\(\int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx\) [547]

Optimal result

Integrand size = 22, antiderivative size = 197 \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=-\frac {3 a (b B+c C) \arctan \left (\frac {c+(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{5/2}}+\frac {B c-b C-a C \cos (x)+a B \sin (x)}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2}+\frac {a (B c-b C)-\left (2 b B c+\left (a^2+2 c^2\right ) C\right ) \cos (x)+\left (a^2 B+2 b (b B+c C)\right ) \sin (x)}{2 \left (a^2-b^2-c^2\right )^2 (a+b \cos (x)+c \sin (x))} \]

[Out]

-3*a*(B*b+C*c)*arctan((c+(a-b)*tan(1/2*x))/(a^2-b^2-c^2)^(1/2))/(a^2-b^2-c^2)^(5/2)+1/2*(B*c-b*C-a*C*cos(x)+a*
B*sin(x))/(a^2-b^2-c^2)/(a+b*cos(x)+c*sin(x))^2+1/2*(a*(B*c-C*b)-(2*b*B*c+(a^2+2*c^2)*C)*cos(x)+(B*a^2+2*b*(B*
b+C*c))*sin(x))/(a^2-b^2-c^2)^2/(a+b*cos(x)+c*sin(x))

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {3235, 3232, 3203, 632, 210} \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=-\frac {3 a (b B+c C) \arctan \left (\frac {(a-b) \tan \left (\frac {x}{2}\right )+c}{\sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{5/2}}+\frac {-\cos (x) \left (C \left (a^2+2 c^2\right )+2 b B c\right )+\sin (x) \left (a^2 B+2 b (b B+c C)\right )+a (B c-b C)}{2 \left (a^2-b^2-c^2\right )^2 (a+b \cos (x)+c \sin (x))}+\frac {a B \sin (x)-a C \cos (x)-b C+B c}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2} \]

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 3203

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

Rule 3232

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)
*Sin[d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + Dist[(a*A - b*B - c*C)/(a^2 -
b^2 - c^2), Int[1/(a + b*Cos[d + e*x] + c*Sin[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[
a^2 - b^2 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]

Rule 3235

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_)*((A_.) + cos[(d_.) + (e_.)*(x
_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B -
b*A)*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Dist[
1/((n + 1)*(a^2 - b^2 - c^2)), Int[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C
) + (n + 2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] /; FreeQ[{a, b, c, d, e, A
, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]

Rubi steps \begin{align*} \text {integral}& = \frac {B c-b C-a C \cos (x)+a B \sin (x)}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2}-\frac {\int \frac {2 (b B+c C)-a B \cos (x)-a C \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx}{2 \left (a^2-b^2-c^2\right )} \\ & = \frac {B c-b C-a C \cos (x)+a B \sin (x)}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2}+\frac {a (B c-b C)-\left (2 b B c+\left (a^2+2 c^2\right ) C\right ) \cos (x)+\left (a^2 B+2 b (b B+c C)\right ) \sin (x)}{2 \left (a^2-b^2-c^2\right )^2 (a+b \cos (x)+c \sin (x))}-\frac {(3 a (b B+c C)) \int \frac {1}{a+b \cos (x)+c \sin (x)} \, dx}{2 \left (a^2-b^2-c^2\right )^2} \\ & = \frac {B c-b C-a C \cos (x)+a B \sin (x)}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2}+\frac {a (B c-b C)-\left (2 b B c+\left (a^2+2 c^2\right ) C\right ) \cos (x)+\left (a^2 B+2 b (b B+c C)\right ) \sin (x)}{2 \left (a^2-b^2-c^2\right )^2 (a+b \cos (x)+c \sin (x))}-\frac {(3 a (b B+c C)) \text {Subst}\left (\int \frac {1}{a+b+2 c x+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2-c^2\right )^2} \\ & = \frac {B c-b C-a C \cos (x)+a B \sin (x)}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2}+\frac {a (B c-b C)-\left (2 b B c+\left (a^2+2 c^2\right ) C\right ) \cos (x)+\left (a^2 B+2 b (b B+c C)\right ) \sin (x)}{2 \left (a^2-b^2-c^2\right )^2 (a+b \cos (x)+c \sin (x))}+\frac {(6 a (b B+c C)) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 c+2 (a-b) \tan \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2-c^2\right )^2} \\ & = -\frac {3 a (b B+c C) \arctan \left (\frac {c+(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\left (a^2-b^2-c^2\right )^{5/2}}+\frac {B c-b C-a C \cos (x)+a B \sin (x)}{2 \left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))^2}+\frac {a (B c-b C)-\left (2 b B c+\left (a^2+2 c^2\right ) C\right ) \cos (x)+\left (a^2 B+2 b (b B+c C)\right ) \sin (x)}{2 \left (a^2-b^2-c^2\right )^2 (a+b \cos (x)+c \sin (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.58 \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=\frac {3 a (b B+c C) \text {arctanh}\left (\frac {c+(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2+c^2}}\right )}{\left (-a^2+b^2+c^2\right )^{5/2}}+\frac {9 a^2 b B c+2 a^4 C-4 a^2 b^2 C+2 b^4 C+5 a^2 c^2 C+4 b^2 c^2 C+2 c^4 C+6 a b c (b B+c C) \cos (x)-c \left (a^2+2 \left (b^2+c^2\right )\right ) (b B+c C) \cos (2 x)+4 a^3 b B \sin (x)+2 a b^3 B \sin (x)+8 a b B c^2 \sin (x)+4 a^3 c C \sin (x)+2 a b^2 c C \sin (x)+8 a c^3 C \sin (x)+a^2 b^2 B \sin (2 x)+2 b^4 B \sin (2 x)+2 b^2 B c^2 \sin (2 x)+a^2 b c C \sin (2 x)+2 b^3 c C \sin (2 x)+2 b c^3 C \sin (2 x)}{4 b \left (-a^2+b^2+c^2\right )^2 (a+b \cos (x)+c \sin (x))^2} \]

