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

Optimal. Leaf size=441 \[ -\frac{20 a b x^2 \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{80 a b x^{3/2} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{240 a b x \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \text{PolyLog}\left (6,-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{480 a b \text{PolyLog}\left (6,e^{c+d \sqrt{x}}\right )}{d^6}+\frac{20 b^2 x^{3/2} \text{PolyLog}\left (2,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{30 b^2 x \text{PolyLog}\left (3,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 b^2 \sqrt{x} \text{PolyLog}\left (4,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{15 b^2 \text{PolyLog}\left (5,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{2 b^2 x^{5/2}}{d} \]

[Out]

(-2*b^2*x^(5/2))/d + (a^2*x^3)/3 - (8*a*b*x^(5/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(5/2)*Coth[c + d*Sq
rt[x]])/d + (10*b^2*x^2*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (20*a*b*x^2*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2
+ (20*a*b*x^2*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (20*b^2*x^(3/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (8
0*a*b*x^(3/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (80*a*b*x^(3/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (30*b
^2*x*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (240*a*b*x*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (240*a*b*x*Poly
Log[4, E^(c + d*Sqrt[x])])/d^4 + (30*b^2*Sqrt[x]*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*Pol
yLog[5, -E^(c + d*Sqrt[x])])/d^5 - (480*a*b*Sqrt[x]*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (15*b^2*PolyLog[5, E^
(2*(c + d*Sqrt[x]))])/d^6 - (480*a*b*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (480*a*b*PolyLog[6, E^(c + d*Sqrt[x
])])/d^6

________________________________________________________________________________________

Rubi [A]  time = 0.63134, antiderivative size = 441, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5437, 4190, 4182, 2531, 6609, 2282, 6589, 4184, 3716, 2190} \[ -\frac{20 a b x^2 \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{80 a b x^{3/2} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{240 a b x \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{PolyLog}\left (5,e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \text{PolyLog}\left (6,-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{480 a b \text{PolyLog}\left (6,e^{c+d \sqrt{x}}\right )}{d^6}+\frac{20 b^2 x^{3/2} \text{PolyLog}\left (2,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{30 b^2 x \text{PolyLog}\left (3,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{30 b^2 \sqrt{x} \text{PolyLog}\left (4,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{15 b^2 \text{PolyLog}\left (5,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{2 b^2 x^{5/2}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b^2*x^(5/2))/d + (a^2*x^3)/3 - (8*a*b*x^(5/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(5/2)*Coth[c + d*Sq
rt[x]])/d + (10*b^2*x^2*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (20*a*b*x^2*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2
+ (20*a*b*x^2*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (20*b^2*x^(3/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (8
0*a*b*x^(3/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (80*a*b*x^(3/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (30*b
^2*x*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (240*a*b*x*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (240*a*b*x*Poly
Log[4, E^(c + d*Sqrt[x])])/d^4 + (30*b^2*Sqrt[x]*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (480*a*b*Sqrt[x]*Pol
yLog[5, -E^(c + d*Sqrt[x])])/d^5 - (480*a*b*Sqrt[x]*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (15*b^2*PolyLog[5, E^
(2*(c + d*Sqrt[x]))])/d^6 - (480*a*b*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (480*a*b*PolyLog[6, E^(c + d*Sqrt[x
])])/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 4190

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && 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]

Rubi steps

\begin{align*} \int x^2 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^5 (a+b \text{csch}(c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^5+2 a b x^5 \text{csch}(c+d x)+b^2 x^5 \text{csch}^2(c+d x)\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^3}{3}+(4 a b) \operatorname{Subst}\left (\int x^5 \text{csch}(c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int x^5 \text{csch}^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{(20 a b) \operatorname{Subst}\left (\int x^4 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(20 a b) \operatorname{Subst}\left (\int x^4 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{\left (10 b^2\right ) \operatorname{Subst}\left (\int x^4 \coth (c+d x) \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{(80 a b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{(80 a b) \operatorname{Subst}\left (\int x^3 \text{Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (20 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 (c+d x)} x^4}{1-e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{(240 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}+\frac{(240 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (40 b^2\right ) \operatorname{Subst}\left (\int x^3 \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 b^2 x^{3/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{240 a b x \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{(480 a b) \operatorname{Subst}\left (\int x \text{Li}_4\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{(480 a b) \operatorname{Subst}\left (\int x \text{Li}_4\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{\left (60 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_2\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 b^2 x^{3/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{30 b^2 x \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 a b x \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{(480 a b) \operatorname{Subst}\left (\int \text{Li}_5\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^5}+\frac{(480 a b) \operatorname{Subst}\left (\int \text{Li}_5\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^5}+\frac{\left (60 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_3\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 b^2 x^{3/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{30 b^2 x \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 a b x \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{30 b^2 \sqrt{x} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{(480 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_5(-x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^6}+\frac{(480 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_5(x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^6}-\frac{\left (30 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_4\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 b^2 x^{3/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{30 b^2 x \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 a b x \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{30 b^2 \sqrt{x} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{480 a b \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_4(x)}{x} \, dx,x,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}\\ &=-\frac{2 b^2 x^{5/2}}{d}+\frac{a^2 x^3}{3}-\frac{8 a b x^{5/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{5/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{10 b^2 x^2 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{20 a b x^2 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 a b x^2 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{20 b^2 x^{3/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{80 a b x^{3/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{80 a b x^{3/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{30 b^2 x \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{240 a b x \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{240 a b x \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{30 b^2 \sqrt{x} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{480 a b \sqrt{x} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{480 a b \sqrt{x} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{15 b^2 \text{Li}_5\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{480 a b \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{480 a b \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}\\ \end{align*}

