos ^2(c+d x))}{(a+b \cos (c+d x))^2} \, dx\) [988]

Optimal result

Integrand size = 39, antiderivative size = 168 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {(b B-2 a C) x}{b^3}-\frac {2 \left (A b^4+a^3 b B-2 a b^3 B-2 a^4 C+3 a^2 b^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac {C \sin (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3110, 3102, 2814, 2738, 211} \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {a \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{b^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))}-\frac {2 \left (-2 a^4 C+a^3 b B+3 a^2 b^2 C-2 a b^3 B+A b^4\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac {x (b B-2 a C)}{b^3}+\frac {C \sin (c+d x)}{b^2 d} \]

[In]

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

[Out]

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

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 3110

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

Rubi steps \begin{align*} \text {integral}& = \frac {a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {-b \left (A b^2-a (b B-a C)\right )+\left (a^2-b^2\right ) (b B-a C) \cos (c+d x)+b \left (a^2-b^2\right ) C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{b^2 \left (a^2-b^2\right )} \\ & = \frac {C \sin (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac {\int \frac {-b^2 \left (A b^2-a (b B-a C)\right )+b \left (a^2-b^2\right ) (b B-2 a C) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )} \\ & = \frac {(b B-2 a C) x}{b^3}+\frac {C \sin (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^3 \left (a^2-b^2\right )} \\ & = \frac {(b B-2 a C) x}{b^3}+\frac {C \sin (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac {\left (2 \left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\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 \left (a^2-b^2\right ) d} \\ & = \frac {(b B-2 a C) x}{b^3}-\frac {2 \left (A b^4+a^3 b B-2 a b^3 B-2 a^4 C+3 a^2 b^2 C\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} b^3 (a+b)^{3/2} d}+\frac {C \sin (c+d x)}{b^2 d}+\frac {a \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{b^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\frac {(b B-2 a C) (c+d x)-\frac {2 \left (A b^4+a \left (a^2 b B-2 b^3 B-2 a^3 C+3 a b^2 C\right )\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+b C \sin (c+d x)+\frac {a b \left (A b^2+a (-b B+a C)\right ) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))}}{b^3 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.32

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -2 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 \left (-\frac {a \left (A \,b^{2}-B a b +a^{2} C \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {\left (A \,b^{4}+B \,a^{3} b -2 B a \,b^{3}-2 a^{4} C +3 C \,a^{2} b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3}}}{d}\)	\(221\)
default	\(\frac {\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -2 C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{3}}-\frac {2 \left (-\frac {a \left (A \,b^{2}-B a b +a^{2} C \right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}+\frac {\left (A \,b^{4}+B \,a^{3} b -2 B a \,b^{3}-2 a^{4} C +3 C \,a^{2} b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a -b \right ) \left (a +b \right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3}}}{d}\)	\(221\)
risch	\(\frac {x B}{b^{2}}-\frac {2 a C x}{b^{3}}-\frac {i C \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{2} d}+\frac {i C \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{2} d}-\frac {2 i a \left (A \,b^{2}-B a b +a^{2} C \right ) \left (a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}{b^{3} \left (-a^{2}+b^{2}\right ) d \left ({\mathrm e}^{2 i \left (d x +c \right )} b +2 a \,{\mathrm e}^{i \left (d x +c \right )}+b \right )}-\frac {b \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}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {\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 \,a^{3}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{2}}+\frac {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 a}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}+\frac {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 ) a^{4} C}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {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 \,a^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}+\frac {b \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}}\, \left (a +b \right ) \left (a -b \right ) d}+\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 ) B}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{2}}-\frac {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 a}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d}-\frac {2 a^{4} \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}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {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 \,a^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}\)	\(979\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (162) = 324\).

