3.36 \(\int x^3 (a+b \text{csch}(c+d \sqrt{x}))^2 \, dx\)

Optimal. Leaf size=597 \[ -\frac{28 a b x^3 \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{168 a b x^{5/2} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{840 a b x^2 \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,e^{c+d \sqrt{x}}\right )}{d^5}-\frac{10080 a b x \text{PolyLog}\left (6,-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{PolyLog}\left (6,e^{c+d \sqrt{x}}\right )}{d^6}+\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,-e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \text{PolyLog}\left (8,-e^{c+d \sqrt{x}}\right )}{d^8}+\frac{20160 a b \text{PolyLog}\left (8,e^{c+d \sqrt{x}}\right )}{d^8}+\frac{42 b^2 x^{5/2} \text{PolyLog}\left (2,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{105 b^2 x^2 \text{PolyLog}\left (3,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{PolyLog}\left (4,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{PolyLog}\left (5,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{315 b^2 \sqrt{x} \text{PolyLog}\left (6,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{315 b^2 \text{PolyLog}\left (7,e^{2 \left (c+d \sqrt{x}\right )}\right )}{2 d^8}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{2 b^2 x^{7/2}}{d} \]

[Out]

(-2*b^2*x^(7/2))/d + (a^2*x^4)/4 - (8*a*b*x^(7/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(7/2)*Coth[c + d*Sq
rt[x]])/d + (14*b^2*x^3*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (28*a*b*x^3*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2
+ (28*a*b*x^3*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (42*b^2*x^(5/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (1
68*a*b*x^(5/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (10
5*b^2*x^2*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (840*a*b*x^2*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (840*a*b
*x^2*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (210*b^2*x^(3/2)*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*
x^(3/2)*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (315*b^2*
x*PolyLog[5, E^(2*(c + d*Sqrt[x]))])/d^6 - (10080*a*b*x*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (10080*a*b*x*Pol
yLog[6, E^(c + d*Sqrt[x])])/d^6 + (315*b^2*Sqrt[x]*PolyLog[6, E^(2*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]
*PolyLog[7, -E^(c + d*Sqrt[x])])/d^7 - (20160*a*b*Sqrt[x]*PolyLog[7, E^(c + d*Sqrt[x])])/d^7 - (315*b^2*PolyLo
g[7, E^(2*(c + d*Sqrt[x]))])/(2*d^8) - (20160*a*b*PolyLog[8, -E^(c + d*Sqrt[x])])/d^8 + (20160*a*b*PolyLog[8,
E^(c + d*Sqrt[x])])/d^8

________________________________________________________________________________________

Rubi [A]  time = 0.816731, antiderivative size = 597, normalized size of antiderivative = 1., number of steps used = 30, 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} \[ -\frac{28 a b x^3 \text{PolyLog}\left (2,-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{PolyLog}\left (2,e^{c+d \sqrt{x}}\right )}{d^2}+\frac{168 a b x^{5/2} \text{PolyLog}\left (3,-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{PolyLog}\left (3,e^{c+d \sqrt{x}}\right )}{d^3}-\frac{840 a b x^2 \text{PolyLog}\left (4,-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{PolyLog}\left (4,e^{c+d \sqrt{x}}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{PolyLog}\left (5,e^{c+d \sqrt{x}}\right )}{d^5}-\frac{10080 a b x \text{PolyLog}\left (6,-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{PolyLog}\left (6,e^{c+d \sqrt{x}}\right )}{d^6}+\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,-e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{PolyLog}\left (7,e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \text{PolyLog}\left (8,-e^{c+d \sqrt{x}}\right )}{d^8}+\frac{20160 a b \text{PolyLog}\left (8,e^{c+d \sqrt{x}}\right )}{d^8}+\frac{42 b^2 x^{5/2} \text{PolyLog}\left (2,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}-\frac{105 b^2 x^2 \text{PolyLog}\left (3,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{PolyLog}\left (4,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}-\frac{315 b^2 x \text{PolyLog}\left (5,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}+\frac{315 b^2 \sqrt{x} \text{PolyLog}\left (6,e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^7}-\frac{315 b^2 \text{PolyLog}\left (7,e^{2 \left (c+d \sqrt{x}\right )}\right )}{2 d^8}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{2 b^2 x^{7/2}}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*Csch[c + d*Sqrt[x]])^2,x]