Mathematica [A]  time = 12.2088, size = 833, normalized size = 1.89 \[ \frac{a^2 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \sinh ^2\left (c+d \sqrt{x}\right ) x^3}{3 \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )^2}+\frac{b^2 \text{csch}\left (\frac{c}{2}\right ) \text{csch}\left (\frac{c}{2}+\frac{d \sqrt{x}}{2}\right ) \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \sinh ^2\left (c+d \sqrt{x}\right ) \sinh \left (\frac{d \sqrt{x}}{2}\right ) x^{5/2}}{d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )^2}-\frac{b^2 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \text{sech}\left (\frac{c}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d \sqrt{x}}{2}\right ) \sinh ^2\left (c+d \sqrt{x}\right ) \sinh \left (\frac{d \sqrt{x}}{2}\right ) x^{5/2}}{d \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )^2}+\frac{2 b \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \left (-\frac{2 b x^{5/2} d^5}{-1+e^{2 c}}+2 a x^{5/2} \log \left (1-e^{-c-d \sqrt{x}}\right ) d^5-2 a x^{5/2} \log \left (1+e^{-c-d \sqrt{x}}\right ) d^5+5 b x^2 \log \left (1-e^{-c-d \sqrt{x}}\right ) d^4+5 b x^2 \log \left (1+e^{-c-d \sqrt{x}}\right ) d^4+40 a x^{3/2} \text{PolyLog}\left (3,-e^{-c-d \sqrt{x}}\right ) d^3-40 a x^{3/2} \text{PolyLog}\left (3,e^{-c-d \sqrt{x}}\right ) d^3-60 b x \text{PolyLog}\left (3,-e^{-c-d \sqrt{x}}\right ) d^2-60 b x \text{PolyLog}\left (3,e^{-c-d \sqrt{x}}\right ) d^2+120 a x \text{PolyLog}\left (4,-e^{-c-d \sqrt{x}}\right ) d^2-120 a x \text{PolyLog}\left (4,e^{-c-d \sqrt{x}}\right ) d^2-120 b \sqrt{x} \text{PolyLog}\left (4,-e^{-c-d \sqrt{x}}\right ) d-120 b \sqrt{x} \text{PolyLog}\left (4,e^{-c-d \sqrt{x}}\right ) d+240 a \sqrt{x} \text{PolyLog}\left (5,-e^{-c-d \sqrt{x}}\right ) d-240 a \sqrt{x} \text{PolyLog}\left (5,e^{-c-d \sqrt{x}}\right ) d+10 \left (a d^4 x^2-2 b d^3 x^{3/2}\right ) \text{PolyLog}\left (2,-e^{-c-d \sqrt{x}}\right )-10 \left (a x^2 