Time = 0.35 (sec) , antiderivative size = 830, normalized size of antiderivative = 4.94 \[ \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))^2} \, dx=\left [-\frac {2 \, {\left (2 \, C a^{5} b - B a^{4} b^{2} - 4 \, C a^{3} b^{3} + 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (2 \, C a^{6} - B a^{5} b - 4 \, C a^{4} b^{2} + 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5}\right )} d x + {\left (2 \, C a^{5} - B a^{4} b - 3 \, C a^{3} b^{2} + 2 \, B a^{2} b^{3} - A a b^{4} + {\left (2 \, C a^{4} b - B a^{3} b^{2} - 3 \, C a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\right )} \cos \left (d x + c\right )\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 ) - 2 \, {\left (2 \, C a^{5} b - B a^{4} b^{2} + {\left (A - 3 \, C\right )} a^{3} b^{3} + B a^{2} b^{4} - {\left (A - C\right )} a b^{5} + {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} d\right )}}, -\frac {{\left (2 \, C a^{5} b - B a^{4} b^{2} - 4 \, C a^{3} b^{3} + 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} d x \cos \left (d x + c\right ) + {\left (2 \, C a^{6} - B a^{5} b - 4 \, C a^{4} b^{2} + 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5}\right )} d x - {\left (2 \, C a^{5} - B a^{4} b - 3 \, C a^{3} b^{2} + 2 \, B a^{2} b^{3} - A a b^{4} + {\left (2 \, C a^{4} b - B a^{3} b^{2} - 3 \, C a^{2} b^{3} + 2 \, B a b^{4} - A b^{5}\right )} \cos \left (d x + c\right )\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^{5} b - B a^{4} b^{2} + {\left (A - 3 \, C\right )} a^{3} b^{3} + B a^{2} b^{4} - {\left (A - C\right )} a b^{5} + {\left (C a^{4} b^{2} - 2 \, C a^{2} b^{4} + C b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\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))^2,x, algorithm="fricas")

[Out]

[-1/2*(2*(2*C*a^5*b - B*a^4*b^2 - 4*C*a^3*b^3 + 2*B*a^2*b^4 + 2*C*a*b^5 - B*b^6)*d*x*cos(d*x + c) + 2*(2*C*a^6
- B*a^5*b - 4*C*a^4*b^2 + 2*B*a^3*b^3 + 2*C*a^2*b^4 - B*a*b^5)*d*x + (2*C*a^5 - B*a^4*b - 3*C*a^3*b^2 + 2*B*a
^2*b^3 - A*a*b^4 + (2*C*a^4*b - B*a^3*b^2 - 3*C*a^2*b^3 + 2*B*a*b^4 - A*b^5)*cos(d*x + c))*sqrt(-a^2 + b^2)*lo
g((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*(2*C*a^5*b - B*a^4*b^2 + (A - 3*C)*a^3*b^3 +
B*a^2*b^4 - (A - C)*a*b^5 + (C*a^4*b^2 - 2*C*a^2*b^4 + C*b^6)*cos(d*x + c))*sin(d*x + c))/((a^4*b^4 - 2*a^2*b
^6 + b^8)*d*cos(d*x + c) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*d), -((2*C*a^5*b - B*a^4*b^2 - 4*C*a^3*b^3 + 2*B*a^2*
b^4 + 2*C*a*b^5 - B*b^6)*d*x*cos(d*x + c) + (2*C*a^6 - B*a^5*b - 4*C*a^4*b^2 + 2*B*a^3*b^3 + 2*C*a^2*b^4 - B*a
*b^5)*d*x - (2*C*a^5 - B*a^4*b - 3*C*a^3*b^2 + 2*B*a^2*b^3 - A*a*b^4 + (2*C*a^4*b - B*a^3*b^2 - 3*C*a^2*b^3 +
2*B*a*b^4 - A*b^5)*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)))
- (2*C*a^5*b - B*a^4*b^2 + (A - 3*C)*a^3*b^3 + B*a^2*b^4 - (A - C)*a*b^5 + (C*a^4*b^2 - 2*C*a^2*b^4 + C*b^6)*c
os(d*x + c))*sin(d*x + c))/((a^4*b^4 - 2*a^2*b^6 + b^8)*d*cos(d*x + c) + (a^5*b^3 - 2*a^3*b^5 + a*b^7)*d)]

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate(cos(d*x+c)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^2,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1249 vs. \(2 (162) = 324\).

