Integrand size = 18, antiderivative size = 239 \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=-\frac {i x \log \left (1+\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{2 \sqrt {a+b} \sqrt {a+c}}+\frac {i x \log \left (1+\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{2 \sqrt {a+b} \sqrt {a+c}}-\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 \sqrt {a+b} \sqrt {a+c}}+\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 \sqrt {a+b} \sqrt {a+c}} \]
-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)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1022\) vs. \(2(239)=478\).
Time = 1.23 (sec) , antiderivative size = 1022, normalized size of antiderivative = 4.28 \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\frac {4 x \text {arctanh}\left (\frac {(a+b) \cot (x)}{\sqrt {-((a+b) (a+c))}}\right )-2 \arccos \left (-\frac {2 a+b+c}{b-c}\right ) \text {arctanh}\left (\frac {\sqrt {-((a+b) (a+c))} \tan (x)}{a+b}\right )+\arccos \left (-\frac {2 a+b+c}{b-c}\right ) \log \left (\frac {\sqrt {2} \sqrt {-((a+b) (a+c))} e^{-i x}}{\sqrt {b-c} \sqrt {2 a+b+c+(b-c) \cos (2 x)}}\right )-2 i \text {arctanh}\left (\frac {(a+b) \cot (x)}{\sqrt {-((a+b) (a+c))}}\right ) \log \left (\frac {\sqrt {2} \sqrt {-((a+b) (a+c))} e^{-i x}}{\sqrt {b-c} \sqrt {2 a+b+c+(b-c) \cos (2 x)}}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-((a+b) (a+c))} \tan (x)}{a+b}\right ) \log \left (\frac {\sqrt {2} \sqrt {-((a+b) (a+c))} e^{-i x}}{\sqrt {b-c} \sqrt {2 a+b+c+(b-c) \cos (2 x)}}\right )+\arccos \left (-\frac {2 a+b+c}{b-c}\right ) \log \left (\frac {\sqrt {2} \sqrt {-((a+b) (a+c))} e^{i x}}{\sqrt {b-c} \sqrt {2 a+b+c+(b-c) \cos (2 x)}}\right )+2 i \text {arctanh}\left (\frac {(a+b) \cot (x)}{\sqrt {-((a+b) (a+c))}}\right ) \log \left (\frac {\sqrt {2} \sqrt {-((a+b) (a+c))} e^{i x}}{\sqrt {b-c} \sqrt {2 a+b+c+(b-c) \cos (2 x)}}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-((a+b) (a+c))} \tan (x)}{a+b}\right ) \log \left (\frac {\sqrt {2} \sqrt {-((a+b) (a+c))} e^{i x}}{\sqrt {b-c} \sqrt {2 a+b+c+(b-c) \cos (2 x)}}\right )-\arccos \left (-\frac {2 a+b+c}{b-c}\right ) \log \left (\frac {2 (a+b) \left (-i a-i c+\sqrt {-((a+b) (a+c))}\right ) (-i+\tan (x))}{(b-c) \left (a+b+\sqrt {-((a+b) (a+c))} \tan (x)\right )}\right )+2 i \text {arctanh}\left (\frac {\sqrt {-((a+b) (a+c))} \tan (x)}{a+b}\right ) \log \left (\frac {2 (a+b) \left (-i a-i c+\sqrt {-((a+b) (a+c))}\right ) (-i+\tan (x))}{(b-c) \left (a+b+\sqrt {-((a+b) (a+c))} \tan (x)\right )}\right )-\arccos \left (-\frac {2 a+b+c}{b-c}\right ) \log \left (\frac {2 (a+b) \left (i a+i c+\sqrt {-((a+b) (a+c))}\right ) (i+\tan (x))}{(b-c) \left (a+b+\sqrt {-((a+b) (a+c))} \tan (x)\right )}\right )-2 i \text {arctanh}\left (\frac {\sqrt {-((a+b) (a+c))} \tan (x)}{a+b}\right ) \log \left (\frac {2 (a+b) \left (i a+i c+\sqrt {-((a+b) (a+c))}\right ) (i+\tan (x))}{(b-c) \left (a+b+\sqrt {-((a+b) (a+c))} \tan (x)\right )}\right )+i \operatorname {PolyLog}\left (2,\frac {\left (2 a+b+c-2 i \sqrt {-((a+b) (a+c))}\right ) \left (a+b-\sqrt {-((a+b) (a+c))} \tan (x)\right )}{(b-c) \left (a+b+\sqrt {-((a+b) (a+c))} \tan (x)\right )}\right )-i \operatorname {PolyLog}\left (2,\frac {\left (2 a+b+c+2 i \sqrt {-((a+b) (a+c))}\right ) \left (a+b-\sqrt {-((a+b) (a+c))} \tan (x)\right )}{(b-c) \left (a+b+\sqrt {-((a+b) (a+c))} \tan (x)\right )}\right )}{4 \sqrt {-((a+b) (a+c))}} \]
(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))])*...
Time = 0.88 (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 |
\(\displaystyle 4 \left (\frac {(b-c) \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {(b-c) \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 (b-c)}-\frac {i x \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 (b-c)}\right )}{4 \sqrt {a+b} \sqrt {a+c}}\right )\) |
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]))
3.5.97.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
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]
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 = 0.91 (sec) , antiderivative size = 820, normalized size of antiderivative = 3.43
method | result | size |
risch | \(-\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 )}-\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 )}}\) | \(820\) |
-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-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))
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 = 4.76 (sec) , antiderivative size = 2869, normalized size of antiderivative = 12.00 \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\text {Too large to display} \]
-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...
\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int \frac {x}{a + b \cos ^{2}{\left (x \right )} + c \sin ^{2}{\left (x \right )}}\, dx \]
\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int { \frac {x}{b \cos \left (x\right )^{2} + c \sin \left (x\right )^{2} + a} \,d x } \]
\[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int { \frac {x}{b \cos \left (x\right )^{2} + c \sin \left (x\right )^{2} + a} \,d x } \]
Timed out. \[ \int \frac {x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx=\int \frac {x}{b\,{\cos \left (x\right )}^2+c\,{\sin \left (x\right )}^2+a} \,d x \]