3.248 \(\int \frac{(e+f x)^3 \text{csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=1053 \[ \text{result too large to display} \]

[Out]

(b*(e + f*x)^3)/(a^2*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) + ((e + f*x)^3*ArcTanh[E^(c + d*x)])/
(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)^3*Coth[c + d*x])/(a^2*d) - (3*f*(e + f
*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b^3*(e + f*x)^3*Log[1 +
(b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) - (3*b*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (3*f^3
*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)
^2*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) - (3*f*(e + f*x)^2*PolyLog[2,
E^(c + d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (3*b^3*f*(e + f*x)^2*Poly
Log[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) - (3*b*f^2*(e + f*x)*PolyLog[2, E^(2*(c +
d*x))])/(a^2*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -E^(
c + d*x)])/(a^3*d^3) + (3*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, E^(
c + d*x)])/(a^3*d^3) + (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^
2 + b^2]*d^3) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2
]*d^3) + (3*b*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^4) + (3*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (6*b^2
*f^3*PolyLog[4, -E^(c + d*x)])/(a^3*d^4) - (3*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (6*b^2*f^3*PolyLog[4, E^(
c + d*x)])/(a^3*d^4) - (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d
^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^4)

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Rubi [A]  time = 1.75162, antiderivative size = 1053, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 14, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5575, 4186, 4182, 2279, 2391, 2531, 6609, 2282, 6589, 4184, 3716, 2190, 3322, 2264} \[ -\frac{(e+f x)^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) b^3}{a^3 \sqrt{a^2+b^2} d}+\frac{(e+f x)^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) b^3}{a^3 \sqrt{a^2+b^2} d}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^3}{a^3 \sqrt{a^2+b^2} d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^3}{a^3 \sqrt{a^2+b^2} d^2}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^3}{a^3 \sqrt{a^2+b^2} d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^3}{a^3 \sqrt{a^2+b^2} d^3}-\frac{6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^3}{a^3 \sqrt{a^2+b^2} d^4}+\frac{6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^3}{a^3 \sqrt{a^2+b^2} d^4}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right ) b^2}{a^3 d}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right ) b^2}{a^3 d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right ) b^2}{a^3 d^2}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right ) b^2}{a^3 d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right ) b^2}{a^3 d^3}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right ) b^2}{a^3 d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right ) b^2}{a^3 d^4}+\frac{(e+f x)^3 b}{a^2 d}+\frac{(e+f x)^3 \coth (c+d x) b}{a^2 d}-\frac{3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) b}{a^2 d^2}-\frac{3 f^2 (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right ) b}{a^2 d^3}+\frac{3 f^3 \text{PolyLog}\left (3,e^{2 (c+d x)}\right ) b}{2 a^2 d^4}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{3 f^3 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac{3 f^3 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac{3 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac{3 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

(b*(e + f*x)^3)/(a^2*d) - (6*f^2*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^3) + ((e + f*x)^3*ArcTanh[E^(c + d*x)])/
(a*d) - (2*b^2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^3*d) + (b*(e + f*x)^3*Coth[c + d*x])/(a^2*d) - (3*f*(e + f
*x)^2*Csch[c + d*x])/(2*a*d^2) - ((e + f*x)^3*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b^3*(e + f*x)^3*Log[1 +
(b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2])])/(a^3*Sqrt[a^2 + b^2]*d) - (3*b*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (3*f^3
*PolyLog[2, -E^(c + d*x)])/(a*d^4) + (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(2*a*d^2) - (3*b^2*f*(e + f*x)
^2*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (3*f^3*PolyLog[2, E^(c + d*x)])/(a*d^4) - (3*f*(e + f*x)^2*PolyLog[2,
E^(c + d*x)])/(2*a*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (3*b^3*f*(e + f*x)^2*Poly
Log[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^2) - (3*b*f^2*(e + f*x)*PolyLog[2, E^(2*(c +
d*x))])/(a^2*d^3) - (3*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -E^(
c + d*x)])/(a^3*d^3) + (3*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, E^(
c + d*x)])/(a^3*d^3) + (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^
2 + b^2]*d^3) - (6*b^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2
]*d^3) + (3*b*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^4) + (3*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) - (6*b^2
*f^3*PolyLog[4, -E^(c + d*x)])/(a^3*d^4) - (3*f^3*PolyLog[4, E^(c + d*x)])/(a*d^4) + (6*b^2*f^3*PolyLog[4, E^(
c + d*x)])/(a^3*d^4) - (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d
^4) + (6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*Sqrt[a^2 + b^2]*d^4)

Rule 5575

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(n - 1))/
(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4186

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

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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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]

