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

Optimal. Leaf size=972 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 2.20277, antiderivative size = 972, normalized size of antiderivative = 1., number of steps used = 62, number of rules used = 23, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.821, Rules used = {5569, 3720, 3716, 2190, 2279, 2391, 32, 2531, 6609, 2282, 6589, 5585, 5450, 3296, 2638, 5452, 4182, 5446, 3311, 2635, 8, 5565, 5561} \[ -\frac{b^2 (e+f x)^4}{4 a^3 f}+\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^3 f}-\frac{(e+f x)^4}{4 a f}-\frac{\coth ^2(c+d x) (e+f x)^3}{2 a d}+\frac{b \text{csch}(c+d x) (e+f x)^3}{a^2 d}-\frac{\left (a^2+b^2\right ) \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{a^3 d}-\frac{\left (a^2+b^2\right ) \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{a^3 d}+\frac{b^2 \log \left (1-e^{2 (c+d x)}\right ) (e+f x)^3}{a^3 d}+\frac{\log \left (1-e^{2 (c+d x)}\right ) (e+f x)^3}{a d}+\frac{(e+f x)^3}{2 a d}-\frac{3 f (e+f x)^2}{2 a d^2}+\frac{6 b f \tanh ^{-1}\left (e^{c+d x}\right ) (e+f x)^2}{a^2 d^2}-\frac{3 f \coth (c+d x) (e+f x)^2}{2 a d^2}-\frac{3 \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) (e+f x)^2}{a^3 d^2}-\frac{3 \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) (e+f x)^2}{a^3 d^2}+\frac{3 b^2 f \text{PolyLog}\left (2,e^{2 (c+d x)}\right ) (e+f x)^2}{2 a^3 d^2}+\frac{3 f \text{PolyLog}\left (2,e^{2 (c+d x)}\right ) (e+f x)^2}{2 a d^2}+\frac{3 f^2 \log \left (1-e^{2 (c+d x)}\right ) (e+f x)}{a d^3}+\frac{6 b f^2 \text{PolyLog}\left (2,-e^{c+d x}\right ) (e+f x)}{a^2 d^3}-\frac{6 b f^2 \text{PolyLog}\left (2,e^{c+d x}\right ) (e+f x)}{a^2 d^3}+\frac{6 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) (e+f x)}{a^3 d^3}+\frac{6 \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) (e+f x)}{a^3 d^3}-\frac{3 b^2 f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right ) (e+f x)}{2 a^3 d^3}-\frac{3 f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right ) (e+f x)}{2 a d^3}+\frac{3 f^3 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a d^4}-\frac{6 b f^3 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^4}+\frac{6 b f^3 \text{PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^4}-\frac{6 \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^4}-\frac{6 \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^3 d^4}+\frac{3 b^2 f^3 \text{PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^3 d^4}+\frac{3 f^3 \text{PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5569

Int[(Coth[(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*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]*Coth[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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && 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 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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -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 5585

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Cosh[c + d*x]^(p + 1)*Coth[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] && IGtQ[p, 0]

Rule 5450

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 3296

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5452

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 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 5446

Int[Cosh[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[((c
+ d*x)^m*Sinh[a + b*x]^(n + 1))/(b*(n + 1)), x] - Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Sinh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3311

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

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5565

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rubi steps

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

Mathematica [C]  time = 73.7676, size = 11848, normalized size = 12.19 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.872, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ({\rm coth} \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*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^3*(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x -
4*c) - a^2)*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x
- c) + 1)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c) - 1)/(a^3*d)) + (3*a*f^3*x^2 + 6*a*e*f^2*x + 3*a*e^2*f + 2*(
b*d*f^3*x^3*e^(3*c) + 3*b*d*e*f^2*x^2*e^(3*c) + 3*b*d*e^2*f*x*e^(3*c))*e^(3*d*x) - (2*a*d*f^3*x^3*e^(2*c) + 3*
a*e^2*f*e^(2*c) + 3*(2*d*e*f^2 + f^3)*a*x^2*e^(2*c) + 6*(d*e^2*f + e*f^2)*a*x*e^(2*c))*e^(2*d*x) - 2*(b*d*f^3*
x^3*e^c + 3*b*d*e*f^2*x^2*e^c + 3*b*d*e^2*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*
c) + a^2*d^2) - 3*(b*d*e^2*f + a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f - a*e*f^2)*x/(a^2*d^2) + 3*(b*d*e^2*f + a*e
*f^2)*log(e^(d*x + c) + 1)/(a^2*d^3) - 3*(b*d*e^2*f - a*e*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) + (d^3*x^3*log(e
^(d*x + c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) + 6*polylog(4, -e^(d*x + c)))
*(a^2*f^3 + b^2*f^3)/(a^3*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog
(3, e^(d*x + c)) + 6*polylog(4, e^(d*x + c)))*(a^2*f^3 + b^2*f^3)/(a^3*d^4) + 3*(a^2*d*e*f^2 + b^2*d*e*f^2 + a
*b*f^3)*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))/(a^3*d^4) + 3*
(a^2*d*e*f^2 + b^2*d*e*f^2 - a*b*f^3)*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3,
e^(d*x + c)))/(a^3*d^4) + 3*(b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + (d^2*e^2*f + f^3)*a^2)*(d*x*log(e^(d*x + c) + 1)
+ dilog(-e^(d*x + c)))/(a^3*d^4) + 3*(b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + (d^2*e^2*f + f^3)*a^2)*(d*x*log(-e^(d*x
+ c) + 1) + dilog(e^(d*x + c)))/(a^3*d^4) - 1/4*((a^2*f^3 + b^2*f^3)*d^4*x^4 + 4*(a^2*d*e*f^2 + b^2*d*e*f^2 +
a*b*f^3)*d^3*x^3 + 6*(b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + (d^2*e^2*f + f^3)*a^2)*d^2*x^2)/(a^3*d^4) - 1/4*((a^2*f
^3 + b^2*f^3)*d^4*x^4 + 4*(a^2*d*e*f^2 + b^2*d*e*f^2 - a*b*f^3)*d^3*x^3 + 6*(b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + (
d^2*e^2*f + f^3)*a^2)*d^2*x^2)/(a^3*d^4) + integrate(-2*((a^2*b*f^3 + b^3*f^3)*x^3 + 3*(a^2*b*e*f^2 + b^3*e*f^
2)*x^2 + 3*(a^2*b*e^2*f + b^3*e^2*f)*x - ((a^3*f^3*e^c + a*b^2*f^3*e^c)*x^3 + 3*(a^3*e*f^2*e^c + a*b^2*e*f^2*e
^c)*x^2 + 3*(a^3*e^2*f*e^c + a*b^2*e^2*f*e^c)*x)*e^(d*x))/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) - a^3*b),
x)

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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*coth(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*coth(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*coth(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out