d^4+2 b x^{3/2} d^3\right ) \text{PolyLog}\left (2,e^{-c-d \sqrt{x}}\right )-120 b \text{PolyLog}\left (5,-e^{-c-d \sqrt{x}}\right )-120 b \text{PolyLog}\left (5,e^{-c-d \sqrt{x}}\right )+240 a \text{PolyLog}\left (6,-e^{-c-d \sqrt{x}}\right )-240 a \text{PolyLog}\left (6,e^{-c-d \sqrt{x}}\right )\right ) \sinh ^2\left (c+d \sqrt{x}\right )}{d^6 \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*x^3*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c + d*Sqrt[x]]^2)/(3*(b + a*Sinh[c + d*Sqrt[x]])^2) + (2*b*(a + b*
Csch[c + d*Sqrt[x]])^2*((-2*b*d^5*x^(5/2))/(-1 + E^(2*c)) + 5*b*d^4*x^2*Log[1 - E^(-c - d*Sqrt[x])] + 2*a*d^5*
x^(5/2)*Log[1 - E^(-c - d*Sqrt[x])] + 5*b*d^4*x^2*Log[1 + E^(-c - d*Sqrt[x])] - 2*a*d^5*x^(5/2)*Log[1 + E^(-c
- d*Sqrt[x])] + 10*(-2*b*d^3*x^(3/2) + a*d^4*x^2)*PolyLog[2, -E^(-c - d*Sqrt[x])] - 10*(2*b*d^3*x^(3/2) + a*d^
4*x^2)*PolyLog[2, E^(-c - d*Sqrt[x])] - 60*b*d^2*x*PolyLog[3, -E^(-c - d*Sqrt[x])] + 40*a*d^3*x^(3/2)*PolyLog[
3, -E^(-c - d*Sqrt[x])] - 60*b*d^2*x*PolyLog[3, E^(-c - d*Sqrt[x])] - 40*a*d^3*x^(3/2)*PolyLog[3, E^(-c - d*Sq
rt[x])] - 120*b*d*Sqrt[x]*PolyLog[4, -E^(-c - d*Sqrt[x])] + 120*a*d^2*x*PolyLog[4, -E^(-c - d*Sqrt[x])] - 120*
b*d*Sqrt[x]*PolyLog[4, E^(-c - d*Sqrt[x])] - 120*a*d^2*x*PolyLog[4, E^(-c - d*Sqrt[x])] - 120*b*PolyLog[5, -E^
(-c - d*Sqrt[x])] + 240*a*d*Sqrt[x]*PolyLog[5, -E^(-c - d*Sqrt[x])] - 120*b*PolyLog[5, E^(-c - d*Sqrt[x])] - 2
40*a*d*Sqrt[x]*PolyLog[5, E^(-c - d*Sqrt[x])] + 240*a*PolyLog[6, -E^(-c - d*Sqrt[x])] - 240*a*PolyLog[6, E^(-c
- d*Sqrt[x])])*Sinh[c + d*Sqrt[x]]^2)/(d^6*(b + a*Sinh[c + d*Sqrt[x]])^2) + (b^2*x^(5/2)*Csch[c/2]*Csch[c/2 +
(d*Sqrt[x])/2]*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c + d*Sqrt[x]]^2*Sinh[(d*Sqrt[x])/2])/(d*(b + a*Sinh[c + d*
Sqrt[x]])^2) - (b^2*x^(5/2)*(a + b*Csch[c + d*Sqrt[x]])^2*Sech[c/2]*Sech[c/2 + (d*Sqrt[x])/2]*Sinh[c + d*Sqrt[
x]]^2*Sinh[(d*Sqrt[x])/2])/(d*(b + a*Sinh[c + d*Sqrt[x]])^2)

