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

Optimal. Leaf size=673 \[ -\frac{10 b x^2 \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{10 b x^2 \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{40 b x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{40 b x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{120 b x \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{120 b x \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{240 b \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^5 \sqrt{a^2+b^2}}-\frac{240 b \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^5 \sqrt{a^2+b^2}}-\frac{240 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^6 \sqrt{a^2+b^2}}+\frac{240 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^6 \sqrt{a^2+b^2}}-\frac{2 b x^{5/2} \log \left (\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{2 b x^{5/2} \log \left (\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{x^3}{3 a} \]

[Out]

x^3/(3*a) - (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^
(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (10*b*x^2*PolyLog[2, -((a*
E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[
x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b -
Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 +
b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sq
rt[a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d
^4) + (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (24
0*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (240*b*PolyL
og[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (240*b*PolyLog[6, -((a*E^(c +
d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6)

________________________________________________________________________________________

Rubi [A]  time = 1.16683, antiderivative size = 673, 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 = {5437, 4191, 3322, 2264, 2190, 2531, 6609, 2282, 6589} \[ -\frac{10 b x^2 \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{10 b x^2 \text{PolyLog}\left (2,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{40 b x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{40 b x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{120 b x \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{120 b x \text{PolyLog}\left (4,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{240 b \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^5 \sqrt{a^2+b^2}}-\frac{240 b \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^5 \sqrt{a^2+b^2}}-\frac{240 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a d^6 \sqrt{a^2+b^2}}+\frac{240 b \text{PolyLog}\left (6,-\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}\right )}{a d^6 \sqrt{a^2+b^2}}-\frac{2 b x^{5/2} \log \left (\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{2 b x^{5/2} \log \left (\frac{a e^{c+d \sqrt{x}}}{\sqrt{a^2+b^2}+b}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^3/(3*a) - (2*b*x^(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) + (2*b*x^
(5/2)*Log[1 + (a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) - (10*b*x^2*PolyLog[2, -((a*
E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (10*b*x^2*PolyLog[2, -((a*E^(c + d*Sqrt[
x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b -
Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (40*b*x^(3/2)*PolyLog[3, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 +
b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sq
rt[a^2 + b^2]*d^4) + (120*b*x*PolyLog[4, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d
^4) + (240*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (24
0*b*Sqrt[x]*PolyLog[5, -((a*E^(c + d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^5) - (240*b*PolyL
og[6, -((a*E^(c + d*Sqrt[x]))/(b - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6) + (240*b*PolyLog[6, -((a*E^(c +
d*Sqrt[x]))/(b + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^6)

Rule 5437

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(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 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, 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 \text{csch}\left (c+d \sqrt{x}\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^5}{a+b \text{csch}(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 \sinh (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 \sinh (c+d x)} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{x^3}{3 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{a}\\ &=\frac{x^3}{3 a}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{2 b-2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2+b^2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^5}{2 b+2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx,x,\sqrt{x}\right )}{\sqrt{a^2+b^2}}\\ &=\frac{x^3}{3 a}-\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{(10 b) \operatorname{Subst}\left (\int x^4 \log \left (1+\frac{2 a e^{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 b) \operatorname{Subst}\left (\int x^4 \log \left (1+\frac{2 a e^{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 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{10 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{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{a e^{c+d \sqrt{x}}}{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 a e^{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 a e^{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 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{10 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{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{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{(120 b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-\frac{2 a e^{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 b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-\frac{2 a e^{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 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{10 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{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{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{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 a e^{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 a e^{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 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{10 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{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{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{240 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^5}-\frac{240 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^5}-\frac{(240 b) \operatorname{Subst}\left (\int \text{Li}_5\left (-\frac{2 a e^{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 b) \operatorname{Subst}\left (\int \text{Li}_5\left (-\frac{2 a e^{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 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{10 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{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{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{240 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^5}-\frac{240 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{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{a x}{-b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a \sqrt{a^2+b^2} d^6}+\frac{(240 b) \operatorname{Subst}\left (\int \frac{\text{Li}_5\left (-\frac{a x}{b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{a \sqrt{a^2+b^2} d^6}\\ &=\frac{x^3}{3 a}-\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}+\frac{2 b x^{5/2} \log \left (1+\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d}-\frac{10 b x^2 \text{Li}_2\left (-\frac{a e^{c+d \sqrt{x}}}{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{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^2}+\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{40 b x^{3/2} \text{Li}_3\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^3}-\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{120 b x \text{Li}_4\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^4}+\frac{240 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^5}-\frac{240 b \sqrt{x} \text{Li}_5\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^5}-\frac{240 b \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b-\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^6}+\frac{240 b \text{Li}_6\left (-\frac{a e^{c+d \sqrt{x}}}{b+\sqrt{a^2+b^2}}\right )}{a \sqrt{a^2+b^2} d^6}\\ \end{align*}

