\(\int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx\) [980]

Optimal result

Integrand size = 33, antiderivative size = 97 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {(b B-a C) x}{b^2}+\frac {2 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b} d}+\frac {C \sin (c+d x)}{b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3102, 2814, 2738, 211} \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {2 \left (A b^2-a (b B-a C)\right ) \arctan \left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^2 d \sqrt {a-b} \sqrt {a+b}}+\frac {x (b B-a C)}{b^2}+\frac {C \sin (c+d x)}{b d} \]

[In]

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

[Out]

((b*B - a*C)*x)/b^2 + (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^2*Sqrt[a + b]*d) + (C*Sin[c + d*x])/(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]

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

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {(b B-a C) (c+d x)-\frac {2 \left (A b^2+a (-b B+a C)\right ) \text {arctanh}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+b C \sin (c+d x)}{b^2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21

method	result	size
derivativedivides	\(\frac {\frac {2 \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2}}}{d}\)	\(117\)
default	\(\frac {\frac {2 \left (A \,b^{2}-B a b +a^{2} C \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{2} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {\frac {2 C b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (B b -C a \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{2}}}{d}\)	\(117\)
risch	\(\frac {x B}{b}-\frac {a C x}{b^{2}}-\frac {i {\mathrm e}^{i \left (d x +c \right )} C}{2 b d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C}{2 b d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) B a}{\sqrt {-a^{2}+b^{2}}\, d b}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a^{2}-i b^{2}+\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{2} C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) A}{\sqrt {-a^{2}+b^{2}}\, d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) B a}{\sqrt {-a^{2}+b^{2}}\, d b}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i a^{2}-i b^{2}-\sqrt {-a^{2}+b^{2}}\, a}{b \sqrt {-a^{2}+b^{2}}}\right ) a^{2} C}{\sqrt {-a^{2}+b^{2}}\, d \,b^{2}}\)	\(491\)

[In]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.41 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\left [-\frac {2 \, {\left (C a^{3} - B a^{2} b - C a b^{2} + B b^{3}\right )} d x + {\left (C a^{2} - B a b + A b^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (C a^{2} b - C b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} d}, -\frac {{\left (C a^{3} - B a^{2} b - C a b^{2} + B b^{3}\right )} d x - {\left (C a^{2} - B a b + A b^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (C a^{2} b - C b^{3}\right )} \sin \left (d x + c\right )}{{\left (a^{2} b^{2} - b^{4}\right )} d}\right ] \]

[In]

integrate((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^3 - B*a^2*b - C*a*b^2 + B*b^3)*d*x + (C*a^2 - B*a*b + 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^2*b - C*b^3)*sin(d*x + c))/((a^2*b^2 - b^4)*d), -((C*a^
3 - B*a^2*b - C*a*b^2 + B*b^3)*d*x - (C*a^2 - B*a*b + A*b^2)*sqrt(a^2 - b^2)*arctan(-(a*cos(d*x + c) + b)/(sqr
t(a^2 - b^2)*sin(d*x + c))) - (C*a^2*b - C*b^3)*sin(d*x + c))/((a^2*b^2 - b^4)*d)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3966 vs. \(2 (83) = 166\).