Time = 0.45 (sec) , antiderivative size = 1249, normalized size of antiderivative = 7.43 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((4*C*a^6*b^2 - 2*B*a^5*b^3 - 2*C*a^5*b^3 + B*a^4*b^4 - 9*C*a^4*b^4 + 5*B*a^3*b^5 + 4*C*a^3*b^5 - A*a^2*b^6 -
2*B*a^2*b^6 + 5*C*a^2*b^6 - 3*B*a*b^7 - 2*C*a*b^7 + A*b^8 + B*b^8 + 2*C*a^3*abs(-a^2*b^3 + b^5) - B*a^2*b*abs(
-a^2*b^3 + b^5) - C*a^2*b*abs(-a^2*b^3 + b^5) + B*a*b^2*abs(-a^2*b^3 + b^5) - 2*C*a*b^2*abs(-a^2*b^3 + b^5) -
A*b^3*abs(-a^2*b^3 + b^5) + B*b^3*abs(-a^2*b^3 + b^5))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*sqrt(1/2)*
tan(1/2*d*x + 1/2*c)/sqrt((2*a^3*b^2 - 2*a*b^4 + sqrt(-4*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5)*(a^3*b^2 - a^2*b^3
- a*b^4 + b^5) + 4*(a^3*b^2 - a*b^4)^2))/(a^3*b^2 - a^2*b^3 - a*b^4 + b^5))))/(a^3*b^2*abs(-a^2*b^3 + b^5) - a
*b^4*abs(-a^2*b^3 + b^5) + (a^2*b^3 - b^5)^2) + (sqrt(a^2 - b^2)*A*b^3*abs(-a^2*b^3 + b^5)*abs(-a + b) + (a^2*
b - a*b^2 - b^3)*sqrt(a^2 - b^2)*B*abs(-a^2*b^3 + b^5)*abs(-a + b) - (2*a^3 - a^2*b - 2*a*b^2)*sqrt(a^2 - b^2)
*C*abs(-a^2*b^3 + b^5)*abs(-a + b) - (a^2*b^6 - b^8)*sqrt(a^2 - b^2)*A*abs(-a + b) - (2*a^5*b^3 - a^4*b^4 - 5*
a^3*b^5 + 2*a^2*b^6 + 3*a*b^7 - b^8)*sqrt(a^2 - b^2)*B*abs(-a + b) + (4*a^6*b^2 - 2*a^5*b^3 - 9*a^4*b^4 + 4*a^
3*b^5 + 5*a^2*b^6 - 2*a*b^7)*sqrt(a^2 - b^2)*C*abs(-a + b))*(pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*sqrt(
1/2)*tan(1/2*d*x + 1/2*c)/sqrt((2*a^3*b^2 - 2*a*b^4 - sqrt(-4*(a^3*b^2 + a^2*b^3 - a*b^4 - b^5)*(a^3*b^2 - a^2
*b^3 - a*b^4 + b^5) + 4*(a^3*b^2 - a*b^4)^2))/(a^3*b^2 - a^2*b^3 - a*b^4 + b^5))))/((a^2*b^3 - b^5)^2*(a^2 - 2
*a*b + b^2) - (a^5*b^2 - 2*a^4*b^3 + 2*a^2*b^5 - a*b^6)*abs(-a^2*b^3 + b^5)) + 2*(2*C*a^3*tan(1/2*d*x + 1/2*c)
^3 - B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + A*a*b^2*tan(1/2*d*x + 1/2*c)^3 - C*a*b^
2*tan(1/2*d*x + 1/2*c)^3 + C*b^3*tan(1/2*d*x + 1/2*c)^3 + 2*C*a^3*tan(1/2*d*x + 1/2*c) - B*a^2*b*tan(1/2*d*x +
1/2*c) + C*a^2*b*tan(1/2*d*x + 1/2*c) + A*a*b^2*tan(1/2*d*x + 1/2*c) - C*a*b^2*tan(1/2*d*x + 1/2*c) - C*b^3*t
an(1/2*d*x + 1/2*c))/((a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 + 2*a*tan(1/2*d*x + 1/2*c)^2 + a +
b)*(a^2*b^2 - b^4)))/d

Mupad [B] (verification not implemented)