[Out]

(-2*b^2*x^(7/2))/d + (a^2*x^4)/4 - (8*a*b*x^(7/2)*ArcTanh[E^(c + d*Sqrt[x])])/d - (2*b^2*x^(7/2)*Coth[c + d*Sq
rt[x]])/d + (14*b^2*x^3*Log[1 - E^(2*(c + d*Sqrt[x]))])/d^2 - (28*a*b*x^3*PolyLog[2, -E^(c + d*Sqrt[x])])/d^2
+ (28*a*b*x^3*PolyLog[2, E^(c + d*Sqrt[x])])/d^2 + (42*b^2*x^(5/2)*PolyLog[2, E^(2*(c + d*Sqrt[x]))])/d^3 + (1
68*a*b*x^(5/2)*PolyLog[3, -E^(c + d*Sqrt[x])])/d^3 - (168*a*b*x^(5/2)*PolyLog[3, E^(c + d*Sqrt[x])])/d^3 - (10
5*b^2*x^2*PolyLog[3, E^(2*(c + d*Sqrt[x]))])/d^4 - (840*a*b*x^2*PolyLog[4, -E^(c + d*Sqrt[x])])/d^4 + (840*a*b
*x^2*PolyLog[4, E^(c + d*Sqrt[x])])/d^4 + (210*b^2*x^(3/2)*PolyLog[4, E^(2*(c + d*Sqrt[x]))])/d^5 + (3360*a*b*
x^(3/2)*PolyLog[5, -E^(c + d*Sqrt[x])])/d^5 - (3360*a*b*x^(3/2)*PolyLog[5, E^(c + d*Sqrt[x])])/d^5 - (315*b^2*
x*PolyLog[5, E^(2*(c + d*Sqrt[x]))])/d^6 - (10080*a*b*x*PolyLog[6, -E^(c + d*Sqrt[x])])/d^6 + (10080*a*b*x*Pol
yLog[6, E^(c + d*Sqrt[x])])/d^6 + (315*b^2*Sqrt[x]*PolyLog[6, E^(2*(c + d*Sqrt[x]))])/d^7 + (20160*a*b*Sqrt[x]
*PolyLog[7, -E^(c + d*Sqrt[x])])/d^7 - (20160*a*b*Sqrt[x]*PolyLog[7, E^(c + d*Sqrt[x])])/d^7 - (315*b^2*PolyLo
g[7, E^(2*(c + d*Sqrt[x]))])/(2*d^8) - (20160*a*b*PolyLog[8, -E^(c + d*Sqrt[x])])/d^8 + (20160*a*b*PolyLog[8,
E^(c + d*Sqrt[x])])/d^8

