3.42 \(\int \frac{x^2}{a+b \csc (c+d \sqrt{x})} \, dx\)

Optimal. Leaf size=807 \[ \frac{x^3}{3 a}+\frac{2 i b \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d}-\frac{2 i b \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d}+\frac{10 b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^2}-\frac{10 b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^2}+\frac{40 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^3}-\frac{40 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^3}-\frac{120 b \text{PolyLog}\left (4,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^4}+\frac{120 b \text{PolyLog}\left (4,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^4}-\frac{240 i b \text{PolyLog}\left (5,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^5}+\frac{240 i b \text{PolyLog}\left (5,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^5}+\frac{240 b \text{PolyLog}\left (6,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^6}-\frac{240 b \text{PolyLog}\left (6,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^6} \]

[Out]

x^3/(3*a) + ((2*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*
d) - ((2*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (1
0*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (10*b*x^2*P
olyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((40*I)*b*x^(3/2)*Po
lyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((40*I)*b*x^(3/2)*Pol
yLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, (I
*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) - ((240*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + ((240*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + (240*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x]
)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) - (240*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqr
t[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6)

________________________________________________________________________________________

Rubi [A]  time = 1.21782, antiderivative size = 807, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {4205, 4191, 3323, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac{x^3}{3 a}+\frac{2 i b \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d}-\frac{2 i b \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{5/2}}{a \sqrt{b^2-a^2} d}+\frac{10 b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^2}-\frac{10 b \text{PolyLog}\left (2,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^2}{a \sqrt{b^2-a^2} d^2}+\frac{40 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^3}-\frac{40 i b \text{PolyLog}\left (3,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x^{3/2}}{a \sqrt{b^2-a^2} d^3}-\frac{120 b \text{PolyLog}\left (4,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^4}+\frac{120 b \text{PolyLog}\left (4,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) x}{a \sqrt{b^2-a^2} d^4}-\frac{240 i b \text{PolyLog}\left (5,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^5}+\frac{240 i b \text{PolyLog}\left (5,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right ) \sqrt{x}}{a \sqrt{b^2-a^2} d^5}+\frac{240 b \text{PolyLog}\left (6,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^6}-\frac{240 b \text{PolyLog}\left (6,\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{b^2-a^2}}\right )}{a \sqrt{b^2-a^2} d^6} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*Csc[c + d*Sqrt[x]]),x]

[Out]

x^3/(3*a) + ((2*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*
d) - ((2*I)*b*x^(5/2)*Log[1 - (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d) + (1
0*b*x^2*PolyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) - (10*b*x^2*P
olyLog[2, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^2) + ((40*I)*b*x^(3/2)*Po
lyLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - ((40*I)*b*x^(3/2)*Pol
yLog[3, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, (I
*a*E^(I*(c + d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^4) - ((240*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + ((240*I)*b*Sqrt[x]*PolyLog[5, (I*a*E^(I*(c
+ d*Sqrt[x])))/(b + Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^5) + (240*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x]
)))/(b - Sqrt[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6) - (240*b*PolyLog[6, (I*a*E^(I*(c + d*Sqrt[x])))/(b + Sqr
t[-a^2 + b^2])])/(a*Sqrt[-a^2 + b^2]*d^6)

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
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 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, 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 \csc \left (c+d \sqrt{x}\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{a+b \csc (c+d x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{x^5}{a}-\frac{b x^5}{a (b+a \sin (c+d x))}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{x^3}{3 a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^5}{b+a \sin (c+d x)} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{x^3}{3 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^5}{i a+2 b e^{i (c+d x)}-i a e^{2 i (c+d x)}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{x^3}{3 a}+\frac{(4 i b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^5}{2 b-2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}-\frac{(4 i b) \operatorname{Subst}\left (\int \frac{e^{i (c+d x)} x^5}{2 b+2 \sqrt{-a^2+b^2}-2 i a e^{i (c+d x)}} \, dx,x,\sqrt{x}\right )}{\sqrt{-a^2+b^2}}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{(10 i b) \operatorname{Subst}\left (\int x^4 \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}+\frac{(10 i b) \operatorname{Subst}\left (\int x^4 \log \left (1-\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{(40 b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{(40 b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^2}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{(120 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^3}+\frac{(120 i b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^3}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{(240 b) \operatorname{Subst}\left (\int x \text{Li}_4\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{(240 b) \operatorname{Subst}\left (\int x \text{Li}_4\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^4}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{240 i b \sqrt{x} \text{Li}_5\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{240 i b \sqrt{x} \text{Li}_5\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{(240 i b) \operatorname{Subst}\left (\int \text{Li}_5\left (\frac{2 i a e^{i (c+d x)}}{2 b-2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^5}-\frac{(240 i b) \operatorname{Subst}\left (\int \text{Li}_5\left (\frac{2 i a e^{i (c+d x)}}{2 b+2 \sqrt{-a^2+b^2}}\right ) \, dx,x,\sqrt{x}\right )}{a \sqrt{-a^2+b^2} d^5}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{240 i b \sqrt{x} \text{Li}_5\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{240 i b \sqrt{x} \text{Li}_5\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{(240 b) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (\frac{i a x}{b-\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{(240 b) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (\frac{i a x}{b+\sqrt{-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt{x}\right )}\right )}{a \sqrt{-a^2+b^2} d^6}\\ &=\frac{x^3}{3 a}+\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}-\frac{2 i b x^{5/2} \log \left (1-\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d}+\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}-\frac{10 b x^2 \text{Li}_2\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^2}+\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{40 i b x^{3/2} \text{Li}_3\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^4}-\frac{240 i b \sqrt{x} \text{Li}_5\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{240 i b \sqrt{x} \text{Li}_5\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^5}+\frac{240 b \text{Li}_6\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b-\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}-\frac{240 b \text{Li}_6\left (\frac{i a e^{i \left (c+d \sqrt{x}\right )}}{b+\sqrt{-a^2+b^2}}\right )}{a \sqrt{-a^2+b^2} d^6}\\ \end{align*}

