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

Optimal. Leaf size=365 \[ -\frac {x \text {Li}_2\left (-\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{2 \sqrt {a+b} \sqrt {a+c}}+\frac {x \text {Li}_2\left (-\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{2 \sqrt {a+b} \sqrt {a+c}}-\frac {i \text {Li}_3\left (-\frac {(b-c) e^{2 i x}}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 \sqrt {a+b} \sqrt {a+c}}+\frac {i \text {Li}_3\left (-\frac {(b-c) e^{2 i x}}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {i x^2 \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 \sqrt {a+b} \sqrt {a+c}}+\frac {i x^2 \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 \sqrt {a+b} \sqrt {a+c}} \]

[Out]

-1/2*I*x^2*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^2*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*x*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)+1/2*x*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*I*polylog(3,-(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*I*polylog(3,-(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)

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Rubi [A]  time = 0.74, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {4587, 3321, 2264, 2190, 2531, 2282, 6589} \[ -\frac {x \text {PolyLog}\left (2,-\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 \sqrt {a+b} \sqrt {a+c}}+\frac {x \text {PolyLog}\left (2,-\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 \sqrt {a+b} \sqrt {a+c}}-\frac {i \text {PolyLog}\left (3,-\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{4 \sqrt {a+b} \sqrt {a+c}}+\frac {i \text {PolyLog}\left (3,-\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{4 \sqrt {a+b} \sqrt {a+c}}-\frac {i x^2 \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 \sqrt {a+b} \sqrt {a+c}}+\frac {i x^2 \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {a+b} \sqrt {a+c}+2 a+b+c}\right )}{2 \sqrt {a+b} \sqrt {a+c}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/2)*x^2*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^2*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]) - (x*PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(2*Sqrt[a + b]*Sqrt[
a + c]) + (x*PolyLog[2, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(2*Sqrt[a + b]*Sq
rt[a + c]) - ((I/4)*PolyLog[3, -(((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[a + b]*Sqrt[a + c]))])/(Sqrt[a +
b]*Sqrt[a + c]) + ((I/4)*PolyLog[3, -(((b - c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[a + b]*Sqrt[a + c]))])/(Sqrt
[a + b]*Sqrt[a + c])

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 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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 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 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 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {x^2}{a+b \cos ^2(x)+c \sin ^2(x)} \, dx &=2 \int \frac {x^2}{2 a+b+c+(b-c) \cos (2 x)} \, dx\\ &=4 \int \frac {e^{2 i x} x^2}{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^2}{-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^2}{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^2 \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^2 \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 x \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}{\sqrt {a+b} \sqrt {a+c}}-\frac {i \int x \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}{\sqrt {a+b} \sqrt {a+c}}\\ &=-\frac {i x^2 \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^2 \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 {x \text {Li}_2\left (-\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 {x \text {Li}_2\left (-\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 {\int \text {Li}_2\left (-\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 {\int \text {Li}_2\left (-\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^2 \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^2 \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 {x \text {Li}_2\left (-\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 {x \text {Li}_2\left (-\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 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {(-b+c) x}{2 a+b+c-2 \sqrt {a+b} \sqrt {a+c}}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt {a+b} \sqrt {a+c}}+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {(-b+c) x}{2 a+b+c+2 \sqrt {a+b} \sqrt {a+c}}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt {a+b} \sqrt {a+c}}\\ &=-\frac {i x^2 \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^2 \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 {x \text {Li}_2\left (-\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 {x \text {Li}_2\left (-\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 \text {Li}_3\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 {i \text {Li}_3\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*}