[In]

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

[Out]

(3*a*(b*B + c*C)*ArcTanh[(c + (a - b)*Tan[x/2])/Sqrt[-a^2 + b^2 + c^2]])/(-a^2 + b^2 + c^2)^(5/2) + (9*a^2*b*B
*c + 2*a^4*C - 4*a^2*b^2*C + 2*b^4*C + 5*a^2*c^2*C + 4*b^2*c^2*C + 2*c^4*C + 6*a*b*c*(b*B + c*C)*Cos[x] - c*(a
^2 + 2*(b^2 + c^2))*(b*B + c*C)*Cos[2*x] + 4*a^3*b*B*Sin[x] + 2*a*b^3*B*Sin[x] + 8*a*b*B*c^2*Sin[x] + 4*a^3*c*
C*Sin[x] + 2*a*b^2*c*C*Sin[x] + 8*a*c^3*C*Sin[x] + a^2*b^2*B*Sin[2*x] + 2*b^4*B*Sin[2*x] + 2*b^2*B*c^2*Sin[2*x
] + a^2*b*c*C*Sin[2*x] + 2*b^3*c*C*Sin[2*x] + 2*b*c^3*C*Sin[2*x])/(4*b*(-a^2 + b^2 + c^2)^2*(a + b*Cos[x] + c*
Sin[x])^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(794\) vs. \(2(187)=374\).

Time = 1.61 (sec) , antiderivative size = 795, normalized size of antiderivative = 4.04