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^3 \left (a+b \text{csch}\left (c+d \sqrt{x}\right )\right )^2 \, dx &=2 \operatorname{Subst}\left (\int x^7 (a+b \text{csch}(c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (a^2 x^7+2 a b x^7 \text{csch}(c+d x)+b^2 x^7 \text{csch}^2(c+d x)\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^4}{4}+(4 a b) \operatorname{Subst}\left (\int x^7 \text{csch}(c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int x^7 \text{csch}^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{(28 a b) \operatorname{Subst}\left (\int x^6 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{(28 a b) \operatorname{Subst}\left (\int x^6 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d}+\frac{\left (14 b^2\right ) \operatorname{Subst}\left (\int x^6 \coth (c+d x) \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{(168 a b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{(168 a b) \operatorname{Subst}\left (\int x^5 \text{Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^2}-\frac{\left (28 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 (c+d x)} x^6}{1-e^{2 (c+d x)}} \, dx,x,\sqrt{x}\right )}{d}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{(840 a b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}+\frac{(840 a b) \operatorname{Subst}\left (\int x^4 \text{Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^3}-\frac{\left (84 b^2\right ) \operatorname{Subst}\left (\int x^5 \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^2}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{(3360 a b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{(3360 a b) \operatorname{Subst}\left (\int x^3 \text{Li}_4\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^4}-\frac{\left (210 b^2\right ) \operatorname{Subst}\left (\int x^4 \text{Li}_2\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^3}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{(10080 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^5}+\frac{(10080 a b) \operatorname{Subst}\left (\int x^2 \text{Li}_5\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^5}+\frac{\left (420 b^2\right ) \operatorname{Subst}\left (\int x^3 \text{Li}_3\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^4}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{10080 a b x \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}+\frac{(20160 a b) \operatorname{Subst}\left (\int x \text{Li}_6\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^6}-\frac{(20160 a b) \operatorname{Subst}\left (\int x \text{Li}_6\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^6}-\frac{\left (630 b^2\right ) \operatorname{Subst}\left (\int x^2 \text{Li}_4\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^5}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 a b x \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{c+d \sqrt{x}}\right )}{d^7}-\frac{(20160 a b) \operatorname{Subst}\left (\int \text{Li}_7\left (-e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^7}+\frac{(20160 a b) \operatorname{Subst}\left (\int \text{Li}_7\left (e^{c+d x}\right ) \, dx,x,\sqrt{x}\right )}{d^7}+\frac{\left (630 b^2\right ) \operatorname{Subst}\left (\int x \text{Li}_5\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^6}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 a b x \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}+\frac{315 b^2 \sqrt{x} \text{Li}_6\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{c+d \sqrt{x}}\right )}{d^7}-\frac{(20160 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_7(-x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^8}+\frac{(20160 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_7(x)}{x} \, dx,x,e^{c+d \sqrt{x}}\right )}{d^8}-\frac{\left (315 b^2\right ) \operatorname{Subst}\left (\int \text{Li}_6\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt{x}\right )}{d^7}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 a b x \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}+\frac{315 b^2 \sqrt{x} \text{Li}_6\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \text{Li}_8\left (-e^{c+d \sqrt{x}}\right )}{d^8}+\frac{20160 a b \text{Li}_8\left (e^{c+d \sqrt{x}}\right )}{d^8}-\frac{\left (315 b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_6(x)}{x} \, dx,x,e^{2 \left (c+d \sqrt{x}\right )}\right )}{2 d^8}\\ &=-\frac{2 b^2 x^{7/2}}{d}+\frac{a^2 x^4}{4}-\frac{8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt{x}}\right )}{d}-\frac{2 b^2 x^{7/2} \coth \left (c+d \sqrt{x}\right )}{d}+\frac{14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^2}-\frac{28 a b x^3 \text{Li}_2\left (-e^{c+d \sqrt{x}}\right )}{d^2}+\frac{28 a b x^3 \text{Li}_2\left (e^{c+d \sqrt{x}}\right )}{d^2}+\frac{42 b^2 x^{5/2} \text{Li}_2\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^3}+\frac{168 a b x^{5/2} \text{Li}_3\left (-e^{c+d \sqrt{x}}\right )}{d^3}-\frac{168 a b x^{5/2} \text{Li}_3\left (e^{c+d \sqrt{x}}\right )}{d^3}-\frac{105 b^2 x^2 \text{Li}_3\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^4}-\frac{840 a b x^2 \text{Li}_4\left (-e^{c+d \sqrt{x}}\right )}{d^4}+\frac{840 a b x^2 \text{Li}_4\left (e^{c+d \sqrt{x}}\right )}{d^4}+\frac{210 b^2 x^{3/2} \text{Li}_4\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^5}+\frac{3360 a b x^{3/2} \text{Li}_5\left (-e^{c+d \sqrt{x}}\right )}{d^5}-\frac{3360 a b x^{3/2} \text{Li}_5\left (e^{c+d \sqrt{x}}\right )}{d^5}-\frac{315 b^2 x \text{Li}_5\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^6}-\frac{10080 a b x \text{Li}_6\left (-e^{c+d \sqrt{x}}\right )}{d^6}+\frac{10080 a b x \text{Li}_6\left (e^{c+d \sqrt{x}}\right )}{d^6}+\frac{315 b^2 \sqrt{x} \text{Li}_6\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{d^7}+\frac{20160 a b \sqrt{x} \text{Li}_7\left (-e^{c+d \sqrt{x}}\right )}{d^7}-\frac{20160 a b \sqrt{x} \text{Li}_7\left (e^{c+d \sqrt{x}}\right )}{d^7}-\frac{315 b^2 \text{Li}_7\left (e^{2 \left (c+d \sqrt{x}\right )}\right )}{2 d^8}-\frac{20160 a b \text{Li}_8\left (-e^{c+d \sqrt{x}}\right )}{d^8}+\frac{20160 a b \text{Li}_8\left (e^{c+d \sqrt{x}}\right )}{d^8}\\ \end{align*}

