3.497 \(\int \frac{x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx\)
Optimal. Leaf size=239 \[ -\frac{\text{PolyLog}\left (2,-\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{\text{PolyLog}\left (2,-\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{4 \sqrt{a+b} \sqrt{a+c}}-\frac{i x \log \left (1+\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{2 \sqrt{a+b} \sqrt{a+c}}+\frac{i x \log \left (1+\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{2 \sqrt{a+b} \sqrt{a+c}} \]
[Out]
((-I/2)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c])
+ ((I/2)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c])
- PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c])
+ PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c])
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Rubi [A] time = 0.491629, antiderivative size = 239, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used =
{4587, 3321, 2264, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{4 \sqrt{a+b} \sqrt{a+c}}+\frac{\text{PolyLog}\left (2,-\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{4 \sqrt{a+b} \sqrt{a+c}}-\frac{i x \log \left (1+\frac{e^{2 i x} (b-c)}{-2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{2 \sqrt{a+b} \sqrt{a+c}}+\frac{i x \log \left (1+\frac{e^{2 i x} (b-c)}{2 \sqrt{a+b} \sqrt{a+c}+2 a+b+c}\right )}{2 \sqrt{a+b} \sqrt{a+c}} \]
Antiderivative was successfully verified.
[In]
Int[x/(a + b*Cos[x]^2 + c*Sin[x]^2),x]
[Out]
((-I/2)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c])
+ ((I/2)*x*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c])])/(Sqrt[a + b]*Sqrt[a + c])
- PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c])
+ PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))]/(4*Sqrt[a + b]*Sqrt[a + c])
Rule 4587
Int[((f_.) + (g_.)*(x_))^(m_.)/((a_.) + Cos[(d_.) + (e_.)*(x_)]^2*(b_.) + (c_.)*Sin[(d_.) + (e_.)*(x_)]^2), x_
Symbol] :> Dist[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]
Rule 3321
Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[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 2264
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]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(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 2190
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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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 2279
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2391
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]
Rubi steps
\begin{align*} \int \frac{x}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx &=2 \int \frac{x}{2 a+b+c+(b-c) \cos (2 x)} \, dx\\ &=4 \int \frac{e^{2 i x} x}{b-c+2 (2 a+b+c) e^{2 i x}+(b-c) e^{4 i x}} \, dx\\ &=\frac{(2 (b-c)) \int \frac{e^{2 i x} x}{-4 \sqrt{a+b} \sqrt{a+c}+2 (2 a+b+c)+2 (b-c) e^{2 i x}} \, dx}{\sqrt{a+b} \sqrt{a+c}}-\frac{(2 (b-c)) \int \frac{e^{2 i x} x}{4 \sqrt{a+b} \sqrt{a+c}+2 (2 a+b+c)+2 (b-c) e^{2 i x}} \, dx}{\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{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 \int \log \left (1+\frac{2 (b-c) e^{2 i x}}{-4 \sqrt{a+b} \sqrt{a+c}+2 (2 a+b+c)}\right ) \, dx}{2 \sqrt{a+b} \sqrt{a+c}}-\frac{i \int \log \left (1+\frac{2 (b-c) e^{2 i x}}{4 \sqrt{a+b} \sqrt{a+c}+2 (2 a+b+c)}\right ) \, dx}{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{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{Subst}\left (\int \frac{\log \left (1+\frac{2 (b-c) x}{-4 \sqrt{a+b} \sqrt{a+c}+2 (2 a+b+c)}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt{a+b} \sqrt{a+c}}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (b-c) x}{4 \sqrt{a+b} \sqrt{a+c}+2 (2 a+b+c)}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \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{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{\text{Li}_2\left (-\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{\text{Li}_2\left (-\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}}\\ \end{align*}
