3.2 \(\int \frac{x^2}{a+b \sin ^2(x)} \, dx\)
Optimal. Leaf size=309 \[ -\frac{x \text{PolyLog}\left (2,\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{PolyLog}\left (2,\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \text{PolyLog}\left (3,\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{i \text{PolyLog}\left (3,\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}} \]
[Out]
((-I/2)*x^2*Log[1 - (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) + ((I/2)*x^2*Log
[1 - (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, (b*E^((2*I)*x))
/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt
[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - ((I/4)*PolyLog[3, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b
])])/(Sqrt[a]*Sqrt[a + b]) + ((I/4)*PolyLog[3, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sq
rt[a + b])
________________________________________________________________________________________
Rubi [A] time = 0.588276, antiderivative size = 309, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used =
{4585, 3321, 2264, 2190, 2531, 2282, 6589} \[ -\frac{x \text{PolyLog}\left (2,\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{PolyLog}\left (2,\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \text{PolyLog}\left (3,\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{i \text{PolyLog}\left (3,\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{4 \sqrt{a} \sqrt{a+b}}-\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )}{2 \sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
[In]
Int[x^2/(a + b*Sin[x]^2),x]
[Out]
((-I/2)*x^2*Log[1 - (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) + ((I/2)*x^2*Log
[1 - (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sqrt[a + b]) - (x*PolyLog[2, (b*E^((2*I)*x))
/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) + (x*PolyLog[2, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt
[a]*Sqrt[a + b])])/(2*Sqrt[a]*Sqrt[a + b]) - ((I/4)*PolyLog[3, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b
])])/(Sqrt[a]*Sqrt[a + b]) + ((I/4)*PolyLog[3, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])])/(Sqrt[a]*Sq
rt[a + b])
Rule 4585
Int[(x_)^(m_.)*((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]^2)^(n_), x_Symbol] :> Dist[1/2^n, Int[x^m*(2*a + b - b*Co
s[2*c + 2*d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a + b, 0] && IGtQ[m, 0] && ILtQ[n, 0] && (EqQ[n, -1
] || (EqQ[m, 1] && EqQ[n, -2]))
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 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 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 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 \sin ^2(x)} \, dx &=2 \int \frac{x^2}{2 a+b-b \cos (2 x)} \, dx\\ &=4 \int \frac{e^{2 i x} x^2}{-b+2 (2 a+b) e^{2 i x}-b e^{4 i x}} \, dx\\ &=-\frac{(2 b) \int \frac{e^{2 i x} x^2}{-4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)-2 b e^{2 i x}} \, dx}{\sqrt{a} \sqrt{a+b}}+\frac{(2 b) \int \frac{e^{2 i x} x^2}{4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)-2 b e^{2 i x}} \, dx}{\sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i \int x \log \left (1-\frac{2 b e^{2 i x}}{-4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right ) \, dx}{\sqrt{a} \sqrt{a+b}}-\frac{i \int x \log \left (1-\frac{2 b e^{2 i x}}{4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right ) \, dx}{\sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{Li}_2\left (\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left (\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{\int \text{Li}_2\left (\frac{2 b e^{2 i x}}{-4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt{a} \sqrt{a+b}}-\frac{\int \text{Li}_2\left (\frac{2 b e^{2 i x}}{4 \sqrt{a} \sqrt{a+b}+2 (2 a+b)}\right ) \, dx}{2 \sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{Li}_2\left (\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left (\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{x} \, dx,x,e^{2 i x}\right )}{4 \sqrt{a} \sqrt{a+b}}\\ &=-\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{i x^2 \log \left (1-\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{x \text{Li}_2\left (\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}+\frac{x \text{Li}_2\left (\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{2 \sqrt{a} \sqrt{a+b}}-\frac{i \text{Li}_3\left (\frac{b e^{2 i x}}{2 a+b-2 \sqrt{a} \sqrt{a+b}}\right )}{4 \sqrt{a} \sqrt{a+b}}+\frac{i \text{Li}_3\left (\frac{b e^{2 i x}}{2 a+b+2 \sqrt{a} \sqrt{a+b}}\right )}{4 \sqrt{a} \sqrt{a+b}}\\ \end{align*}
Mathematica [A] time = 2.64492, size = 237, normalized size = 0.77 \[ -\frac{i \left (-2 i x \text{PolyLog}\left (2,\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )+2 i x \text{PolyLog}\left (2,\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )+\text{PolyLog}\left (3,\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )-\text{PolyLog}\left (3,\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )+2 x^2 \log \left (1-\frac{b e^{2 i x}}{-2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )-2 x^2 \log \left (1-\frac{b e^{2 i x}}{2 \sqrt{a} \sqrt{a+b}+2 a+b}\right )\right )}{4 \sqrt{a} \sqrt{a+b}} \]
Antiderivative was successfully verified.