Mathematica [A]  time = 12.9163, size = 1077, normalized size = 1.8 \[ \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^4}{4 \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^{7/2}}{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^{7/2}}{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^{7/2} d^7}{-1+e^{2 c}}+2 a x^{7/2} \log \left (1-e^{-c-d \sqrt{x}}\right ) d^7-2 a x^{7/2} \log \left (1+e^{-c-d \sqrt{x}}\right ) d^7+7 b x^3 \log \left (1-e^{-c-d \sqrt{x}}\right ) d^6+7 b x^3 \log \left (1+e^{-c-d \sqrt{x}}\right ) d^6-42 b \left (x^{5/2} \text{PolyLog}\left (2,-e^{-c-d \sqrt{x}}\right ) d^5+5 x^2 \text{PolyLog}\left (3,-e^{-c-d \sqrt{x}}\right ) d^4+20 \left (x^{3/2} \text{PolyLog}\left (4,-e^{-c-d \sqrt{x}}\right ) d^3+3 x \text{PolyLog}\left (5,-e^{-c-d \sqrt{x}}\right ) d^2+6 \left (d \sqrt{x} \text{PolyLog}\left (6,-e^{-c-d \sqrt{x}}\right )+\text{PolyLog}\left (7,-e^{-c-d \sqrt{x}}\right )\right )\right )\right )-42 b \left (x^{5/2} \text{PolyLog}\left (2,e^{-c-d \sqrt{x}}\right ) d^5+5 x^2 \text{PolyLog}\left (3,e^{-c-d \sqrt{x}}\right ) d^4+20 \left (x^{3/2} \text{PolyLog}\left (4,e^{-c-d \sqrt{x}}\right ) d^3+3 x \text{PolyLog}\left (5,e^{-c-d \sqrt{x}}\right ) d^2+6 \left (d \sqrt{x} \text{PolyLog}\left (6,e^{-c-d \sqrt{x}}\right )+\text{PolyLog}\left (7,e^{-c-d \sqrt{x}}\right )\right )\right )\right )+14 a \left (x^3 \text{PolyLog}\left (2,-e^{-c-d \sqrt{x}}\right ) d^6+6 \left (x^{5/2} \text{PolyLog}\left (3,-e^{-c-d \sqrt{x}}\right ) d^5+5 x^2 \text{PolyLog}\left (4,-e^{-c-d \sqrt{x}}\right ) d^4+20 \left (x^{3/2} \text{PolyLog}\left (5,-e^{-c-d \sqrt{x}}\right ) d^3+3 x \text{PolyLog}\left (6,-e^{-c-d \sqrt{x}}\right ) d^2+6 \sqrt{x} \text{PolyLog}\left (7,-e^{-c-d \sqrt{x}}\right ) d+6 \text{PolyLog}\left (8,-e^{-c-d \sqrt{x}}\right )\right )\right )\right )-14 a \left (x^3 \text{PolyLog}\left (2,e^{-c-d \sqrt{x}}\right ) d^6+6 \left (x^{5/2} \text{PolyLog}\left (3,e^{-c-d \sqrt{x}}\right ) d^5+5 x^2 \text{PolyLog}\left (4,e^{-c-d \sqrt{x}}\right ) d^4+20 \left (x^{3/2} \text{PolyLog}\left (5,e^{-c-d \sqrt{x}}\right ) d^3+3 x \text{PolyLog}\left (6,e^{-c-d \sqrt{x}}\right ) d^2+6 \sqrt{x} \text{PolyLog}\left (7,e^{-c-d \sqrt{x}}\right ) d+6 \text{PolyLog}\left (8,e^{-c-d \sqrt{x}}\right )\right )\right )\right )\right ) \sinh ^2\left (c+d \sqrt{x}\right )}{d^8 \left (b+a \sinh \left (c+d \sqrt{x}\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*Csch[c + d*Sqrt[x]])^2,x]