Mathematica [B] time = 3.1501, size = 507, normalized size = 2.12 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right ) \left (2 x+\frac{i \left (-\text{PolyLog}\left (2,\frac{\sqrt{a+b}-i \sqrt{a+c} \tan (x)}{\sqrt{a+b}-\sqrt{a+c}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{a+b}-i \sqrt{a+c} \tan (x)}{\sqrt{a+b}+\sqrt{a+c}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{a+b}+i \sqrt{a+c} \tan (x)}{\sqrt{a+b}-\sqrt{a+c}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{a+b}+i \sqrt{a+c} \tan (x)}{\sqrt{a+b}+\sqrt{a+c}}\right )+\log \left (\frac{\sqrt{a+c} (1+i \tan (x))}{\sqrt{a+b}+\sqrt{a+c}}\right ) \log \left (1-\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )-\log \left (\frac{i \sqrt{a+c} (\tan (x)+i)}{\sqrt{a+b}-\sqrt{a+c}}\right ) \log \left (1-\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )+\log \left (\frac{\sqrt{a+c} (1-i \tan (x))}{\sqrt{a+b}+\sqrt{a+c}}\right ) \log \left (1+\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )-\log \left (\frac{\sqrt{a+c} (1+i \tan (x))}{\sqrt{a+c}-\sqrt{a+b}}\right ) \log \left (1+\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )\right )}{\log \left (1-\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )-\log \left (1+\frac{i \sqrt{a+c} \tan (x)}{\sqrt{a+b}}\right )}\right )}{2 \sqrt{a+b} \sqrt{a+c}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[x/(a + b*Cos[x]^2 + c*Sin[x]^2),x]
[Out]
(ArcTan[(Sqrt[a + c]*Tan[x])/Sqrt[a + b]]*(2*x + (I*(Log[(Sqrt[a + c]*(1 + I*Tan[x]))/(Sqrt[a + b] + Sqrt[a +
c])]*Log[1 - (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - Log[(I*Sqrt[a + c]*(I + Tan[x]))/(Sqrt[a + b] - Sqrt[a + c]
)]*Log[1 - (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] + Log[(Sqrt[a + c]*(1 - I*Tan[x]))/(Sqrt[a + b] + Sqrt[a + c])]
*Log[1 + (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - Log[(Sqrt[a + c]*(1 + I*Tan[x]))/(-Sqrt[a + b] + Sqrt[a + c])]*
Log[1 + (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - PolyLog[2, (Sqrt[a + b] - I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] - S
qrt[a + c])] + PolyLog[2, (Sqrt[a + b] - I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] + Sqrt[a + c])] - PolyLog[2, (Sqrt
[a + b] + I*Sqrt[a + c]*Tan[x])/(Sqrt[a + b] - Sqrt[a + c])] + PolyLog[2, (Sqrt[a + b] + I*Sqrt[a + c]*Tan[x])
/(Sqrt[a + b] + Sqrt[a + c])]))/(Log[1 - (I*Sqrt[a + c]*Tan[x])/Sqrt[a + b]] - Log[1 + (I*Sqrt[a + c]*Tan[x])/
Sqrt[a + b]])))/(2*Sqrt[a + b]*Sqrt[a + c])
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Maple [B] time = 0.105, size = 820, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x/(a+b*cos(x)^2+c*sin(x)^2),x)
[Out]
-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))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{b \cos \left (x\right )^{2} + c \sin \left (x\right )^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*cos(x)^2+c*sin(x)^2),x, algorithm="maxima")
[Out]
integrate(x/(b*cos(x)^2 + c*sin(x)^2 + a), x)
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Fricas [B] time = 4.13438, size = 7385, normalized size = 30.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*cos(x)^2+c*sin(x)^2),x, algorithm="fricas")
[Out]
1/16*(4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*log(-1/2*((2*(2*a + b + c)*cos(x) + (4*I
*a + 2*I*b + 2*I*c)*sin(x) - 4*((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)) - 2*b +
2*c)/(b - c)) - 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*log(1/2*((2*(2*a + b + c)*cos(