method	result	size
default	\(-\frac {2 \left (-\frac {\left (2 B \,a^{4}-3 B \,a^{3} b +2 B \,a^{2} b^{2}-4 B \,a^{2} c^{2}-3 B a \,b^{3}+2 B \,b^{4}+4 B \,b^{2} c^{2}+2 B \,c^{4}-3 C \,a^{3} c +6 C \,a^{2} b c -3 C a \,b^{2} c \right ) \tan \left (\frac {x}{2}\right )^{3}}{2 \left (a^{4}-2 a^{2} b^{2}-2 a^{2} c^{2}+b^{4}+2 b^{2} c^{2}+c^{4}\right ) \left (a -b \right )}-\frac {\left (2 B \,a^{4} c -9 B \,a^{3} b c +14 B \,a^{2} b^{2} c -4 B \,a^{2} c^{3}-9 B a \,b^{3} c +2 B \,b^{4} c +4 B \,b^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+2 C \,a^{4} b +4 C \,a^{3} b^{2}-5 C \,a^{3} c^{2}-4 C \,a^{2} b^{3}+14 C \,a^{2} b \,c^{2}-2 C a \,b^{4}-13 C a \,b^{2} c^{2}-2 C a \,c^{4}+2 C \,b^{5}+4 C \,b^{3} c^{2}+2 C b \,c^{4}\right ) \tan \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}-2 a^{2} b^{2}-2 a^{2} c^{2}+b^{4}+2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (2 B \,a^{5}-3 B \,a^{4} b +B \,a^{3} b^{2}-4 B \,a^{3} c^{2}+B \,a^{2} b^{3}-8 B \,a^{2} b \,c^{2}-3 B a \,b^{4}+8 B a \,b^{2} c^{2}+2 B a \,c^{4}+2 B \,b^{5}+4 B \,b^{3} c^{2}+2 B b \,c^{4}-5 C \,a^{4} c +5 C \,a^{3} b c +5 C \,a^{2} b^{2} c -4 C \,a^{2} c^{3}-5 C a \,b^{3} c +4 C a b \,c^{3}\right ) \tan \left (\frac {x}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}-2 a^{2} c^{2}+b^{4}+2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 B \,a^{2} b c -5 B \,b^{3} c -2 B b \,c^{3}+2 C \,a^{4}-4 C \,a^{2} b^{2}+C \,a^{2} c^{2}+2 C \,b^{4}-C \,b^{2} c^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}-2 a^{2} c^{2}+b^{4}+2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\tan \left (\frac {x}{2}\right )^{2} a -\tan \left (\frac {x}{2}\right )^{2} b +2 c \tan \left (\frac {x}{2}\right )+a +b \right )^{2}}-\frac {3 a \left (B b +C c \right ) \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}-2 a^{2} c^{2}+b^{4}+2 b^{2} c^{2}+c^{4}\right ) \sqrt {a^{2}-b^{2}-c^{2}}}\)	\(795\)
risch	\(\frac {i \left (5 i B a b \,c^{2} {\mathrm e}^{i x}-3 i B a b \,c^{2} {\mathrm e}^{3 i x}+9 i C \,a^{2} b c \,{\mathrm e}^{2 i x}+3 i C a \,b^{2} c \,{\mathrm e}^{3 i x}+5 i C a \,b^{2} c \,{\mathrm e}^{i x}+2 \,{\mathrm e}^{2 i x} a^{4} C +2 C \,b^{4} {\mathrm e}^{2 i x}+2 i B \,b^{4}+2 C \,c^{4} {\mathrm e}^{2 i x}-2 C \,c^{4}+5 C \,a^{2} c^{2} {\mathrm e}^{2 i x}+4 C \,b^{2} c^{2} {\mathrm e}^{2 i x}+2 i {\mathrm e}^{2 i x} a^{4} B +2 i B \,b^{4} {\mathrm e}^{2 i x}+2 i B \,c^{4} {\mathrm e}^{2 i x}+2 i B \,b^{2} c^{2}+2 i C \,b^{3} c +2 i C b \,c^{3}-4 C \,a^{2} b^{2} {\mathrm e}^{2 i x}+i B \,a^{2} b^{2}-2 B \,b^{3} c -2 B b \,c^{3}-C \,a^{2} c^{2}-2 C \,b^{2} c^{2}-B \,a^{2} b c +4 i B \,a^{3} b \,{\mathrm e}^{i x}+5 i B a \,b^{3} {\mathrm e}^{i x}+4 i C \,a^{3} c \,{\mathrm e}^{i x}+5 i C a \,c^{3} {\mathrm e}^{i x}+i C \,a^{2} b c +6 C a b \,c^{2} {\mathrm e}^{3 i x}+3 i B a \,b^{3} {\mathrm e}^{3 i x}+5 i B \,a^{2} b^{2} {\mathrm e}^{2 i x}+9 B \,a^{2} b c \,{\mathrm e}^{2 i x}+6 B a \,b^{2} c \,{\mathrm e}^{3 i x}-3 i C a \,c^{3} {\mathrm e}^{3 i x}-4 i B \,a^{2} c^{2} {\mathrm e}^{2 i x}+4 i B \,b^{2} c^{2} {\mathrm e}^{2 i x}\right )}{\left (-i c \,{\mathrm e}^{2 i x}+b \,{\mathrm e}^{2 i x}+i c +2 a \,{\mathrm e}^{i x}+b \right )^{2} \left (a^{2}-b^{2}-c^{2}\right )^{2} \left (i b +c \right )}-\frac {3 \ln \left ({\mathrm e}^{i x}+\frac {i a c \sqrt {-a^{2}+b^{2}+c^{2}}-i a^{2} b +i b^{3}+i b \,c^{2}+a b \sqrt {-a^{2}+b^{2}+c^{2}}+a^{2} c -b^{2} c -c^{3}}{\left (b^{2}+c^{2}\right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right ) a b B}{2 \sqrt {-a^{2}+b^{2}+c^{2}}\, \left (a^{2}-b^{2}-c^{2}\right )^{2}}-\frac {3 a \ln \left ({\mathrm e}^{i x}+\frac {i a c \sqrt {-a^{2}+b^{2}+c^{2}}-i a^{2} b +i b^{3}+i b \,c^{2}+a b \sqrt {-a^{2}+b^{2}+c^{2}}+a^{2} c -b^{2} c -c^{3}}{\left (b^{2}+c^{2}\right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right ) C c}{2 \sqrt {-a^{2}+b^{2}+c^{2}}\, \left (a^{2}-b^{2}-c^{2}\right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}+\frac {i a c \sqrt {-a^{2}+b^{2}+c^{2}}+i a^{2} b -i b^{3}-i b \,c^{2}+a b \sqrt {-a^{2}+b^{2}+c^{2}}-a^{2} c +b^{2} c +c^{3}}{\left (b^{2}+c^{2}\right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right ) a b B}{2 \sqrt {-a^{2}+b^{2}+c^{2}}\, \left (a^{2}-b^{2}-c^{2}\right )^{2}}+\frac {3 a \ln \left ({\mathrm e}^{i x}+\frac {i a c \sqrt {-a^{2}+b^{2}+c^{2}}+i a^{2} b -i b^{3}-i b \,c^{2}+a b \sqrt {-a^{2}+b^{2}+c^{2}}-a^{2} c +b^{2} c +c^{3}}{\left (b^{2}+c^{2}\right ) \sqrt {-a^{2}+b^{2}+c^{2}}}\right ) C c}{2 \sqrt {-a^{2}+b^{2}+c^{2}}\, \left (a^{2}-b^{2}-c^{2}\right )^{2}}\)	\(1055\)

[In]

int((B*cos(x)+C*sin(x))/(a+b*cos(x)+c*sin(x))^3,x,method=_RETURNVERBOSE)

[Out]