Mathematica [A]  time = 1.9, size = 716, normalized size = 1.06 \[ \frac{-30 b e^c d^4 x^2 \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )+30 b e^c d^4 x^2 \text{PolyLog}\left (2,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+b e^c}\right )+120 b e^c d^3 x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )-120 b e^c d^3 x^{3/2} \text{PolyLog}\left (3,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+b e^c}\right )-360 b e^c d^2 x \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )+360 b e^c d^2 x \text{PolyLog}\left (4,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+b e^c}\right )+720 b e^c d \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )-720 b e^c d \sqrt{x} \text{PolyLog}\left (5,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+b e^c}\right )-720 b e^c \text{PolyLog}\left (6,-\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}\right )+720 b e^c \text{PolyLog}\left (6,-\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+b e^c}\right )+d^6 x^3 \sqrt{e^{2 c} \left (a^2+b^2\right )}-6 b e^c d^5 x^{5/2} \log \left (\frac{a e^{2 c+d \sqrt{x}}}{b e^c-\sqrt{e^{2 c} \left (a^2+b^2\right )}}+1\right )+6 b e^c d^5 x^{5/2} \log \left (\frac{a e^{2 c+d \sqrt{x}}}{\sqrt{e^{2 c} \left (a^2+b^2\right )}+b e^c}+1\right )}{3 a d^6 \sqrt{e^{2 c} \left (a^2+b^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^6*Sqrt[(a^2 + b^2)*E^(2*c)]*x^3 - 6*b*d^5*E^c*x^(5/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 +
b^2)*E^(2*c)])] + 6*b*d^5*E^c*x^(5/2)*Log[1 + (a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])] - 3
0*b*d^4*E^c*x^2*PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 30*b*d^4*E^c*x^2*
PolyLog[2, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] + 120*b*d^3*E^c*x^(3/2)*PolyLog[3,
-((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] - 120*b*d^3*E^c*x^(3/2)*PolyLog[3, -((a*E^(2*c
+ d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 360*b*d^2*E^c*x*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b
*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 360*b*d^2*E^c*x*PolyLog[4, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2
+ b^2)*E^(2*c)]))] + 720*b*d*E^c*Sqrt[x]*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c
)]))] - 720*b*d*E^c*Sqrt[x]*PolyLog[5, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))] - 720*b
*E^c*PolyLog[6, -((a*E^(2*c + d*Sqrt[x]))/(b*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))] + 720*b*E^c*PolyLog[6, -((a*E^
(2*c + d*Sqrt[x]))/(b*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/(3*a*d^6*Sqrt[(a^2 + b^2)*E^(2*c)])

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b{\rm csch} \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*csch(c+d*x^(1/2))),x)

[Out]

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

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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*csch(c+d*x^(1/2))),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2}}{b \operatorname{csch}\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*csch(c+d*x^(1/2))),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{a + b \operatorname{csch}{\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*csch(c+d*x**(1/2))),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{b \operatorname{csch}\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*csch(c+d*x^(1/2))),x, algorithm="giac")

[Out]

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