Time = 171.15 (sec) , antiderivative size = 3966, normalized size of antiderivative = 40.89 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((zoo*x*(A + B*cos(c) + C*cos(c)**2)/cos(c), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (A*tan(c/2 + d*x/2)**3/
(b*d*tan(c/2 + d*x/2)**2 + b*d) + A*tan(c/2 + d*x/2)/(b*d*tan(c/2 + d*x/2)**2 + b*d) + B*d*x*tan(c/2 + d*x/2)*
*2/(b*d*tan(c/2 + d*x/2)**2 + b*d) + B*d*x/(b*d*tan(c/2 + d*x/2)**2 + b*d) - B*tan(c/2 + d*x/2)**3/(b*d*tan(c/
2 + d*x/2)**2 + b*d) - B*tan(c/2 + d*x/2)/(b*d*tan(c/2 + d*x/2)**2 + b*d) - C*d*x*tan(c/2 + d*x/2)**2/(b*d*tan
(c/2 + d*x/2)**2 + b*d) - C*d*x/(b*d*tan(c/2 + d*x/2)**2 + b*d) + C*tan(c/2 + d*x/2)**3/(b*d*tan(c/2 + d*x/2)*
*2 + b*d) + 3*C*tan(c/2 + d*x/2)/(b*d*tan(c/2 + d*x/2)**2 + b*d), Eq(a, b)), (A*tan(c/2 + d*x/2)**2/(b*d*tan(c
/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + A/(b*d*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + B*d*x*tan(c/2 +
d*x/2)**3/(b*d*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + B*d*x*tan(c/2 + d*x/2)/(b*d*tan(c/2 + d*x/2)**3 +
b*d*tan(c/2 + d*x/2)) + B*tan(c/2 + d*x/2)**2/(b*d*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + B/(b*d*tan(c
/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + C*d*x*tan(c/2 + d*x/2)**3/(b*d*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*
x/2)) + C*d*x*tan(c/2 + d*x/2)/(b*d*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + 3*C*tan(c/2 + d*x/2)**2/(b*d
*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)) + C/(b*d*tan(c/2 + d*x/2)**3 + b*d*tan(c/2 + d*x/2)), Eq(a, -b)),
((A*x + B*sin(c + d*x)/d + C*x*sin(c + d*x)**2/2 + C*x*cos(c + d*x)**2/2 + C*sin(c + d*x)*cos(c + d*x)/(2*d))
/a, Eq(b, 0)), (x*(A + B*cos(c) + C*cos(c)**2)/(a + b*cos(c)), Eq(d, 0)), (A*b**2*log(-sqrt(-a/(a - b) - b/(a
- b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b
**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/
(a - b) - b/(a - b))) + A*b**2*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b**2*d*sqrt(-a/(a - b)
- b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b)
)*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) - A*b**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c
/2 + d*x/2))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a
/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(
a - b))) - A*b**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*
tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*
x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + B*a*b*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2/(a
*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt
(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + B*a*b*d*x*sqrt(-a/(a - b
) - b/(a - b))/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a -
b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) - B*a*b*l
og(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b)
)*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 +
d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) - B*a*b*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))
/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*s
qrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + B*a*b*log(sqrt(-a/(a
- b) - b/(a - b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x
/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**
3*d*sqrt(-a/(a - b) - b/(a - b))) + B*a*b*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b**2*d*sqrt(
-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) -
b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) - B*b**2*d*x*sqrt(-a/(a - b) - b/(a - b
))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) -
b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) -
B*b**2*d*x*sqrt(-a/(a - b) - b/(a - b))/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d
*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a -
b) - b/(a - b))) - C*a**2*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(
a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(
c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) - C*a**2*d*x*sqrt(-a/(a - b) - b/(a - b))/(a*b**2*d*sqr
t(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b)
- b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + C*a**2*log(-sqrt(-a/(a - b) - b/(a
- b)) + tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b
**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/
(a - b) - b/(a - b))) + C*a**2*log(-sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b**2*d*sqrt(-a/(a - b)
- b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b)
)*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) - C*a**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c
/2 + d*x/2))*tan(c/2 + d*x/2)**2/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a
/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(
a - b))) - C*a**2*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*
tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*
x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + C*a*b*d*x*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2/(a
*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt
(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + C*a*b*d*x*sqrt(-a/(a - b
) - b/(a - b))/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a -
b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))) + 2*C*a*b
*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)/(a*b**2*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 + a*
b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)**2 - b**3*d*sqrt(-a
/(a - b) - b/(a - b))) - 2*C*b**2*sqrt(-a/(a - b) - b/(a - b))*tan(c/2 + d*x/2)/(a*b**2*d*sqrt(-a/(a - b) - b/
(a - b))*tan(c/2 + d*x/2)**2 + a*b**2*d*sqrt(-a/(a - b) - b/(a - b)) - b**3*d*sqrt(-a/(a - b) - b/(a - b))*tan
(c/2 + d*x/2)**2 - b**3*d*sqrt(-a/(a - b) - b/(a - b))), True))

Maxima [F(-2)]

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

[In]