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Mathematica [A]  time = 3.79, size = 258, normalized size = 0.71 \[ -\frac {i \left (-2 i x \text {Li}_2\left (\frac {(c-b) e^{2 i x}}{2 a+b+c-2 \sqrt {(a+b) (a+c)}}\right )+2 i x \text {Li}_2\left (\frac {(c-b) e^{2 i x}}{2 a+b+c+2 \sqrt {(a+b) (a+c)}}\right )+\text {Li}_3\left (\frac {(c-b) e^{2 i x}}{2 a+b+c-2 \sqrt {(a+b) (a+c)}}\right )-\text {Li}_3\left (\frac {(c-b) e^{2 i x}}{2 a+b+c+2 \sqrt {(a+b) (a+c)}}\right )+2 x^2 \log \left (1+\frac {e^{2 i x} (b-c)}{-2 \sqrt {(a+b) (a+c)}+2 a+b+c}\right )-2 x^2 \log \left (1+\frac {e^{2 i x} (b-c)}{2 \sqrt {(a+b) (a+c)}+2 a+b+c}\right )\right )}{4 \sqrt {(a+b) (a+c)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-1/4*I)*(2*x^2*Log[1 + ((b - c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[(a + b)*(a + c)])] - 2*x^2*Log[1 + ((b -
c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[(a + b)*(a + c)])] - (2*I)*x*PolyLog[2, ((-b + c)*E^((2*I)*x))/(2*a + b
+ c - 2*Sqrt[(a + b)*(a + c)])] + (2*I)*x*PolyLog[2, ((-b + c)*E^((2*I)*x))/(2*a + b + c + 2*Sqrt[(a + b)*(a +
 c)])] + PolyLog[3, ((-b + c)*E^((2*I)*x))/(2*a + b + c - 2*Sqrt[(a + b)*(a + c)])] - PolyLog[3, ((-b + c)*E^(
(2*I)*x))/(2*a + b + c + 2*Sqrt[(a + b)*(a + c)])]))/Sqrt[(a + b)*(a + c)]

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fricas [C]  time = 2.22, size = 4357, normalized size = 11.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(4*I*(b - c)*x^2*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^2*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^2*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^2*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^2*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)*s
in(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^2*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^2*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)*co
s(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^2*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)) + 8*(b - c)*x*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) + 8*(
b - c)*x*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) + 8*(b - c)*x*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) + 8*(b - c)*x*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) - 8*(b - c)*x*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) - 8*(b - c)*x*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) - 8*(b - c)*x*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) - 8*(b - c)*x*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*(2*I*b - 2*I*c)*sqrt
((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, -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))/(b - c)) + 4*(-2*I*b + 2*I
*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, 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))/(b - c)) + 4*(-2*I*
b + 2*I*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, -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))/(b - c)) +
4*(2*I*b - 2*I*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, 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))/(b -
c)) + 4*(-2*I*b + 2*I*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, -1/2*(2*(2*a + b + c)*co
s(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))
/(b - c)) + 4*(2*I*b - 2*I*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, 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))/(b - c)) + 4*(2*I*b - 2*I*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, -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))/(b - c)) + 4*(-2*I*b + 2*I*c)*sqrt((a^2 + a*b + (a + b)*c)/(b^2 - 2*b*c + c^2))*polylog(3, 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))/(b - c)))/(a^2 + a*b + (a + b)*c)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{b \cos \relax (x)^{2} + c \sin \relax (x)^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.29, size = 1161, normalized size = 3.18 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*x*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))-1/4*I/((a+b
)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*c*polylog(3,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))
-1/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*a*x*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)
-2*a-b-c))-1/2*I/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*b*x^2*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*polylog(2,(b-c)*exp(2*I*x)/(2*((a+b)*(a+c))^(1/2)-2*a-b-c))-1/
4*I/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*b*polylog(3,(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)*c*x*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(
a+c))^(1/2)-2*a-b-c))-1/2*I/((a+b)*(a+c))^(1/2)*x^2*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)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*b*x*polylog(2,(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2
*a-b-c))-1/4*I/((a+b)*(a+c))^(1/2)*polylog(3,(b-c)*exp(2*I*x)/(2*((a+b)*(a+c))^(1/2)-2*a-b-c))-2/3/((a+b)*(a+c
))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*a*x^3-1/2*I/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*polylog(3,(b-c)*exp(2*I
*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))-2/3/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*x^3-I/((a+b)*(a+c))^(1/2)/(-2*((a+b
)*(a+c))^(1/2)-2*a-b-c)*a*x^2*ln(1-(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))-1/3/((a+b)*(a+c))^(1/2)/
(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*b*x^3-1/2*I/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*c*x^2*ln(1-(
b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))-1/3/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*c*x
^3-I/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*x^2*ln(1-(b-c)*exp(2*I*x)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c))-1/3/((a+b)*(
a+c))^(1/2)*x^3-1/2*I/((a+b)*(a+c))^(1/2)/(-2*((a+b)*(a+c))^(1/2)-2*a-b-c)*a*polylog(3,(b-c)*exp(2*I*x)/(-2*((
a+b)*(a+c))^(1/2)-2*a-b-c))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{b \cos \relax (x)^{2} + c \sin \relax (x)^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{b\,{\cos \relax (x)}^2+c\,{\sin \relax (x)}^2+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{a + b \cos ^{2}{\relax (x )} + c \sin ^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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