Time = 7.51 (sec) , antiderivative size = 3816, normalized size of antiderivative = 22.71 \[ \int \frac {\cos (c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*tan(c/2 + (d*x)/2)*(2*C*a^3 - C*b^3 + A*a*b^2 - B*a^2*b - C*a*b^2 + C*a^2*b))/(b^2*(a + b)*(a - b)) + (2*t
an(c/2 + (d*x)/2)^3*(2*C*a^3 + C*b^3 + A*a*b^2 - B*a^2*b - C*a*b^2 - C*a^2*b))/(b^2*(a + b)*(a - b)))/(d*(a +
b + tan(c/2 + (d*x)/2)^4*(a - b) + 2*a*tan(c/2 + (d*x)/2)^2)) + (log(tan(c/2 + (d*x)/2) + 1i)*(B*b - 2*C*a)*1i
)/(b^3*d) - (log(tan(c/2 + (d*x)/2) - 1i)*(B*b*1i - C*a*2i))/(b^3*d) - (atan((((-(a + b)^3*(a - b)^3)^(1/2)*((
32*tan(c/2 + (d*x)/2)*(A^2*b^8 + B^2*b^8 + 8*C^2*a^8 - 2*B^2*a*b^7 - 8*C^2*a^7*b + 3*B^2*a^2*b^6 + 4*B^2*a^3*b
^5 - 5*B^2*a^4*b^4 - 2*B^2*a^5*b^3 + 2*B^2*a^6*b^2 + 4*C^2*a^2*b^6 - 8*C^2*a^3*b^5 + 5*C^2*a^4*b^4 + 16*C^2*a^
5*b^3 - 16*C^2*a^6*b^2 - 4*A*B*a*b^7 - 4*B*C*a*b^7 - 8*B*C*a^7*b + 2*A*B*a^3*b^5 + 6*A*C*a^2*b^6 - 4*A*C*a^4*b
^4 + 8*B*C*a^2*b^6 - 8*B*C*a^3*b^5 - 16*B*C*a^4*b^4 + 18*B*C*a^5*b^3 + 8*B*C*a^6*b^2))/(a*b^6 + b^7 - a^2*b^5
- a^3*b^4) + (((32*(A*a^2*b^10 - B*b^12 - A*b^12 - A*a^3*b^9 + B*a^2*b^10 - 3*B*a^3*b^9 + B*a^5*b^7 - 3*C*a^2*
b^10 - 3*C*a^3*b^9 + 5*C*a^4*b^8 + C*a^5*b^7 - 2*C*a^6*b^6 + A*a*b^11 + 2*B*a*b^11 + 2*C*a*b^11))/(a*b^8 + b^9
- a^2*b^7 - a^3*b^6) - (32*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2
*B*a*b^3 + B*a^3*b)*(2*a*b^11 - 2*a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 - 2*a^6*b^6))/((a*b^6 + b^7 - a
^2*b^5 - a^3*b^4)*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^4 - 2*C*a^4 + 3*
C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(A*b^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*
B*a*b^3 + B*a^3*b)*1i)/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3) + ((-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 +
(d*x)/2)*(A^2*b^8 + B^2*b^8 + 8*C^2*a^8 - 2*B^2*a*b^7 - 8*C^2*a^7*b + 3*B^2*a^2*b^6 + 4*B^2*a^3*b^5 - 5*B^2*a^
4*b^4 - 2*B^2*a^5*b^3 + 2*B^2*a^6*b^2 + 4*C^2*a^2*b^6 - 8*C^2*a^3*b^5 + 5*C^2*a^4*b^4 + 16*C^2*a^5*b^3 - 16*C^
2*a^6*b^2 - 4*A*B*a*b^7 - 4*B*C*a*b^7 - 8*B*C*a^7*b + 2*A*B*a^3*b^5 + 6*A*C*a^2*b^6 - 4*A*C*a^4*b^4 + 8*B*C*a^
2*b^6 - 8*B*C*a^3*b^5 - 16*B*C*a^4*b^4 + 18*B*C*a^5*b^3 + 8*B*C*a^6*b^2))/(a*b^6 + b^7 - a^2*b^5 - a^3*b^4) -
(((32*(A*a^2*b^10 - B*b^12 - A*b^12 - A*a^3*b^9 + B*a^2*b^10 - 3*B*a^3*b^9 + B*a^5*b^7 - 3*C*a^2*b^10 - 3*C*a^
3*b^9 + 5*C*a^4*b^8 + C*a^5*b^7 - 2*C*a^6*b^6 + A*a*b^11 + 2*B*a*b^11 + 2*C*a*b^11))/(a*b^8 + b^9 - a^2*b^7 -
a^3*b^6) + (32*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*
a^3*b)*(2*a*b^11 - 2*a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 - 2*a^6*b^6))/((a*b^6 + b^7 - a^2*b^5 - a^3*
b^4)*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2
*B*a*b^3 + B*a^3*b))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(A*b^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*a
^3*b)*1i)/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))/((64*(8*C^3*a^8 + A*B^2*b^8 - A^2*B*b^8 - 2*B^3*a*b^7 - 4*C
^3*a^7*b - 2*B^3*a^2*b^6 + 3*B^3*a^3*b^5 + B^3*a^4*b^4 - B^3*a^5*b^3 + 12*C^3*a^4*b^4 + 6*C^3*a^5*b^3 - 20*C^3