Mathematica [A]  time = 1.84069, size = 898, normalized size = 1.11 \[ \frac{\csc \left (c+d \sqrt{x}\right ) \left (x^3-\frac{6 b e^{i c} \left (x^{5/2} \log \left (\frac{e^{i \left (2 c+d \sqrt{x}\right )} a}{i b e^{i c}-\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}+1\right ) d^5-x^{5/2} \log \left (\frac{e^{i \left (2 c+d \sqrt{x}\right )} a}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}+1\right ) d^5-5 i x^2 \text{PolyLog}\left (2,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d^4+5 i x^2 \text{PolyLog}\left (2,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d^4+20 x^{3/2} \text{PolyLog}\left (3,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d^3-20 x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d^3+60 i x \text{PolyLog}\left (4,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d^2-60 i x \text{PolyLog}\left (4,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d^2-120 \sqrt{x} \text{PolyLog}\left (5,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d+120 \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) d-120 i \text{PolyLog}\left (6,\frac{i a e^{i \left (2 c+d \sqrt{x}\right )}}{e^{i c} b+i \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right )+120 i \text{PolyLog}\left (6,-\frac{a e^{i \left (2 c+d \sqrt{x}\right )}}{i e^{i c} b+\sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right )\right )}{d^6 \sqrt{\left (a^2-b^2\right ) e^{2 i c}}}\right ) \left (b+a \sin \left (c+d \sqrt{x}\right )\right )}{3 a \left (a+b \csc \left (c+d \sqrt{x}\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*Csc[c + d*Sqrt[x]]),x]

[Out]

(Csc[c + d*Sqrt[x]]*(x^3 - (6*b*E^(I*c)*(d^5*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) - Sqrt[(
a^2 - b^2)*E^((2*I)*c)])] - d^5*x^(5/2)*Log[1 + (a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^
((2*I)*c)])] - (5*I)*d^4*x^2*PolyLog[2, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)
*c)])] + (5*I)*d^4*x^2*PolyLog[2, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))
] + 20*d^3*x^(3/2)*PolyLog[3, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2
0*d^3*x^(3/2)*PolyLog[3, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] + (60*I
)*d^2*x*PolyLog[4, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - (60*I)*d^2*x
*PolyLog[4, -((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - 120*d*Sqrt[x]*Poly
Log[5, (I*a*E^(I*(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 120*d*Sqrt[x]*PolyLog[5,
-((a*E^(I*(2*c + d*Sqrt[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))] - (120*I)*PolyLog[6, (I*a*E^(I*
(2*c + d*Sqrt[x])))/(b*E^(I*c) + I*Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (120*I)*PolyLog[6, -((a*E^(I*(2*c + d*Sqr
t[x])))/(I*b*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)]))]))/(d^6*Sqrt[(a^2 - b^2)*E^((2*I)*c)]))*(b + a*Sin[c +
d*Sqrt[x]]))/(3*a*(a + b*Csc[c + d*Sqrt[x]]))

________________________________________________________________________________________

Maple [F]  time = 0.144, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\csc \left ( c+d\sqrt{x} \right ) \right ) ^{-1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b*csc(c+d*x^(1/2))),x)

[Out]

int(x^2/(a+b*csc(c+d*x^(1/2))),x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*csc(c+d*x^(1/2))),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{b \csc \left (d \sqrt{x} + c\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*csc(c+d*x^(1/2))),x, algorithm="fricas")

[Out]

integral(x^2/(b*csc(d*sqrt(x) + c) + a), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{a + b \csc{\left (c + d \sqrt{x} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b*csc(c+d*x**(1/2))),x)

[Out]

Integral(x**2/(a + b*csc(c + d*sqrt(x))), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b \csc \left (d \sqrt{x} + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b*csc(c+d*x^(1/2))),x, algorithm="giac")

[Out]

integrate(x^2/(b*csc(d*sqrt(x) + c) + a), x)