-2*(-1/2*(2*B*a^4-3*B*a^3*b+2*B*a^2*b^2-4*B*a^2*c^2-3*B*a*b^3+2*B*b^4+4*B*b^2*c^2+2*B*c^4-3*C*a^3*c+6*C*a^2*b*
c-3*C*a*b^2*c)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a-b)*tan(1/2*x)^3-1/2*(2*B*a^4*c-9*B*a^3*b*c+14*B*
a^2*b^2*c-4*B*a^2*c^3-9*B*a*b^3*c+2*B*b^4*c+4*B*b^2*c^3+2*B*c^5-2*C*a^5+2*C*a^4*b+4*C*a^3*b^2-5*C*a^3*c^2-4*C*
a^2*b^3+14*C*a^2*b*c^2-2*C*a*b^4-13*C*a*b^2*c^2-2*C*a*c^4+2*C*b^5+4*C*b^3*c^2+2*C*b*c^4)/(a^4-2*a^2*b^2-2*a^2*
c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*x)^2-1/2*(2*B*a^5-3*B*a^4*b+B*a^3*b^2-4*B*a^3*c^2+B*a^2*b^3-8*B
*a^2*b*c^2-3*B*a*b^4+8*B*a*b^2*c^2+2*B*a*c^4+2*B*b^5+4*B*b^3*c^2+2*B*b*c^4-5*C*a^4*c+5*C*a^3*b*c+5*C*a^2*b^2*c
-4*C*a^2*c^3-5*C*a*b^3*c+4*C*a*b*c^3)/(a^4-2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tan(1/2*x)+1
/2*a*(5*B*a^2*b*c-5*B*b^3*c-2*B*b*c^3+2*C*a^4-4*C*a^2*b^2+C*a^2*c^2+2*C*b^4-C*b^2*c^2)/(a^4-2*a^2*b^2-2*a^2*c^
2+b^4+2*b^2*c^2+c^4)/(a^2-2*a*b+b^2))/(tan(1/2*x)^2*a-tan(1/2*x)^2*b+2*c*tan(1/2*x)+a+b)^2-3*a*(B*b+C*c)/(a^4-
2*a^2*b^2-2*a^2*c^2+b^4+2*b^2*c^2+c^4)/(a^2-b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tan(1/2*x)+2*c)/(a^2-b^2-c^2)^(
1/2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1551 vs. \(2 (187) = 374\).

