\(\int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx\) [440]

Optimal result

Integrand size = 18, antiderivative size = 239 \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=-\frac {i x \log \left (1+\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{2 \sqrt {a+b} \sqrt {a+c}}+\frac {i x \log \left (1+\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{2 \sqrt {a+b} \sqrt {a+c}}-\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 \sqrt {a+b} \sqrt {a+c}}+\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 \sqrt {a+b} \sqrt {a+c}} \] Output:

-1/2*I*x*ln(1+(b-c)*exp(2*I*x)/(2*a+b+c-2*(a+b)^(1/2)*(a+c)^(1/2)))/(a+b)^ 
(1/2)/(a+c)^(1/2)+1/2*I*x*ln(1+(b-c)*exp(2*I*x)/(2*a+b+c+2*(a+b)^(1/2)*(a+ 
c)^(1/2)))/(a+b)^(1/2)/(a+c)^(1/2)-1/4*polylog(2,-(b-c)*exp(2*I*x)/(2*a+b+ 
c-2*(a+b)^(1/2)*(a+c)^(1/2)))/(a+b)^(1/2)/(a+c)^(1/2)+1/4*polylog(2,-(b-c) 
*exp(2*I*x)/(2*a+b+c+2*(a+b)^(1/2)*(a+c)^(1/2)))/(a+b)^(1/2)/(a+c)^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1022\) vs. \(2(239)=478\).

Time = 0.91 (sec) , antiderivative size = 1022, normalized size of antiderivative = 4.28 \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx =\text {Too large to display} \] Input:

Integrate[x/(a + b*Cos[x]^2 + c*Sin[x]^2),x]
 

Output:

(4*x*ArcTanh[((a + b)*Cot[x])/Sqrt[-((a + b)*(a + c))]] - 2*ArcCos[-((2*a 
+ b + c)/(b - c))]*ArcTanh[(Sqrt[-((a + b)*(a + c))]*Tan[x])/(a + b)] + Ar 
cCos[-((2*a + b + c)/(b - c))]*Log[(Sqrt[2]*Sqrt[-((a + b)*(a + c))])/(Sqr 
t[b - c]*E^(I*x)*Sqrt[2*a + b + c + (b - c)*Cos[2*x]])] - (2*I)*ArcTanh[(( 
a + b)*Cot[x])/Sqrt[-((a + b)*(a + c))]]*Log[(Sqrt[2]*Sqrt[-((a + b)*(a + 
c))])/(Sqrt[b - c]*E^(I*x)*Sqrt[2*a + b + c + (b - c)*Cos[2*x]])] + (2*I)* 
ArcTanh[(Sqrt[-((a + b)*(a + c))]*Tan[x])/(a + b)]*Log[(Sqrt[2]*Sqrt[-((a 
+ b)*(a + c))])/(Sqrt[b - c]*E^(I*x)*Sqrt[2*a + b + c + (b - c)*Cos[2*x]]) 
] + ArcCos[-((2*a + b + c)/(b - c))]*Log[(Sqrt[2]*Sqrt[-((a + b)*(a + c))] 
*E^(I*x))/(Sqrt[b - c]*Sqrt[2*a + b + c + (b - c)*Cos[2*x]])] + (2*I)*ArcT 
anh[((a + b)*Cot[x])/Sqrt[-((a + b)*(a + c))]]*Log[(Sqrt[2]*Sqrt[-((a + b) 
*(a + c))]*E^(I*x))/(Sqrt[b - c]*Sqrt[2*a + b + c + (b - c)*Cos[2*x]])] - 
(2*I)*ArcTanh[(Sqrt[-((a + b)*(a + c))]*Tan[x])/(a + b)]*Log[(Sqrt[2]*Sqrt 
[-((a + b)*(a + c))]*E^(I*x))/(Sqrt[b - c]*Sqrt[2*a + b + c + (b - c)*Cos[ 
2*x]])] - ArcCos[-((2*a + b + c)/(b - c))]*Log[(2*(a + b)*((-I)*a - I*c + 
Sqrt[-((a + b)*(a + c))])*(-I + Tan[x]))/((b - c)*(a + b + Sqrt[-((a + b)* 
(a + c))]*Tan[x]))] + (2*I)*ArcTanh[(Sqrt[-((a + b)*(a + c))]*Tan[x])/(a + 
 b)]*Log[(2*(a + b)*((-I)*a - I*c + Sqrt[-((a + b)*(a + c))])*(-I + Tan[x] 
))/((b - c)*(a + b + Sqrt[-((a + b)*(a + c))]*Tan[x]))] - ArcCos[-((2*a + 
b + c)/(b - c))]*Log[(2*(a + b)*(I*a + I*c + Sqrt[-((a + b)*(a + c))])*...
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5098, 3042, 3802, 2694, 27, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx\)

\(\Big \downarrow \) 5098

\(\displaystyle 2 \int \frac {x}{2 a+b+c+(b-c) \cos (2 x)}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle 2 \int \frac {x}{2 a+b+c+(b-c) \sin \left (2 x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 3802