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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \text{csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \text{csch}^3(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \text{csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{\int (e+f x)^3 \text{csch}(c+d x) \, dx}{2 a}-\frac{b \int (e+f x)^3 \text{csch}^2(c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x)^3 \text{csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac{\left (3 f^2\right ) \int (e+f x) \text{csch}(c+d x) \, dx}{a d^2}\\ &=-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{b^2 \int (e+f x)^3 \text{csch}(c+d x) \, dx}{a^3}-\frac{b^3 \int \frac{(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac{(3 f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{2 a d}-\frac{(3 f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{2 a d}-\frac{(3 b f) \int (e+f x)^2 \coth (c+d x) \, dx}{a^2 d}-\frac{\left (3 f^3\right ) \int \log \left (1-e^{c+d x}\right ) \, dx}{a d^3}+\frac{\left (3 f^3\right ) \int \log \left (1+e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}-\frac{\left (2 b^3\right ) \int \frac{e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^3}+\frac{(6 b f) \int \frac{e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a^2 d}-\frac{\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a^3 d}+\frac{\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a^3 d}-\frac{\left (3 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (3 f^2\right ) \int (e+f x) \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^2}-\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{\left (2 b^4\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \sqrt{a^2+b^2}}+\frac{\left (2 b^4\right ) \int \frac{e^{c+d x} (e+f x)^3}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a^3 \sqrt{a^2+b^2}}+\frac{\left (6 b f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}+\frac{\left (6 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a^3 d^2}-\frac{\left (6 b^2 f^2\right ) \int (e+f x) \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a^3 d^2}+\frac{\left (3 f^3\right ) \int \text{Li}_3\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac{\left (3 f^3\right ) \int \text{Li}_3\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}+\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}-\frac{3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{3 b f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \sqrt{a^2+b^2} d}-\frac{\left (3 b^3 f\right ) \int (e+f x)^2 \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \sqrt{a^2+b^2} d}+\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac{\left (3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}+\frac{\left (3 b f^3\right ) \int \text{Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a^2 d^3}-\frac{\left (6 b^2 f^3\right ) \int \text{Li}_3\left (-e^{c+d x}\right ) \, dx}{a^3 d^3}+\frac{\left (6 b^2 f^3\right ) \int \text{Li}_3\left (e^{c+d x}\right ) \, dx}{a^3 d^3}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}+\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}-\frac{3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}+\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}-\frac{3 b f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{3 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{\left (6 b^3 f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \sqrt{a^2+b^2} d^2}-\frac{\left (6 b^3 f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \sqrt{a^2+b^2} d^2}+\frac{\left (3 b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^2 d^4}-\frac{\left (6 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}+\frac{\left (6 b^2 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{c+d x}\right )}{a^3 d^4}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}+\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}-\frac{3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}+\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}-\frac{3 b f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{6 b^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^3}-\frac{6 b^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^3}+\frac{3 b f^3 \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac{3 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a^3 d^4}-\frac{3 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{6 b^2 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a^3 d^4}-\frac{\left (6 b^3 f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \sqrt{a^2+b^2} d^3}+\frac{\left (6 b^3 f^3\right ) \int \text{Li}_3\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^3 \sqrt{a^2+b^2} d^3}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}+\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}-\frac{3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}+\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}-\frac{3 b f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{6 b^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^3}-\frac{6 b^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^3}+\frac{3 b f^3 \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac{3 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a^3 d^4}-\frac{3 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{6 b^2 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a^3 d^4}-\frac{\left (6 b^3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \sqrt{a^2+b^2} d^4}+\frac{\left (6 b^3 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \sqrt{a^2+b^2} d^4}\\ &=\frac{b (e+f x)^3}{a^2 d}-\frac{6 f^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^3}+\frac{(e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{2 b^2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}+\frac{b (e+f x)^3 \coth (c+d x)}{a^2 d}-\frac{3 f (e+f x)^2 \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x)^3 \coth (c+d x) \text{csch}(c+d x)}{2 a d}-\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}+\frac{b^3 (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d}-\frac{3 b f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}-\frac{3 f^3 \text{Li}_2\left (-e^{c+d x}\right )}{a d^4}+\frac{3 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{2 a d^2}-\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (-e^{c+d x}\right )}{a^3 d^2}+\frac{3 f^3 \text{Li}_2\left (e^{c+d x}\right )}{a d^4}-\frac{3 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{2 a d^2}+\frac{3 b^2 f (e+f x)^2 \text{Li}_2\left (e^{c+d x}\right )}{a^3 d^2}-\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}+\frac{3 b^3 f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^2}-\frac{3 b f^2 (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{3 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a d^3}+\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (-e^{c+d x}\right )}{a^3 d^3}+\frac{3 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a d^3}-\frac{6 b^2 f^2 (e+f x) \text{Li}_3\left (e^{c+d x}\right )}{a^3 d^3}+\frac{6 b^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^3}-\frac{6 b^3 f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^3}+\frac{3 b f^3 \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^4}+\frac{3 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a d^4}-\frac{6 b^2 f^3 \text{Li}_4\left (-e^{c+d x}\right )}{a^3 d^4}-\frac{3 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a d^4}+\frac{6 b^2 f^3 \text{Li}_4\left (e^{c+d x}\right )}{a^3 d^4}-\frac{6 b^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^4}+\frac{6 b^3 f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 \sqrt{a^2+b^2} d^4}\\ \end{align*}