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b{\rm csch} \left (c+d\sqrt{x}\right ) \right ) ^{2}\, 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)))^2,x)

[Out]

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

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Maxima [A]  time = 1.98784, size = 670, normalized size = 1.52 \begin{align*} \frac{1}{3} \, a^{2} x^{3} - \frac{4 \, b^{2} x^{\frac{5}{2}}}{d e^{\left (2 \, d \sqrt{x} + 2 \, c\right )} - d} - \frac{4 \,{\left (d^{5} x^{\frac{5}{2}} \log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 5 \, d^{4} x^{2}{\rm Li}_2\left (-e^{\left (d \sqrt{x} + c\right )}\right ) - 20 \, d^{3} x^{\frac{3}{2}}{\rm Li}_{3}(-e^{\left (d \sqrt{x} + c\right )}) + 60 \, d^{2} x{\rm Li}_{4}(-e^{\left (d \sqrt{x} + c\right )}) - 120 \, d \sqrt{x}{\rm Li}_{5}(-e^{\left (d \sqrt{x} + c\right )}) + 120 \,{\rm Li}_{6}(-e^{\left (d \sqrt{x} + c\right )})\right )} a b}{d^{6}} + \frac{4 \,{\left (d^{5} x^{\frac{5}{2}} \log \left (-e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 5 \, d^{4} x^{2}{\rm Li}_2\left (e^{\left (d \sqrt{x} + c\right )}\right ) - 20 \, d^{3} x^{\frac{3}{2}}{\rm Li}_{3}(e^{\left (d \sqrt{x} + c\right )}) + 60 \, d^{2} x{\rm Li}_{4}(e^{\left (d \sqrt{x} + c\right )}) - 120 \, d \sqrt{x}{\rm Li}_{5}(e^{\left (d \sqrt{x} + c\right )}) + 120 \,{\rm Li}_{6}(e^{\left (d \sqrt{x} + c\right )})\right )} a b}{d^{6}} + \frac{10 \,{\left (d^{4} x^{2} \log \left (e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 4 \, d^{3} x^{\frac{3}{2}}{\rm Li}_2\left (-e^{\left (d \sqrt{x} + c\right )}\right ) - 12 \, d^{2} x{\rm Li}_{3}(-e^{\left (d \sqrt{x} + c\right )}) + 24 \, d \sqrt{x}{\rm Li}_{4}(-e^{\left (d \sqrt{x} + c\right )}) - 24 \,{\rm Li}_{5}(-e^{\left (d \sqrt{x} + c\right )})\right )} b^{2}}{d^{6}} + \frac{10 \,{\left (d^{4} x^{2} \log \left (-e^{\left (d \sqrt{x} + c\right )} + 1\right ) + 4 \, d^{3} x^{\frac{3}{2}}{\rm Li}_2\left (e^{\left (d \sqrt{x} + c\right )}\right ) - 12 \, d^{2} x{\rm Li}_{3}(e^{\left (d \sqrt{x} + c\right )}) + 24 \, d \sqrt{x}{\rm Li}_{4}(e^{\left (d \sqrt{x} + c\right )}) - 24 \,{\rm Li}_{5}(e^{\left (d \sqrt{x} + c\right )})\right )} b^{2}}{d^{6}} - \frac{2 \,{\left (a b d^{6} x^{3} + 3 \, b^{2} d^{5} x^{\frac{5}{2}}\right )}}{3 \, d^{6}} + \frac{2 \,{\left (a b d^{6} x^{3} - 3 \, b^{2} d^{5} x^{\frac{5}{2}}\right )}}{3 \, d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3 - 4*b^2*x^(5/2)/(d*e^(2*d*sqrt(x) + 2*c) - d) - 4*(d^5*x^(5/2)*log(e^(d*sqrt(x) + c) + 1) + 5*d^4*
x^2*dilog(-e^(d*sqrt(x) + c)) - 20*d^3*x^(3/2)*polylog(3, -e^(d*sqrt(x) + c)) + 60*d^2*x*polylog(4, -e^(d*sqrt
(x) + c)) - 120*d*sqrt(x)*polylog(5, -e^(d*sqrt(x) + c)) + 120*polylog(6, -e^(d*sqrt(x) + c)))*a*b/d^6 + 4*(d^
5*x^(5/2)*log(-e^(d*sqrt(x) + c) + 1) + 5*d^4*x^2*dilog(e^(d*sqrt(x) + c)) - 20*d^3*x^(3/2)*polylog(3, e^(d*sq
rt(x) + c)) + 60*d^2*x*polylog(4, e^(d*sqrt(x) + c)) - 120*d*sqrt(x)*polylog(5, e^(d*sqrt(x) + c)) + 120*polyl
og(6, e^(d*sqrt(x) + c)))*a*b/d^6 + 10*(d^4*x^2*log(e^(d*sqrt(x) + c) + 1) + 4*d^3*x^(3/2)*dilog(-e^(d*sqrt(x)
+ c)) - 12*d^2*x*polylog(3, -e^(d*sqrt(x) + c)) + 24*d*sqrt(x)*polylog(4, -e^(d*sqrt(x) + c)) - 24*polylog(5,
-e^(d*sqrt(x) + c)))*b^2/d^6 + 10*(d^4*x^2*log(-e^(d*sqrt(x) + c) + 1) + 4*d^3*x^(3/2)*dilog(e^(d*sqrt(x) + c
)) - 12*d^2*x*polylog(3, e^(d*sqrt(x) + c)) + 24*d*sqrt(x)*polylog(4, e^(d*sqrt(x) + c)) - 24*polylog(5, e^(d*
sqrt(x) + c)))*b^2/d^6 - 2/3*(a*b*d^6*x^3 + 3*b^2*d^5*x^(5/2))/d^6 + 2/3*(a*b*d^6*x^3 - 3*b^2*d^5*x^(5/2))/d^6

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

[Out]

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

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

[Out]

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

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

[Out]

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