integrate((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 = 147, normalized size of antiderivative = 1.52 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=-\frac {\frac {{\left (C a - B b\right )} {\left (d x + c\right )}}{b^{2}} - \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} b} + \frac {2 \, {\left (C a^{2} - B a b + A b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{2}}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.25 (sec) , antiderivative size = 4410, normalized size of antiderivative = 45.46 \[ \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(2*B*b^3*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(d*(b^4 - a^2*b^2)) + (2*C*a^3*atan(sin(c/2 + (d*x)/2)/c
os(c/2 + (d*x)/2)))/(d*(b^4 - a^2*b^2)) + (C*b^3*sin(c + d*x))/(d*(b^4 - a^2*b^2)) - (C*a^2*b*sin(c + d*x))/(d
*(b^4 - a^2*b^2)) + (A*b^2*atan((C^2*a^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - B^2*b^7*sin(c/2 + (d*x)/2)*
(b^2 - a^2)^(1/2)*1i - A^2*b^7*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^7*sin(c/2 + (d*x)/2)*(b^2 - a^2
)^(1/2)*2i + A^2*a*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - A^2*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2
)*1i + B^2*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + A^2*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i
+ A^2*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + B^2*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i +
B^2*a^3*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - B^2*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*3i + B
^2*a^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - C^2*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2
*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^4*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i - C^2*a
^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*3i + A*B*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + B*C*a*b^6
*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^4*b*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*4i - B*C*a^6*b*sin(c
/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*4i - A*B*a^2*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*4i + A*B*a^2*b^5*sin(c/2
+ (d*x)/2)*(b^2 - a^2)^(1/2)*2i - A*B*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - A*B*a^4*b^3*sin(c/2 +
(d*x)/2)*(b^2 - a^2)^(1/2)*2i - A*C*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + A*C*a^3*b^2*sin(c/2 + (
d*x)/2)*(b^2 - a^2)^(3/2)*4i - A*C*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + A*C*a^4*b^3*sin(c/2 + (d*
x)/2)*(b^2 - a^2)^(1/2)*2i + A*C*a^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^2*b^5*sin(c/2 + (d*x)
/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + B*C*a^4*b^3*sin(c/2 + (d*x)/2
)*(b^2 - a^2)^(1/2)*6i)/(A^2*b^8*cos(c/2 + (d*x)/2) + B^2*b^8*cos(c/2 + (d*x)/2) - 2*A^2*a^2*b^6*cos(c/2 + (d*
x)/2) + A^2*a^4*b^4*cos(c/2 + (d*x)/2) - 2*B^2*a^2*b^6*cos(c/2 + (d*x)/2) + B^2*a^4*b^4*cos(c/2 + (d*x)/2) + C
^2*a^2*b^6*cos(c/2 + (d*x)/2) - 2*C^2*a^4*b^4*cos(c/2 + (d*x)/2) + C^2*a^6*b^2*cos(c/2 + (d*x)/2) + 4*A*B*a^3*
b^5*cos(c/2 + (d*x)/2) - 2*A*B*a^5*b^3*cos(c/2 + (d*x)/2) + 2*A*C*a^2*b^6*cos(c/2 + (d*x)/2) - 4*A*C*a^4*b^4*c
os(c/2 + (d*x)/2) + 2*A*C*a^6*b^2*cos(c/2 + (d*x)/2) + 4*B*C*a^3*b^5*cos(c/2 + (d*x)/2) - 2*B*C*a^5*b^3*cos(c/
2 + (d*x)/2) - 2*A*B*a*b^7*cos(c/2 + (d*x)/2) - 2*B*C*a*b^7*cos(c/2 + (d*x)/2)))*(b^2 - a^2)^(1/2)*2i)/(d*(b^4
- a^2*b^2)) + (C*a^2*atan((C^2*a^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - B^2*b^7*sin(c/2 + (d*x)/2)*(b^2
- a^2)^(1/2)*1i - A^2*b^7*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^7*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/
2)*2i + A^2*a*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - A^2*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i
+ B^2*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + A^2*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + A^
2*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + B^2*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + B^2*
a^3*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - B^2*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*3i + B^2*a^
5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - C^2*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^3*
b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^4*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i - C^2*a^5*b^
2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*3i + A*B*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + B*C*a*b^6*sin(