x) - (4*I*a + 2*I*b + 2*I*c)*sin(x) - 4*((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))
+ 2*b - 2*c)/(b - c)) - 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*log(-1/2*((2*(2*a + b
+ c)*cos(x) + (-4*I*a - 2*I*b - 2*I*c)*sin(x) - 4*((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)) - 2*b + 2*c)/(b - c)) + 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*log(1/2*((2
*(2*a + b + c)*cos(x) - (-4*I*a - 2*I*b - 2*I*c)*sin(x) - 4*((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)) + 2*b - 2*c)/(b - c)) - 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*lo
g(-1/2*((2*(2*a + b + c)*cos(x) + (4*I*a + 2*I*b + 2*I*c)*sin(x) + 4*((b - c)*cos(x) + (I*b - I*c)*sin(x))*sqr
t((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)) - 2*b + 2*c)/(b - c)) + 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c +
c^2))*log(1/2*((2*(2*a + b + c)*cos(x) - (4*I*a + 2*I*b + 2*I*c)*sin(x) + 4*((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)) + 2*b - 2*c)/(b - c)) + 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2
*b*c + c^2))*log(-1/2*((2*(2*a + b + c)*cos(x) + (-4*I*a - 2*I*b - 2*I*c)*sin(x) + 4*((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)) - 2*b + 2*c)/(b - c)) - 4*I*(b - c)*x*sqrt((a^2 + a*b + (a + b)*c
)/(b^2 - 2*b*c + c^2))*log(1/2*((2*(2*a + b + c)*cos(x) - (-4*I*a - 2*I*b - 2*I*c)*sin(x) + 4*((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)) + 2*b - 2*c)/(b - c)) + 4*(b - c)*sqrt((a^2 + a*b + (a +
b)*c)/(b^2 - 2*b*c + c^2))*dilog(1/2*((2*(2*a + b + c)*cos(x) + (4*I*a + 2*I*b + 2*I*c)*sin(x) - 4*((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)) - 2*b + 2*c)/(b - c) + 1) + 4*(b - c)*sqrt((a^2 +
a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(-1/2*((2*(2*a + b + c)*cos(x) - (4*I*a + 2*I*b + 2*I*c)*sin(x) -
4*((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)*s
qrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2)) + 2*a + b + c)/(b - c)) + 2*b - 2*c)/(b - c) + 1) + 4*(b - c)
*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(1/2*((2*(2*a + b + c)*cos(x) + (-4*I*a - 2*I*b - 2*I*
c)*sin(x) - 4*((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)) - 2*b + 2*c)/(b - c) + 1)
+ 4*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(-1/2*((2*(2*a + b + c)*cos(x) - (-4*I*a -
2*I*b - 2*I*c)*sin(x) - 4*((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)) + 2*b - 2*c)/(
b - c) + 1) - 4*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(1/2*((2*(2*a + b + c)*cos(x) +
(4*I*a + 2*I*b + 2*I*c)*sin(x) + 4*((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)) - 2*b
+ 2*c)/(b - c) + 1) - 4*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(-1/2*((2*(2*a + b + c
)*cos(x) - (4*I*a + 2*I*b + 2*I*c)*sin(x) + 4*((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)) + 2*b - 2*c)/(b - c) + 1) - 4*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(1/2*((2*(2*
a + b + c)*cos(x) + (-4*I*a - 2*I*b - 2*I*c)*sin(x) + 4*((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)) - 2*b + 2*c)/(b - c) + 1) - 4*(b - c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*dilog(
-1/2*((2*(2*a + b + c)*cos(x) - (-4*I*a - 2*I*b - 2*I*c)*sin(x) + 4*((b - c)*cos(x) - (-I*b + I*c)*sin(x))*sqr
t((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)) + 2*b - 2*c)/(b - c) + 1))/(a^2 + a*b + (a + b)*c)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{a + b \cos ^{2}{\left (x \right )} + c \sin ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*cos(x)**2+c*sin(x)**2),x)
[Out]
Integral(x/(a + b*cos(x)**2 + c*sin(x)**2), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{b \cos \left (x\right )^{2} + c \sin \left (x\right )^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x/(a+b*cos(x)^2+c*sin(x)^2),x, algorithm="giac")
[Out]
integrate(x/(b*cos(x)^2 + c*sin(x)^2 + a), x)