[In]
Integrate[x^2/(a + b*Sin[x]^2),x]
[Out]
((-I/4)*(2*x^2*Log[1 - (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] - 2*x^2*Log[1 - (b*E^((2*I)*x))/(2*a
+ b + 2*Sqrt[a]*Sqrt[a + b])] - (2*I)*x*PolyLog[2, (b*E^((2*I)*x))/(2*a + b - 2*Sqrt[a]*Sqrt[a + b])] + (2*I)
*x*PolyLog[2, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])] + PolyLog[3, (b*E^((2*I)*x))/(2*a + b - 2*Sqr
t[a]*Sqrt[a + b])] - PolyLog[3, (b*E^((2*I)*x))/(2*a + b + 2*Sqrt[a]*Sqrt[a + b])]))/(Sqrt[a]*Sqrt[a + b])
________________________________________________________________________________________
Maple [B] time = 0.109, size = 660, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(a+b*sin(x)^2),x)
[Out]
2/3/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*a*x^3-1/4*I/((a+b)*a)^(1/2)*polylog(3,b*exp(2*I*x)/(-2*((a+b)*a)
^(1/2)+2*a+b))-1/3/((a+b)*a)^(1/2)*x^3+1/4*I/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*b*polylog(3,b*exp(2*I*x
)/(2*((a+b)*a)^(1/2)+2*a+b))+1/2/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*b*x*polylog(2,b*exp(2*I*x)/(2*((a+b
)*a)^(1/2)+2*a+b))+I/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*a*x^2*ln(1-b*exp(2*I*x)/(2*((a+b)*a)^(1/2)+2*a+
b))+2/3/(2*((a+b)*a)^(1/2)+2*a+b)*x^3-1/2*I/((a+b)*a)^(1/2)*x^2*ln(1-b*exp(2*I*x)/(-2*((a+b)*a)^(1/2)+2*a+b))+
1/(2*((a+b)*a)^(1/2)+2*a+b)*x*polylog(2,b*exp(2*I*x)/(2*((a+b)*a)^(1/2)+2*a+b))+1/2*I/(2*((a+b)*a)^(1/2)+2*a+b
)*polylog(3,b*exp(2*I*x)/(2*((a+b)*a)^(1/2)+2*a+b))+1/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*a*x*polylog(2,
b*exp(2*I*x)/(2*((a+b)*a)^(1/2)+2*a+b))+1/2*I/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*b*x^2*ln(1-b*exp(2*I*x
)/(2*((a+b)*a)^(1/2)+2*a+b))-1/2/((a+b)*a)^(1/2)*x*polylog(2,b*exp(2*I*x)/(-2*((a+b)*a)^(1/2)+2*a+b))+I/(2*((a
+b)*a)^(1/2)+2*a+b)*x^2*ln(1-b*exp(2*I*x)/(2*((a+b)*a)^(1/2)+2*a+b))+1/3/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*
a+b)*b*x^3+1/2*I/((a+b)*a)^(1/2)/(2*((a+b)*a)^(1/2)+2*a+b)*a*polylog(3,b*exp(2*I*x)/(2*((a+b)*a)^(1/2)+2*a+b))
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b \sin \left (x\right )^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(a+b*sin(x)^2),x, algorithm="maxima")
[Out]
integrate(x^2/(b*sin(x)^2 + a), x)
________________________________________________________________________________________
Fricas [C] time = 1.05823, size = 6359, normalized size = 20.58 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(a+b*sin(x)^2),x, algorithm="fricas")
[Out]
1/16*(4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*
b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) - 4*I*b*x^2*sqrt((a^2
+ a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*
b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2
*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt(
(a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (-4
*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a
+ b)/b) - 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*
(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) + 4*I*
b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))
*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 4*I*b*x^2*sqrt((a^2 + a*b)/
b^2)*log(1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))
*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b) - 4*I*b*x^2*sqrt((a^2 + a*b)/b^2)*log(-1/2*((2*(2*a
+ b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2
+ a*b)/b^2) - 2*a - b)/b) - 2*b)/b) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (4*I*a + 2
*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)
+ 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos
(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) + 8*b*x*s
qrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqr
t((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) + 2*b)/b + 1) + 8*b*x*sqrt((a^2 + a*b)/b^2)*
dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sq
rt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b) - 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a +
b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a
*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (4*I*a + 2
*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)
- 2*b)/b + 1) - 8*b*x*sqrt((a^2 + a*b)/b^2)*dilog(-1/2*((2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x) + 4*(b*
cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) + 2*b)/b + 1) - 8*b
*x*sqrt((a^2 + a*b)/b^2)*dilog(1/2*((2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*
sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b) - 2*b)/b + 1) + 8*I*b*sqrt((a^2 + a*b)/b
^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b
^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a +
b)*cos(x) - (4*I*a + 2*I*b)*sin(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*
b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*s
in(x) - 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) + 8*
I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) - 4*(b*cos(x) + I*b*si
n(x))*sqrt((a^2 + a*b)/b^2))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)/b) - 8*I*b*sqrt((a^2 + a*b)/b^2)*po
lylog(3, 1/2*(2*(2*a + b)*cos(x) + (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*s
qrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*co
s(x) - (4*I*a + 2*I*b)*sin(x) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b
^2) - 2*a - b)/b)/b) + 8*I*b*sqrt((a^2 + a*b)/b^2)*polylog(3, 1/2*(2*(2*a + b)*cos(x) + (-4*I*a - 2*I*b)*sin(x
) + 4*(b*cos(x) - I*b*sin(x))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b) - 8*I*b
*sqrt((a^2 + a*b)/b^2)*polylog(3, -1/2*(2*(2*a + b)*cos(x) - (-4*I*a - 2*I*b)*sin(x) + 4*(b*cos(x) + I*b*sin(x
))*sqrt((a^2 + a*b)/b^2))*sqrt(-(2*b*sqrt((a^2 + a*b)/b^2) - 2*a - b)/b)/b))/(a^2 + a*b)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{a + b \sin ^{2}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2/(a+b*sin(x)**2),x)
[Out]
Integral(x**2/(a + b*sin(x)**2), x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b \sin \left (x\right )^{2} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2/(a+b*sin(x)^2),x, algorithm="giac")
[Out]
integrate(x^2/(b*sin(x)^2 + a), x)