c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^4*b*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*4i - B*C*a^6*b*sin(c/2 +
(d*x)/2)*(b^2 - a^2)^(1/2)*4i - A*B*a^2*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*4i + A*B*a^2*b^5*sin(c/2 + (d
*x)/2)*(b^2 - a^2)^(1/2)*2i - A*B*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - A*B*a^4*b^3*sin(c/2 + (d*x
)/2)*(b^2 - a^2)^(1/2)*2i - A*C*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + A*C*a^3*b^2*sin(c/2 + (d*x)/
2)*(b^2 - a^2)^(3/2)*4i - A*C*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + A*C*a^4*b^3*sin(c/2 + (d*x)/2)
*(b^2 - a^2)^(1/2)*2i + A*C*a^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^2*b^5*sin(c/2 + (d*x)/2)*(
b^2 - a^2)^(1/2)*2i - B*C*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + B*C*a^4*b^3*sin(c/2 + (d*x)/2)*(b^
2 - a^2)^(1/2)*6i)/(A^2*b^8*cos(c/2 + (d*x)/2) + B^2*b^8*cos(c/2 + (d*x)/2) - 2*A^2*a^2*b^6*cos(c/2 + (d*x)/2)
+ A^2*a^4*b^4*cos(c/2 + (d*x)/2) - 2*B^2*a^2*b^6*cos(c/2 + (d*x)/2) + B^2*a^4*b^4*cos(c/2 + (d*x)/2) + C^2*a^
2*b^6*cos(c/2 + (d*x)/2) - 2*C^2*a^4*b^4*cos(c/2 + (d*x)/2) + C^2*a^6*b^2*cos(c/2 + (d*x)/2) + 4*A*B*a^3*b^5*c
os(c/2 + (d*x)/2) - 2*A*B*a^5*b^3*cos(c/2 + (d*x)/2) + 2*A*C*a^2*b^6*cos(c/2 + (d*x)/2) - 4*A*C*a^4*b^4*cos(c/
2 + (d*x)/2) + 2*A*C*a^6*b^2*cos(c/2 + (d*x)/2) + 4*B*C*a^3*b^5*cos(c/2 + (d*x)/2) - 2*B*C*a^5*b^3*cos(c/2 + (
d*x)/2) - 2*A*B*a*b^7*cos(c/2 + (d*x)/2) - 2*B*C*a*b^7*cos(c/2 + (d*x)/2)))*(b^2 - a^2)^(1/2)*2i)/(d*(b^4 - a^
2*b^2)) - (2*B*a^2*b*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(d*(b^4 - a^2*b^2)) - (2*C*a*b^2*atan(sin(c/
2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(d*(b^4 - a^2*b^2)) - (B*a*b*atan((C^2*a^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3
/2)*2i - B^2*b^7*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i - A^2*b^7*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C
^2*a^7*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + A^2*a*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - A^2*a*b^6
*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + B^2*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + A^2*a^2*b^5*sin
(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + A^2*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + B^2*a^2*b^5*sin(c
/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + B^2*a^3*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*2i - B^2*a^3*b^4*sin(c/2
+ (d*x)/2)*(b^2 - a^2)^(1/2)*3i + B^2*a^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - C^2*a^2*b^5*sin(c/2 +
(d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*1i + C^2*a^4*b^3*sin(c/2 + (
d*x)/2)*(b^2 - a^2)^(1/2)*1i - C^2*a^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*3i + A*B*a*b^6*sin(c/2 + (d*x)
/2)*(b^2 - a^2)^(1/2)*2i + B*C*a*b^6*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^4*b*sin(c/2 + (d*x)/2)*(b
^2 - a^2)^(3/2)*4i - B*C*a^6*b*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*4i - A*B*a^2*b^3*sin(c/2 + (d*x)/2)*(b^2 -
a^2)^(3/2)*4i + A*B*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - A*B*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a
^2)^(1/2)*2i - A*B*a^4*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - A*C*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2
)^(1/2)*2i + A*C*a^3*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(3/2)*4i - A*C*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^
(1/2)*2i + A*C*a^4*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i + A*C*a^5*b^2*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1
/2)*2i - B*C*a^2*b^5*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*2i - B*C*a^3*b^4*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2
)*2i + B*C*a^4*b^3*sin(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2)*6i)/(A^2*b^8*cos(c/2 + (d*x)/2) + B^2*b^8*cos(c/2 + (d
*x)/2) - 2*A^2*a^2*b^6*cos(c/2 + (d*x)/2) + A^2*a^4*b^4*cos(c/2 + (d*x)/2) - 2*B^2*a^2*b^6*cos(c/2 + (d*x)/2)
+ B^2*a^4*b^4*cos(c/2 + (d*x)/2) + C^2*a^2*b^6*cos(c/2 + (d*x)/2) - 2*C^2*a^4*b^4*cos(c/2 + (d*x)/2) + C^2*a^6
*b^2*cos(c/2 + (d*x)/2) + 4*A*B*a^3*b^5*cos(c/2 + (d*x)/2) - 2*A*B*a^5*b^3*cos(c/2 + (d*x)/2) + 2*A*C*a^2*b^6*
cos(c/2 + (d*x)/2) - 4*A*C*a^4*b^4*cos(c/2 + (d*x)/2) + 2*A*C*a^6*b^2*cos(c/2 + (d*x)/2) + 4*B*C*a^3*b^5*cos(c
/2 + (d*x)/2) - 2*B*C*a^5*b^3*cos(c/2 + (d*x)/2) - 2*A*B*a*b^7*cos(c/2 + (d*x)/2) - 2*B*C*a*b^7*cos(c/2 + (d*x
)/2)))*(b^2 - a^2)^(1/2)*2i)/(d*(b^4 - a^2*b^2))