Time = 0.47 (sec) , antiderivative size = 3264, normalized size of antiderivative = 16.57 \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/4*(2*C*a^6*b - 6*C*a^4*b^3 + 6*C*a^2*b^5 - 2*C*b^7 - 6*C*b*c^6 + 2*B*c^7 - 2*(3*B*a^2 - B*b^2)*c^5 + 2*(4*C
*a^2*b - 7*C*b^3)*c^4 + 2*(3*B*a^4 - 5*B*a^2*b^2 - B*b^4)*c^3 - 2*(2*C*a^4*b - 7*C*a^2*b^3 + 5*C*b^5)*c^2 + 4*
(2*B*b^2*c^5 + 2*C*b*c^6 - (C*a^2*b - 4*C*b^3)*c^4 - (B*a^2*b^2 - 4*B*b^4)*c^3 - (C*a^4*b + C*a^2*b^3 - 2*C*b^
5)*c^2 - (B*a^4*b^2 + B*a^2*b^4 - 2*B*b^6)*c)*cos(x)^2 - 3*(B*a^3*b^3 + C*a^3*b^2*c + B*a*b*c^4 + C*a*c^5 + (C
*a^3 + C*a*b^2)*c^3 + (B*a^3*b + B*a*b^3)*c^2 + (B*a*b^5 + C*a*b^4*c - B*a*b*c^4 - C*a*c^5)*cos(x)^2 + 2*(B*a^
2*b^4 + C*a^2*b^3*c + B*a^2*b^2*c^2 + C*a^2*b*c^3)*cos(x) + 2*(B*a^2*b^3*c + C*a^2*b^2*c^2 + B*a^2*b*c^3 + C*a
^2*c^4 + (B*a*b^4*c + C*a*b^3*c^2 + B*a*b^2*c^3 + C*a*b*c^4)*cos(x))*sin(x))*sqrt(-a^2 + b^2 + c^2)*log((a^2*b
^2 - 2*b^4 - c^4 - (a^2 + 3*b^2)*c^2 - (2*a^2*b^2 - b^4 - 2*a^2*c^2 + c^4)*cos(x)^2 - 2*(a*b^3 + a*b*c^2)*cos(
x) - 2*(a*b^2*c + a*c^3 - (b*c^3 - (2*a^2*b - b^3)*c)*cos(x))*sin(x) - 2*(2*a*b*c*cos(x)^2 - a*b*c + (b^2*c +
c^3)*cos(x) - (b^3 + b*c^2 + (a*b^2 - a*c^2)*cos(x))*sin(x))*sqrt(-a^2 + b^2 + c^2))/(2*a*b*cos(x) + (b^2 - c^
2)*cos(x)^2 + a^2 + c^2 + 2*(b*c*cos(x) + a*c)*sin(x))) - 2*(B*a^6 - 4*B*a^4*b^2 + 2*B*a^2*b^4 + B*b^6)*c + 2*
(B*a*b*c^5 + C*a*c^6 + (C*a^3 + 2*C*a*b^2)*c^4 + (B*a^3*b + 2*B*a*b^3)*c^3 - (2*C*a^5 - C*a^3*b^2 - C*a*b^4)*c
^2 - (2*B*a^5*b - B*a^3*b^3 - B*a*b^5)*c)*cos(x) + 2*(2*B*a^5*b^2 - B*a^3*b^4 - B*a*b^6 - B*a*b^2*c^4 - C*a*b*
c^5 - (C*a^3*b + 2*C*a*b^3)*c^3 - (B*a^3*b^2 + 2*B*a*b^4)*c^2 + (2*C*a^5*b - C*a^3*b^3 - C*a*b^5)*c + (B*a^4*b
^3 + B*a^2*b^5 - 2*B*b^7 + 2*B*b*c^6 + 2*C*c^7 - (C*a^2 - 2*C*b^2)*c^5 - (B*a^2*b - 2*B*b^3)*c^4 - (C*a^4 + 2*
C*b^4)*c^3 - (B*a^4*b + 2*B*b^5)*c^2 + (C*a^4*b^2 + C*a^2*b^4 - 2*C*b^6)*c)*cos(x))*sin(x))/(a^8*b^2 - 3*a^6*b
^4 + 3*a^4*b^6 - a^2*b^8 - c^10 + 2*(a^2 - 2*b^2)*c^8 + (5*a^2*b^2 - 6*b^4)*c^6 - (2*a^6 - 3*a^4*b^2 - 3*a^2*b
^4 + 4*b^6)*c^4 + (a^8 - 5*a^6*b^2 + 6*a^4*b^4 - a^2*b^6 - b^8)*c^2 + (a^6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10
+ c^10 - 3*(a^2 - b^2)*c^8 + (3*a^4 - 6*a^2*b^2 + 2*b^4)*c^6 - (a^6 - 3*a^4*b^2 + 2*b^6)*c^4 - 3*(a^4*b^4 - 2*
a^2*b^6 + b^8)*c^2)*cos(x)^2 + 2*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9 - a*b*c^8 + (3*a^3*b - 4*a*b^3)*c^6
- 3*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*c^4 + (a^7*b - 6*a^5*b^3 + 9*a^3*b^5 - 4*a*b^7)*c^2)*cos(x) - 2*(a*c^9 - (3*
a^3 - 4*a*b^2)*c^7 + 3*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^7 - 6*a^5*b^2 + 9*a^3*b^4 - 4*a*b^6)*c^3 - (a^7*b^