Mathematica [B]  time = 45.3315, size = 2800, normalized size = 2.66 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^3*Csch[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-(-12*a*b*d^3*e^2*E^(2*c)*f*x - 12*a*b*d^3*e*E^(2*c)*f^2*x^2 - 4*a*b*d^3*E^(2*c)*f^3*x^3 + 2*a^2*d^3*e^3*ArcTa
nh[E^(c + d*x)] - 4*b^2*d^3*e^3*ArcTanh[E^(c + d*x)] - 2*a^2*d^3*e^3*E^(2*c)*ArcTanh[E^(c + d*x)] + 4*b^2*d^3*
e^3*E^(2*c)*ArcTanh[E^(c + d*x)] - 12*a^2*d*e*f^2*ArcTanh[E^(c + d*x)] + 12*a^2*d*e*E^(2*c)*f^2*ArcTanh[E^(c +
d*x)] - 3*a^2*d^3*e^2*f*x*Log[1 - E^(c + d*x)] + 6*b^2*d^3*e^2*f*x*Log[1 - E^(c + d*x)] + 3*a^2*d^3*e^2*E^(2*
c)*f*x*Log[1 - E^(c + d*x)] - 6*b^2*d^3*e^2*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 6*a^2*d*f^3*x*Log[1 - E^(c + d*
x)] - 6*a^2*d*E^(2*c)*f^3*x*Log[1 - E^(c + d*x)] - 3*a^2*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] + 6*b^2*d^3*e*f^2*
x^2*Log[1 - E^(c + d*x)] + 3*a^2*d^3*e*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] - 6*b^2*d^3*e*E^(2*c)*f^2*x^2*Log[
1 - E^(c + d*x)] - a^2*d^3*f^3*x^3*Log[1 - E^(c + d*x)] + 2*b^2*d^3*f^3*x^3*Log[1 - E^(c + d*x)] + a^2*d^3*E^(
2*c)*f^3*x^3*Log[1 - E^(c + d*x)] - 2*b^2*d^3*E^(2*c)*f^3*x^3*Log[1 - E^(c + d*x)] + 3*a^2*d^3*e^2*f*x*Log[1 +
E^(c + d*x)] - 6*b^2*d^3*e^2*f*x*Log[1 + E^(c + d*x)] - 3*a^2*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(c + d*x)] + 6*b^
2*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 6*a^2*d*f^3*x*Log[1 + E^(c + d*x)] + 6*a^2*d*E^(2*c)*f^3*x*Log[1
+ E^(c + d*x)] + 3*a^2*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] - 6*b^2*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] - 3*a^2*d
^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] + 6*b^2*d^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] + a^2*d^3*f^3*x^3
*Log[1 + E^(c + d*x)] - 2*b^2*d^3*f^3*x^3*Log[1 + E^(c + d*x)] - a^2*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(c + d*x)]
+ 2*b^2*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(c + d*x)] - 6*a*b*d^2*e^2*f*Log[1 - E^(2*(c + d*x))] + 6*a*b*d^2*e^2*E^
(2*c)*f*Log[1 - E^(2*(c + d*x))] - 12*a*b*d^2*e*f^2*x*Log[1 - E^(2*(c + d*x))] + 12*a*b*d^2*e*E^(2*c)*f^2*x*Lo
g[1 - E^(2*(c + d*x))] - 6*a*b*d^2*f^3*x^2*Log[1 - E^(2*(c + d*x))] + 6*a*b*d^2*E^(2*c)*f^3*x^2*Log[1 - E^(2*(
c + d*x))] - 3*(-1 + E^(2*c))*f*(-2*b^2*d^2*(e + f*x)^2 + a^2*(-2*f^2 + d^2*(e + f*x)^2))*PolyLog[2, -E^(c + d
*x)] + 3*(-1 + E^(2*c))*f*(-2*b^2*d^2*(e + f*x)^2 + a^2*(-2*f^2 + d^2*(e + f*x)^2))*PolyLog[2, E^(c + d*x)] -
6*a*b*d*e*f^2*PolyLog[2, E^(2*(c + d*x))] + 6*a*b*d*e*E^(2*c)*f^2*PolyLog[2, E^(2*(c + d*x))] - 6*a*b*d*f^3*x*
PolyLog[2, E^(2*(c + d*x))] + 6*a*b*d*E^(2*c)*f^3*x*PolyLog[2, E^(2*(c + d*x))] - 6*a^2*d*e*f^2*PolyLog[3, -E^