\(\displaystyle 4 \int \frac {e^{2 i x} x}{b+2 (2 a+b+c) e^{2 i x}+(b-c) e^{4 i x}-c}dx\)

\(\Big \downarrow \) 2694

\(\displaystyle 4 \left (\frac {(b-c) \int \frac {e^{2 i x} x}{2 \left (2 a+(b-c) e^{2 i x}+b+c-2 \sqrt {a+b} \sqrt {a+c}\right )}dx}{2 \sqrt {a+b} \sqrt {a+c}}-\frac {(b-c) \int \frac {e^{2 i x} x}{2 \left (2 a+(b-c) e^{2 i x}+b+c+2 \sqrt {a+b} \sqrt {a+c}\right )}dx}{2 \sqrt {a+b} \sqrt {a+c}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 4 \left (\frac {(b-c) \int \frac {e^{2 i x} x}{2 a+(b-c) e^{2 i x}+b+c-2 \sqrt {a+b} \sqrt {a+c}}dx}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {(b-c) \int \frac {e^{2 i x} x}{2 a+(b-c) e^{2 i x}+b+c+2 \sqrt {a+b} \sqrt {a+c}}dx}{4 \sqrt {a+b} \sqrt {a+c}}\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle 4 \left (\frac {(b-c) \left (\frac {i \int \log \left (\frac {e^{2 i x} (b-c)}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}+1\right )dx}{2 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {(b-c) \left (\frac {i \int \log \left (\frac {e^{2 i x} (b-c)}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}+1\right )dx}{2 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}\right )\)

\(\Big \downarrow \) 2715

\(\displaystyle 4 \left (\frac {(b-c) \left (\frac {\int e^{-2 i x} \log \left (\frac {e^{2 i x} (b-c)}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}+1\right )de^{2 i x}}{4 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {(b-c) \left (\frac {\int e^{-2 i x} \log \left (\frac {e^{2 i x} (b-c)}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}+1\right )de^{2 i x}}{4 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle 4 \left (\frac {(b-c) \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {(b-c) \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}\right )\)

Input:

Int[x/(a + b*Cos[x]^2 + c*Sin[x]^2),x]
 

Output:

4*(((b - c)*(((-1/2*I)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sq 
rt[a + b]*Sqrt[a + c])])/(b - c) - PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a 
 + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*(b - c))))/(4*Sqrt[a + b]*Sqrt[ 
a + c]) - ((b - c)*(((-1/2*I)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c 
 + 2*Sqrt[a + b]*Sqrt[a + c])])/(b - c) - PolyLog[2, -(((b - c)*E^((2*I)*x 
))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*(b - c))))/(4*Sqrt[a + b 
]*Sqrt[a + c]))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3802

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*( 
x_)]), x_Symbol] :> Simp[2   Int[(c + d*x)^m*E^(I*Pi*(k - 1/2))*(E^(I*(e + 
f*x))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2*I*( 
e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 5098

Int[((f_.) + (g_.)*(x_))^(m_.)/((a_.) + Cos[(d_.) + (e_.)*(x_)]^2*(b_.) + ( 
c_.)*Sin[(d_.) + (e_.)*(x_)]^2), x_Symbol] :> Simp[2   Int[(f + g*x)^m/(2*a 
 + b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g} 
, x] && IGtQ[m, 0] && NeQ[a + b, 0] && NeQ[a + c, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (189 ) = 378\).

Time = 0.39 (sec) , antiderivative size = 820, normalized size of antiderivative = 3.43

method	result	size
risch	\(-\frac {i x \ln \left (1-\frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right )}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}}-\frac {x^{2}}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}}-\frac {\operatorname {polylog}\left (2, \frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right )}{4 \sqrt {\left (a +b \right ) \left (a +c \right )}}-\frac {i \ln \left (1-\frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) x}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}-\frac {i \ln \left (1-\frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) a x}{\sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {i \ln \left (1-\frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) b x}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {i \ln \left (1-\frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) c x}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {x^{2}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}-\frac {a \,x^{2}}{\sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {b \,x^{2}}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {c \,x^{2}}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right )}{2 \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) a}{2 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) b}{4 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}-\frac {\operatorname {polylog}\left (2, \frac {\left (b -c \right ) {\mathrm e}^{2 i x}}{-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c}\right ) c}{4 \sqrt {\left (a +b \right ) \left (a +c \right )}\, \left (-2 \sqrt {\left (a +b \right ) \left (a +c \right )}-2 a -b -c \right )}\)	\(820\)