2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c + (b*c^9 - (3*a^2*b - 4*b^3)*c^7 + 3*(a^4*b - 3*a^2*b^3 + 2*b^5)*c^5 - (a
^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*c^3 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*b^7 - b^9)*c)*cos(x))*sin(x)), 1/2*(C
*a^6*b - 3*C*a^4*b^3 + 3*C*a^2*b^5 - C*b^7 - 3*C*b*c^6 + B*c^7 - (3*B*a^2 - B*b^2)*c^5 + (4*C*a^2*b - 7*C*b^3)
*c^4 + (3*B*a^4 - 5*B*a^2*b^2 - B*b^4)*c^3 - (2*C*a^4*b - 7*C*a^2*b^3 + 5*C*b^5)*c^2 + 2*(2*B*b^2*c^5 + 2*C*b*
c^6 - (C*a^2*b - 4*C*b^3)*c^4 - (B*a^2*b^2 - 4*B*b^4)*c^3 - (C*a^4*b + C*a^2*b^3 - 2*C*b^5)*c^2 - (B*a^4*b^2 +
B*a^2*b^4 - 2*B*b^6)*c)*cos(x)^2 - 3*(B*a^3*b^3 + C*a^3*b^2*c + B*a*b*c^4 + C*a*c^5 + (C*a^3 + C*a*b^2)*c^3 +
(B*a^3*b + B*a*b^3)*c^2 + (B*a*b^5 + C*a*b^4*c - B*a*b*c^4 - C*a*c^5)*cos(x)^2 + 2*(B*a^2*b^4 + C*a^2*b^3*c +
B*a^2*b^2*c^2 + C*a^2*b*c^3)*cos(x) + 2*(B*a^2*b^3*c + C*a^2*b^2*c^2 + B*a^2*b*c^3 + C*a^2*c^4 + (B*a*b^4*c +
C*a*b^3*c^2 + B*a*b^2*c^3 + C*a*b*c^4)*cos(x))*sin(x))*sqrt(a^2 - b^2 - c^2)*arctan(-(a*b*cos(x) + a*c*sin(x)
+ b^2 + c^2)*sqrt(a^2 - b^2 - c^2)/((c^3 - (a^2 - b^2)*c)*cos(x) + (a^2*b - b^3 - b*c^2)*sin(x))) - (B*a^6 -
4*B*a^4*b^2 + 2*B*a^2*b^4 + B*b^6)*c + (B*a*b*c^5 + C*a*c^6 + (C*a^3 + 2*C*a*b^2)*c^4 + (B*a^3*b + 2*B*a*b^3)*
c^3 - (2*C*a^5 - C*a^3*b^2 - C*a*b^4)*c^2 - (2*B*a^5*b - B*a^3*b^3 - B*a*b^5)*c)*cos(x) + (2*B*a^5*b^2 - B*a^3
*b^4 - B*a*b^6 - B*a*b^2*c^4 - C*a*b*c^5 - (C*a^3*b + 2*C*a*b^3)*c^3 - (B*a^3*b^2 + 2*B*a*b^4)*c^2 + (2*C*a^5*
b - C*a^3*b^3 - C*a*b^5)*c + (B*a^4*b^3 + B*a^2*b^5 - 2*B*b^7 + 2*B*b*c^6 + 2*C*c^7 - (C*a^2 - 2*C*b^2)*c^5 -
(B*a^2*b - 2*B*b^3)*c^4 - (C*a^4 + 2*C*b^4)*c^3 - (B*a^4*b + 2*B*b^5)*c^2 + (C*a^4*b^2 + C*a^2*b^4 - 2*C*b^6)*
c)*cos(x))*sin(x))/(a^8*b^2 - 3*a^6*b^4 + 3*a^4*b^6 - a^2*b^8 - c^10 + 2*(a^2 - 2*b^2)*c^8 + (5*a^2*b^2 - 6*b^
4)*c^6 - (2*a^6 - 3*a^4*b^2 - 3*a^2*b^4 + 4*b^6)*c^4 + (a^8 - 5*a^6*b^2 + 6*a^4*b^4 - a^2*b^6 - b^8)*c^2 + (a^
6*b^4 - 3*a^4*b^6 + 3*a^2*b^8 - b^10 + c^10 - 3*(a^2 - b^2)*c^8 + (3*a^4 - 6*a^2*b^2 + 2*b^4)*c^6 - (a^6 - 3*a
^4*b^2 + 2*b^6)*c^4 - 3*(a^4*b^4 - 2*a^2*b^6 + b^8)*c^2)*cos(x)^2 + 2*(a^7*b^3 - 3*a^5*b^5 + 3*a^3*b^7 - a*b^9
- a*b*c^8 + (3*a^3*b - 4*a*b^3)*c^6 - 3*(a^5*b - 3*a^3*b^3 + 2*a*b^5)*c^4 + (a^7*b - 6*a^5*b^3 + 9*a^3*b^5 -
4*a*b^7)*c^2)*cos(x) - 2*(a*c^9 - (3*a^3 - 4*a*b^2)*c^7 + 3*(a^5 - 3*a^3*b^2 + 2*a*b^4)*c^5 - (a^7 - 6*a^5*b^2
+ 9*a^3*b^4 - 4*a*b^6)*c^3 - (a^7*b^2 - 3*a^5*b^4 + 3*a^3*b^6 - a*b^8)*c + (b*c^9 - (3*a^2*b - 4*b^3)*c^7 + 3
*(a^4*b - 3*a^2*b^3 + 2*b^5)*c^5 - (a^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*c^3 - (a^6*b^3 - 3*a^4*b^5 + 3*a^2*
b^7 - b^9)*c)*cos(x))*sin(x))]