*a^6*b^2 + 3*A*B^2*a*b^7 + 2*A^2*C*a*b^7 - 12*B*C^2*a^7*b - A*B^2*a^2*b^6 - A*B^2*a^3*b^5 + 4*A*C^2*a^2*b^6 +
8*A*C^2*a^3*b^5 - 4*A*C^2*a^4*b^4 - 4*A*C^2*a^5*b^3 - 20*B*C^2*a^3*b^5 - 13*B*C^2*a^4*b^4 + 32*B*C^2*a^5*b^3 +
8*B*C^2*a^6*b^2 + 11*B^2*C*a^2*b^6 + 9*B^2*C*a^3*b^5 - 17*B^2*C*a^4*b^4 - 5*B^2*C*a^5*b^3 + 6*B^2*C*a^6*b^2 -
4*A*B*C*a*b^7 - 10*A*B*C*a^2*b^6 + 4*A*B*C*a^3*b^5 + 4*A*B*C*a^4*b^4))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - ((
-(a + b)^3*(a - b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(A^2*b^8 + B^2*b^8 + 8*C^2*a^8 - 2*B^2*a*b^7 - 8*C^2*a^7*b
+ 3*B^2*a^2*b^6 + 4*B^2*a^3*b^5 - 5*B^2*a^4*b^4 - 2*B^2*a^5*b^3 + 2*B^2*a^6*b^2 + 4*C^2*a^2*b^6 - 8*C^2*a^3*b
^5 + 5*C^2*a^4*b^4 + 16*C^2*a^5*b^3 - 16*C^2*a^6*b^2 - 4*A*B*a*b^7 - 4*B*C*a*b^7 - 8*B*C*a^7*b + 2*A*B*a^3*b^5
+ 6*A*C*a^2*b^6 - 4*A*C*a^4*b^4 + 8*B*C*a^2*b^6 - 8*B*C*a^3*b^5 - 16*B*C*a^4*b^4 + 18*B*C*a^5*b^3 + 8*B*C*a^6
*b^2))/(a*b^6 + b^7 - a^2*b^5 - a^3*b^4) + (((32*(A*a^2*b^10 - B*b^12 - A*b^12 - A*a^3*b^9 + B*a^2*b^10 - 3*B*
a^3*b^9 + B*a^5*b^7 - 3*C*a^2*b^10 - 3*C*a^3*b^9 + 5*C*a^4*b^8 + C*a^5*b^7 - 2*C*a^6*b^6 + A*a*b^11 + 2*B*a*b^
11 + 2*C*a*b^11))/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) - (32*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b
^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b)*(2*a*b^11 - 2*a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 -
2*a^6*b^6))/((a*b^6 + b^7 - a^2*b^5 - a^3*b^4)*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*(-(a + b)^3*(a - b)^
3)^(1/2)*(A*b^4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(A*b^
4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3) + ((-(a + b)^3*(a -
b)^3)^(1/2)*((32*tan(c/2 + (d*x)/2)*(A^2*b^8 + B^2*b^8 + 8*C^2*a^8 - 2*B^2*a*b^7 - 8*C^2*a^7*b + 3*B^2*a^2*b^6
+ 4*B^2*a^3*b^5 - 5*B^2*a^4*b^4 - 2*B^2*a^5*b^3 + 2*B^2*a^6*b^2 + 4*C^2*a^2*b^6 - 8*C^2*a^3*b^5 + 5*C^2*a^4*b
^4 + 16*C^2*a^5*b^3 - 16*C^2*a^6*b^2 - 4*A*B*a*b^7 - 4*B*C*a*b^7 - 8*B*C*a^7*b + 2*A*B*a^3*b^5 + 6*A*C*a^2*b^6
- 4*A*C*a^4*b^4 + 8*B*C*a^2*b^6 - 8*B*C*a^3*b^5 - 16*B*C*a^4*b^4 + 18*B*C*a^5*b^3 + 8*B*C*a^6*b^2))/(a*b^6 +
b^7 - a^2*b^5 - a^3*b^4) - (((32*(A*a^2*b^10 - B*b^12 - A*b^12 - A*a^3*b^9 + B*a^2*b^10 - 3*B*a^3*b^9 + B*a^5*
b^7 - 3*C*a^2*b^10 - 3*C*a^3*b^9 + 5*C*a^4*b^8 + C*a^5*b^7 - 2*C*a^6*b^6 + A*a*b^11 + 2*B*a*b^11 + 2*C*a*b^11)
)/(a*b^8 + b^9 - a^2*b^7 - a^3*b^6) + (32*tan(c/2 + (d*x)/2)*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^4 - 2*C*a^4 + 3
*C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b)*(2*a*b^11 - 2*a^2*b^10 - 4*a^3*b^9 + 4*a^4*b^8 + 2*a^5*b^7 - 2*a^6*b^6))/((a
*b^6 + b^7 - a^2*b^5 - a^3*b^4)*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^4
- 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))*(A*b^4 - 2*C*a^4 + 3*
C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b))/(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3)))*(-(a + b)^3*(a - b)^3)^(1/2)*(A*b^
4 - 2*C*a^4 + 3*C*a^2*b^2 - 2*B*a*b^3 + B*a^3*b)*2i)/(d*(b^9 - 3*a^2*b^7 + 3*a^4*b^5 - a^6*b^3))