(c + d*x)] + 12*b^2*d*e*f^2*PolyLog[3, -E^(c + d*x)] + 6*a^2*d*e*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] - 12*b^2
*d*e*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] - 6*a^2*d*f^3*x*PolyLog[3, -E^(c + d*x)] + 12*b^2*d*f^3*x*PolyLog[3,
-E^(c + d*x)] + 6*a^2*d*E^(2*c)*f^3*x*PolyLog[3, -E^(c + d*x)] - 12*b^2*d*E^(2*c)*f^3*x*PolyLog[3, -E^(c + d*
x)] + 6*a^2*d*e*f^2*PolyLog[3, E^(c + d*x)] - 12*b^2*d*e*f^2*PolyLog[3, E^(c + d*x)] - 6*a^2*d*e*E^(2*c)*f^2*P
olyLog[3, E^(c + d*x)] + 12*b^2*d*e*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)] + 6*a^2*d*f^3*x*PolyLog[3, E^(c + d*x)
] - 12*b^2*d*f^3*x*PolyLog[3, E^(c + d*x)] - 6*a^2*d*E^(2*c)*f^3*x*PolyLog[3, E^(c + d*x)] + 12*b^2*d*E^(2*c)*
f^3*x*PolyLog[3, E^(c + d*x)] + 3*a*b*f^3*PolyLog[3, E^(2*(c + d*x))] - 3*a*b*E^(2*c)*f^3*PolyLog[3, E^(2*(c +
d*x))] + 6*a^2*f^3*PolyLog[4, -E^(c + d*x)] - 12*b^2*f^3*PolyLog[4, -E^(c + d*x)] - 6*a^2*E^(2*c)*f^3*PolyLog
[4, -E^(c + d*x)] + 12*b^2*E^(2*c)*f^3*PolyLog[4, -E^(c + d*x)] - 6*a^2*f^3*PolyLog[4, E^(c + d*x)] + 12*b^2*f
^3*PolyLog[4, E^(c + d*x)] + 6*a^2*E^(2*c)*f^3*PolyLog[4, E^(c + d*x)] - 12*b^2*E^(2*c)*f^3*PolyLog[4, E^(c +
d*x)])/(2*a^3*d^4*(-1 + E^(2*c))) + (b^3*(2*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - 3*d^3*e^2*f
*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^
2])] - d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] +
3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*e*f^2*PolyLog[3, (b*E^(c + d*x)
)/(-a + Sqrt[a^2 + b^2])] + 6*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*e*f^2*PolyLog[3
, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] -
6*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))]))/(a^3*Sqrt[a^2 + b^2]*d^4) + (Csch[c]*Csch[c + d*x]^2*(2*b*d*e^3*Cosh[c] + 6*b*d*e^2*f*x*Cosh[c] + 6
*b*d*e*f^2*x^2*Cosh[c] + 2*b*d*f^3*x^3*Cosh[c] + 3*a*e^2*f*Cosh[d*x] + 6*a*e*f^2*x*Cosh[d*x] + 3*a*f^3*x^2*Cos
h[d*x] - 3*a*e^2*f*Cosh[2*c + d*x] - 6*a*e*f^2*x*Cosh[2*c + d*x] - 3*a*f^3*x^2*Cosh[2*c + d*x] - 2*b*d*e^3*Cos
h[c + 2*d*x] - 6*b*d*e^2*f*x*Cosh[c + 2*d*x] - 6*b*d*e*f^2*x^2*Cosh[c + 2*d*x] - 2*b*d*f^3*x^3*Cosh[c + 2*d*x]
+ a*d*e^3*Sinh[d*x] + 3*a*d*e^2*f*x*Sinh[d*x] + 3*a*d*e*f^2*x^2*Sinh[d*x] + a*d*f^3*x^3*Sinh[d*x] - a*d*e^3*S
inh[2*c + d*x] - 3*a*d*e^2*f*x*Sinh[2*c + d*x] - 3*a*d*e*f^2*x^2*Sinh[2*c + d*x] - a*d*f^3*x^3*Sinh[2*c + d*x]
))/(4*a^2*d^2)

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Maple [F]  time = 0.806, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ({\rm csch} \left (dx+c\right ) \right ) ^{3}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),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((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*csch(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out