Sympy [F(-1)]

Timed out. \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=\text {Timed out} \]

[In]

integrate((B*cos(x)+C*sin(x))/(a+b*cos(x)+c*sin(x))**3,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((B*cos(x)+C*sin(x))/(a+b*cos(x)+c*sin(x))^3,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(c^2+b^2-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. 1034 vs. \(2 (187) = 374\).

Time = 0.36 (sec) , antiderivative size = 1034, normalized size of antiderivative = 5.25 \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

3*(B*a*b + C*a*c)*(pi*floor(1/2*x/pi + 1/2)*sgn(-2*a + 2*b) + arctan(-(a*tan(1/2*x) - b*tan(1/2*x) + c)/sqrt(a
^2 - b^2 - c^2)))/((a^4 - 2*a^2*b^2 + b^4 - 2*a^2*c^2 + 2*b^2*c^2 + c^4)*sqrt(a^2 - b^2 - c^2)) + (2*B*a^5*tan
(1/2*x)^3 - 5*B*a^4*b*tan(1/2*x)^3 + 5*B*a^3*b^2*tan(1/2*x)^3 - 5*B*a^2*b^3*tan(1/2*x)^3 + 5*B*a*b^4*tan(1/2*x
)^3 - 2*B*b^5*tan(1/2*x)^3 - 3*C*a^4*c*tan(1/2*x)^3 + 9*C*a^3*b*c*tan(1/2*x)^3 - 9*C*a^2*b^2*c*tan(1/2*x)^3 +
3*C*a*b^3*c*tan(1/2*x)^3 - 4*B*a^3*c^2*tan(1/2*x)^3 + 4*B*a^2*b*c^2*tan(1/2*x)^3 + 4*B*a*b^2*c^2*tan(1/2*x)^3
- 4*B*b^3*c^2*tan(1/2*x)^3 + 2*B*a*c^4*tan(1/2*x)^3 - 2*B*b*c^4*tan(1/2*x)^3 - 2*C*a^5*tan(1/2*x)^2 + 2*C*a^4*
b*tan(1/2*x)^2 + 4*C*a^3*b^2*tan(1/2*x)^2 - 4*C*a^2*b^3*tan(1/2*x)^2 - 2*C*a*b^4*tan(1/2*x)^2 + 2*C*b^5*tan(1/
2*x)^2 + 2*B*a^4*c*tan(1/2*x)^2 - 9*B*a^3*b*c*tan(1/2*x)^2 + 14*B*a^2*b^2*c*tan(1/2*x)^2 - 9*B*a*b^3*c*tan(1/2
*x)^2 + 2*B*b^4*c*tan(1/2*x)^2 - 5*C*a^3*c^2*tan(1/2*x)^2 + 14*C*a^2*b*c^2*tan(1/2*x)^2 - 13*C*a*b^2*c^2*tan(1
/2*x)^2 + 4*C*b^3*c^2*tan(1/2*x)^2 - 4*B*a^2*c^3*tan(1/2*x)^2 + 4*B*b^2*c^3*tan(1/2*x)^2 - 2*C*a*c^4*tan(1/2*x
)^2 + 2*C*b*c^4*tan(1/2*x)^2 + 2*B*c^5*tan(1/2*x)^2 + 2*B*a^5*tan(1/2*x) - 3*B*a^4*b*tan(1/2*x) + B*a^3*b^2*ta
n(1/2*x) + B*a^2*b^3*tan(1/2*x) - 3*B*a*b^4*tan(1/2*x) + 2*B*b^5*tan(1/2*x) - 5*C*a^4*c*tan(1/2*x) + 5*C*a^3*b
*c*tan(1/2*x) + 5*C*a^2*b^2*c*tan(1/2*x) - 5*C*a*b^3*c*tan(1/2*x) - 4*B*a^3*c^2*tan(1/2*x) - 8*B*a^2*b*c^2*tan
(1/2*x) + 8*B*a*b^2*c^2*tan(1/2*x) + 4*B*b^3*c^2*tan(1/2*x) - 4*C*a^2*c^3*tan(1/2*x) + 4*C*a*b*c^3*tan(1/2*x)
+ 2*B*a*c^4*tan(1/2*x) + 2*B*b*c^4*tan(1/2*x) - 2*C*a^5 + 4*C*a^3*b^2 - 2*C*a*b^4 - 5*B*a^3*b*c + 5*B*a*b^3*c
- C*a^3*c^2 + C*a*b^2*c^2 + 2*B*a*b*c^3)/((a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*a
^4*c^2 + 4*a^3*b*c^2 - 4*a*b^3*c^2 + 2*b^4*c^2 + a^2*c^4 - 2*a*b*c^4 + b^2*c^4)*(a*tan(1/2*x)^2 - b*tan(1/2*x)
^2 + 2*c*tan(1/2*x) + a + b)^2)

Mupad [B] (verification not implemented)

Time = 30.83 (sec) , antiderivative size = 923, normalized size of antiderivative = 4.69 \[ \int \frac {B \cos (x)+C \sin (x)}{(a+b \cos (x)+c \sin (x))^3} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (2\,B\,a^4-3\,B\,a^3\,b-3\,C\,a^3\,c+2\,B\,a^2\,b^2+6\,C\,a^2\,b\,c-4\,B\,a^2\,c^2-3\,B\,a\,b^3-3\,C\,a\,b^2\,c+2\,B\,b^4+4\,B\,b^2\,c^2+2\,B\,c^4\right )}{\left (a-b\right )\,\left (a^4-2\,a^2\,b^2-2\,a^2\,c^2+b^4+2\,b^2\,c^2+c^4\right )}-\frac {2\,C\,a^5-4\,C\,a^3\,b^2+5\,B\,a^3\,b\,c+C\,a^3\,c^2+2\,C\,a\,b^4-5\,B\,a\,b^3\,c-C\,a\,b^2\,c^2-2\,B\,a\,b\,c^3}{{\left (a-b\right )}^2\,\left (a^4-2\,a^2\,b^2-2\,a^2\,c^2+b^4+2\,b^2\,c^2+c^4\right )}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (-2\,C\,a^5+2\,C\,a^4\,b+2\,B\,a^4\,c+4\,C\,a^3\,b^2-9\,B\,a^3\,b\,c-5\,C\,a^3\,c^2-4\,C\,a^2\,b^3+14\,B\,a^2\,b^2\,c+14\,C\,a^2\,b\,c^2-4\,B\,a^2\,c^3-2\,C\,a\,b^4-9\,B\,a\,b^3\,c-13\,C\,a\,b^2\,c^2-2\,C\,a\,c^4+2\,C\,b^5+2\,B\,b^4\,c+4\,C\,b^3\,c^2+4\,B\,b^2\,c^3+2\,C\,b\,c^4+2\,B\,c^5\right )}{{\left (a-b\right )}^2\,\left (a^4-2\,a^2\,b^2-2\,a^2\,c^2+b^4+2\,b^2\,c^2+c^4\right )}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,B\,a^5-3\,B\,a^4\,b-5\,C\,a^4\,c+B\,a^3\,b^2+5\,C\,a^3\,b\,c-4\,B\,a^3\,c^2+B\,a^2\,b^3+5\,C\,a^2\,b^2\,c-8\,B\,a^2\,b\,c^2-4\,C\,a^2\,c^3-3\,B\,a\,b^4-5\,C\,a\,b^3\,c+8\,B\,a\,b^2\,c^2+4\,C\,a\,b\,c^3+2\,B\,a\,c^4+2\,B\,b^5+4\,B\,b^3\,c^2+2\,B\,b\,c^4\right )}{{\left (a-b\right )}^2\,\left (a^4-2\,a^2\,b^2-2\,a^2\,c^2+b^4+2\,b^2\,c^2+c^4\right )}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^4\,\left (a^2-2\,a\,b+b^2\right )+2\,a\,b+\mathrm {tan}\left (\frac {x}{2}\right )\,\left (4\,a\,c+4\,b\,c\right )+{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (4\,a\,c-4\,b\,c\right )+a^2+b^2+{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2-2\,b^2+4\,c^2\right )}+\frac {3\,a\,\mathrm {atanh}\left (\frac {3\,a\,\left (B\,b+C\,c\right )\,\left (\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,a-2\,b\right )+\frac {2\,a^4\,c-4\,a^2\,b^2\,c-4\,a^2\,c^3+2\,b^4\,c+4\,b^2\,c^3+2\,c^5}{a^4-2\,a^2\,b^2-2\,a^2\,c^2+b^4+2\,b^2\,c^2+c^4}\right )\,\left (a^4-2\,a^2\,b^2-2\,a^2\,c^2+b^4+2\,b^2\,c^2+c^4\right )}{2\,\left (3\,B\,a\,b+3\,C\,a\,c\right )\,{\left (-a^2+b^2+c^2\right )}^{5/2}}\right )\,\left (B\,b+C\,c\right )}{{\left (-a^2+b^2+c^2\right )}^{5/2}} \]