[Out]

(a^2*x^4*(a + b*Csch[c + d*Sqrt[x]])^2*Sinh[c + d*Sqrt[x]]^2)/(4*(b + a*Sinh[c + d*Sqrt[x]])^2) + (2*b*(a + b*
Csch[c + d*Sqrt[x]])^2*((-2*b*d^7*x^(7/2))/(-1 + E^(2*c)) + 7*b*d^6*x^3*Log[1 - E^(-c - d*Sqrt[x])] + 2*a*d^7*
x^(7/2)*Log[1 - E^(-c - d*Sqrt[x])] + 7*b*d^6*x^3*Log[1 + E^(-c - d*Sqrt[x])] - 2*a*d^7*x^(7/2)*Log[1 + E^(-c
- d*Sqrt[x])] - 42*b*(d^5*x^(5/2)*PolyLog[2, -E^(-c - d*Sqrt[x])] + 5*d^4*x^2*PolyLog[3, -E^(-c - d*Sqrt[x])]
+ 20*(d^3*x^(3/2)*PolyLog[4, -E^(-c - d*Sqrt[x])] + 3*d^2*x*PolyLog[5, -E^(-c - d*Sqrt[x])] + 6*(d*Sqrt[x]*Pol
yLog[6, -E^(-c - d*Sqrt[x])] + PolyLog[7, -E^(-c - d*Sqrt[x])]))) - 42*b*(d^5*x^(5/2)*PolyLog[2, E^(-c - d*Sqr
t[x])] + 5*d^4*x^2*PolyLog[3, E^(-c - d*Sqrt[x])] + 20*(d^3*x^(3/2)*PolyLog[4, E^(-c - d*Sqrt[x])] + 3*d^2*x*P
olyLog[5, E^(-c - d*Sqrt[x])] + 6*(d*Sqrt[x]*PolyLog[6, E^(-c - d*Sqrt[x])] + PolyLog[7, E^(-c - d*Sqrt[x])]))
) + 14*a*(d^6*x^3*PolyLog[2, -E^(-c - d*Sqrt[x])] + 6*(d^5*x^(5/2)*PolyLog[3, -E^(-c - d*Sqrt[x])] + 5*d^4*x^2
*PolyLog[4, -E^(-c - d*Sqrt[x])] + 20*(d^3*x^(3/2)*PolyLog[5, -E^(-c - d*Sqrt[x])] + 3*d^2*x*PolyLog[6, -E^(-c
- d*Sqrt[x])] + 6*d*Sqrt[x]*PolyLog[7, -E^(-c - d*Sqrt[x])] + 6*PolyLog[8, -E^(-c - d*Sqrt[x])]))) - 14*a*(d^
6*x^3*PolyLog[2, E^(-c - d*Sqrt[x])] + 6*(d^5*x^(5/2)*PolyLog[3, E^(-c - d*Sqrt[x])] + 5*d^4*x^2*PolyLog[4, E^
(-c - d*Sqrt[x])] + 20*(d^3*x^(3/2)*PolyLog[5, E^(-c - d*Sqrt[x])] + 3*d^2*x*PolyLog[6, E^(-c - d*Sqrt[x])] +
6*d*Sqrt[x]*PolyLog[7, E^(-c - d*Sqrt[x])] + 6*PolyLog[8, E^(-c - d*Sqrt[x])]))))*Sinh[c + d*Sqrt[x]]^2)/(d^8*
(b + a*Sinh[c + d*Sqrt[x]])^2) + (b^2*x^(7/2)*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^(7/2)*(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 = 0.139, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \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^3*(a+b*csch(c+d*x^(1/2)))^2,x)