Input:

int(x/(a+cos(x)^2*b+c*sin(x)^2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*I/((a+b)*(a+c))^(1/2)*x*ln(1-(b-c)*exp(2*I*x)/(2*((a+b)*(a+c))^(1/2)- 
2*a-b-c))-1/2/((a+b)*(a+c))^(1/2)*x^2-1/4/((a+b)*(a+c))^(1/2)*polylog(2,(b 
-c)*exp(2*I*x)/(2*((a+b)*(a+c))^(1/2)-2*a-b-c))-I/(-2*((a+b)*(a+c))^(1/2)- 
2*a-b-c)*ln(1-(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))*x-I/((a+b 
)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*ln(1-(b-c)*exp(2*I*x)/(-2* 
((a+b)*(a+c))^(1/2)-2*a-b-c))*a*x-1/2*I/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+ 
c))^(1/2)-2*a-b-c)*ln(1-(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)) 
*b*x-1/2*I/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*ln(1-(b-c) 
*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))*c*x-1/(-2*((a+b)*(a+c))^(1/2 
)-2*a-b-c)*x^2-1/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*a*x^ 
2-1/2/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*b*x^2-1/2/((a+b 
)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*c*x^2-1/2/(-2*((a+b)*(a+c) 
)^(1/2)-2*a-b-c)*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b- 
c))-1/2/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*polylog(2,(b- 
c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))*a-1/4/((a+b)*(a+c))^(1/2)/ 
(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+ 
c))^(1/2)-2*a-b-c))*b-1/4/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a- 
b-c)*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))*c
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2869 vs. \(2 (189) = 378\).

Time = 4.14 (sec) , antiderivative size = 2869, normalized size of antiderivative = 12.00 \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\text {Too large to display} \] Input:

integrate(x/(a+b*cos(x)^2+c*sin(x)^2),x, algorithm="fricas")
 

Output:

-1/4*(-I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*log(( 
((2*a + b + c)*cos(x) + (2*I*a + I*b + I*c)*sin(x) - 2*((b - c)*cos(x) + ( 
I*b - I*c)*sin(x))*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2)))*sqrt 
(-(2*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2)) + 2*a + b + 
 c)/(b - c)) + b - c)/(b - c)) + I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/ 
(b^2 - 2*b*c + c^2))*log(-(((2*a + b + c)*cos(x) - (2*I*a + I*b + I*c)*sin 
(x) - 2*((b - c)*cos(x) - (I*b - I*c)*sin(x))*sqrt((a^2 + a*b + (a + b)*c) 
/(b^2 - 2*b*c + c^2)))*sqrt(-(2*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 
- 2*b*c + c^2)) + 2*a + b + c)/(b - c)) - b + c)/(b - c)) + I*(b - c)*x*sq 
rt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*log((((2*a + b + c)*cos(x) 
 + (-2*I*a - I*b - I*c)*sin(x) - 2*((b - c)*cos(x) + (-I*b + I*c)*sin(x))* 
sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2)))*sqrt(-(2*(b - c)*sqrt(( 
a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2)) + 2*a + b + c)/(b - c)) + b - 
c)/(b - c)) - I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2) 
)*log(-(((2*a + b + c)*cos(x) - (-2*I*a - I*b - I*c)*sin(x) - 2*((b - c)*c 
os(x) - (-I*b + I*c)*sin(x))*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c 
^2)))*sqrt(-(2*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2)) + 
 2*a + b + c)/(b - c)) - b + c)/(b - c)) + I*(b - c)*x*sqrt((a^2 + a*b + ( 
a + b)*c)/(b^2 - 2*b*c + c^2))*log((((2*a + b + c)*cos(x) + (2*I*a + I*b + 
 I*c)*sin(x) + 2*((b - c)*cos(x) - (-I*b + I*c)*sin(x))*sqrt((a^2 + a*b...
 

Sympy [F]

\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int \frac {x}{a + b \cos ^{2}{\left (x \right )} + c \sin ^{2}{\left (x \right )}}\, dx \] Input:

integrate(x/(a+b*cos(x)**2+c*sin(x)**2),x)
 

Output:

Integral(x/(a + b*cos(x)**2 + c*sin(x)**2), x)
 

Maxima [F]

\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int { \frac {x}{b \cos \left (x\right )^{2} + c \sin \left (x\right )^{2} + a} \,d x } \] Input:

integrate(x/(a+b*cos(x)^2+c*sin(x)^2),x, algorithm="maxima")
 

Output:

integrate(x/(b*cos(x)^2 + c*sin(x)^2 + a), x)
 

Giac [F]

\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int { \frac {x}{b \cos \left (x\right )^{2} + c \sin \left (x\right )^{2} + a} \,d x } \] Input:

integrate(x/(a+b*cos(x)^2+c*sin(x)^2),x, algorithm="giac")
 

Output:

integrate(x/(b*cos(x)^2 + c*sin(x)^2 + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int \frac {x}{b\,{\cos \left (x\right )}^2+c\,{\sin \left (x\right )}^2+a} \,d x \] Input:

int(x/(a + c*sin(x)^2 + b*cos(x)^2),x)
 

Output:

int(x/(a + c*sin(x)^2 + b*cos(x)^2), x)
 

Reduce [F]

\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int \frac {x}{\cos \left (x \right )^{2} b +\sin \left (x \right )^{2} c +a}d x \] Input:

int(x/(a+b*cos(x)^2+c*sin(x)^2),x)
 

Output:

int(x/(cos(x)**2*b + sin(x)**2*c + a),x)