[In]

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

[Out]

((tan(x/2)^3*(2*B*a^4 + 2*B*b^4 + 2*B*c^4 + 2*B*a^2*b^2 - 4*B*a^2*c^2 + 4*B*b^2*c^2 - 3*B*a*b^3 - 3*B*a^3*b -
3*C*a^3*c - 3*C*a*b^2*c + 6*C*a^2*b*c))/((a - b)*(a^4 + b^4 + c^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2)) - (2*C
*a^5 - 4*C*a^3*b^2 + C*a^3*c^2 + 2*C*a*b^4 - 2*B*a*b*c^3 - 5*B*a*b^3*c + 5*B*a^3*b*c - C*a*b^2*c^2)/((a - b)^2
*(a^4 + b^4 + c^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2)) + (tan(x/2)^2*(2*B*c^5 - 2*C*a^5 + 2*C*b^5 - 4*B*a^2*c
^3 - 4*C*a^2*b^3 + 4*C*a^3*b^2 + 4*B*b^2*c^3 - 5*C*a^3*c^2 + 4*C*b^3*c^2 + 2*B*a^4*c - 2*C*a*b^4 + 2*C*a^4*b +
2*B*b^4*c - 2*C*a*c^4 + 2*C*b*c^4 - 9*B*a*b^3*c - 9*B*a^3*b*c + 14*B*a^2*b^2*c - 13*C*a*b^2*c^2 + 14*C*a^2*b*
c^2))/((a - b)^2*(a^4 + b^4 + c^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2)) + (tan(x/2)*(2*B*a^5 + 2*B*b^5 + B*a^2
*b^3 + B*a^3*b^2 - 4*B*a^3*c^2 + 4*B*b^3*c^2 - 4*C*a^2*c^3 - 3*B*a*b^4 - 3*B*a^4*b + 2*B*a*c^4 + 2*B*b*c^4 - 5
*C*a^4*c + 4*C*a*b*c^3 - 5*C*a*b^3*c + 5*C*a^3*b*c + 8*B*a*b^2*c^2 - 8*B*a^2*b*c^2 + 5*C*a^2*b^2*c))/((a - b)^
2*(a^4 + b^4 + c^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2)))/(tan(x/2)^4*(a^2 - 2*a*b + b^2) + 2*a*b + tan(x/2)*(
4*a*c + 4*b*c) + tan(x/2)^3*(4*a*c - 4*b*c) + a^2 + b^2 + tan(x/2)^2*(2*a^2 - 2*b^2 + 4*c^2)) + (3*a*atanh((3*
a*(B*b + C*c)*(tan(x/2)*(2*a - 2*b) + (2*a^4*c + 2*b^4*c + 2*c^5 - 4*a^2*c^3 + 4*b^2*c^3 - 4*a^2*b^2*c)/(a^4 +
b^4 + c^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2))*(a^4 + b^4 + c^4 - 2*a^2*b^2 - 2*a^2*c^2 + 2*b^2*c^2))/(2*(3*
B*a*b + 3*C*a*c)*(b^2 - a^2 + c^2)^(5/2)))*(B*b + C*c))/(b^2 - a^2 + c^2)^(5/2)