[Out]

int(x^3*(a+b*csch(c+d*x^(1/2)))^2,x)

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Maxima [A]  time = 2.06516, size = 875, normalized size = 1.47 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="maxima")

[Out]

1/4*a^2*x^4 - 4*b^2*x^(7/2)/(d*e^(2*d*sqrt(x) + 2*c) - d) - 4*(d^7*x^(7/2)*log(e^(d*sqrt(x) + c) + 1) + 7*d^6*
x^3*dilog(-e^(d*sqrt(x) + c)) - 42*d^5*x^(5/2)*polylog(3, -e^(d*sqrt(x) + c)) + 210*d^4*x^2*polylog(4, -e^(d*s
qrt(x) + c)) - 840*d^3*x^(3/2)*polylog(5, -e^(d*sqrt(x) + c)) + 2520*d^2*x*polylog(6, -e^(d*sqrt(x) + c)) - 50
40*d*sqrt(x)*polylog(7, -e^(d*sqrt(x) + c)) + 5040*polylog(8, -e^(d*sqrt(x) + c)))*a*b/d^8 + 4*(d^7*x^(7/2)*lo
g(-e^(d*sqrt(x) + c) + 1) + 7*d^6*x^3*dilog(e^(d*sqrt(x) + c)) - 42*d^5*x^(5/2)*polylog(3, e^(d*sqrt(x) + c))
+ 210*d^4*x^2*polylog(4, e^(d*sqrt(x) + c)) - 840*d^3*x^(3/2)*polylog(5, e^(d*sqrt(x) + c)) + 2520*d^2*x*polyl
og(6, e^(d*sqrt(x) + c)) - 5040*d*sqrt(x)*polylog(7, e^(d*sqrt(x) + c)) + 5040*polylog(8, e^(d*sqrt(x) + c)))*
a*b/d^8 + 14*(d^6*x^3*log(e^(d*sqrt(x) + c) + 1) + 6*d^5*x^(5/2)*dilog(-e^(d*sqrt(x) + c)) - 30*d^4*x^2*polylo
g(3, -e^(d*sqrt(x) + c)) + 120*d^3*x^(3/2)*polylog(4, -e^(d*sqrt(x) + c)) - 360*d^2*x*polylog(5, -e^(d*sqrt(x)
+ c)) + 720*d*sqrt(x)*polylog(6, -e^(d*sqrt(x) + c)) - 720*polylog(7, -e^(d*sqrt(x) + c)))*b^2/d^8 + 14*(d^6*
x^3*log(-e^(d*sqrt(x) + c) + 1) + 6*d^5*x^(5/2)*dilog(e^(d*sqrt(x) + c)) - 30*d^4*x^2*polylog(3, e^(d*sqrt(x)
+ c)) + 120*d^3*x^(3/2)*polylog(4, e^(d*sqrt(x) + c)) - 360*d^2*x*polylog(5, e^(d*sqrt(x) + c)) + 720*d*sqrt(x
)*polylog(6, e^(d*sqrt(x) + c)) - 720*polylog(7, e^(d*sqrt(x) + c)))*b^2/d^8 - 1/2*(a*b*d^8*x^4 + 4*b^2*d^7*x^
(7/2))/d^8 + 1/2*(a*b*d^8*x^4 - 4*b^2*d^7*x^(7/2))/d^8

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{3} \operatorname{csch}\left (d \sqrt{x} + c\right )^{2} + 2 \, a b x^{3} \operatorname{csch}\left (d \sqrt{x} + c\right ) + a^{2} x^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="fricas")

[Out]

integral(b^2*x^3*csch(d*sqrt(x) + c)^2 + 2*a*b*x^3*csch(d*sqrt(x) + c) + a^2*x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \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**3*(a+b*csch(c+d*x**(1/2)))**2,x)

[Out]

Integral(x**3*(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^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*csch(c+d*x^(1/2)))^2,x, algorithm="giac")

[Out]

integrate((b*csch(d*sqrt(x) + c) + a)^2*x^3, x)