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

Optimal. Leaf size=601 \[ -\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^3}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^4}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^4}+\frac{3 b f^2 (e+f x) \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac{3 b f^3 \text{PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^2 d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac{6 f^3 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^4}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.00297, antiderivative size = 601, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {5587, 5452, 4182, 2531, 2282, 6589, 5569, 3716, 2190, 6609, 5561} \[ -\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^3}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^4}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^4}+\frac{3 b f^2 (e+f x) \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac{3 b f^3 \text{PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^2 d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac{6 f^3 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^4}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5587

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(p - 1)*Coth[c + d*x]^n)/(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 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 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 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 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 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 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 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 (c+d x) \text{csch}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \coth (c+d x) \text{csch}(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}-\frac{b \int (e+f x)^3 \coth (c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac{(3 f) \int (e+f x)^2 \text{csch}(c+d x) \, dx}{a d}\\ &=-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{(2 b) \int \frac{e^{2 (c+d x)} (e+f x)^3}{1-e^{2 (c+d x)}} \, dx}{a^2}+\frac{b^2 \int \frac{e^{c+d x} (e+f x)^3}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^2}+\frac{b^2 \int \frac{e^{c+d x} (e+f x)^3}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a^2}-\frac{\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a d^2}+\frac{\left (6 f^2\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d^2}\\ &=-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f^2 (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^3}-\frac{(3 b f) \int (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a^2 d}-\frac{(3 b f) \int (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a^2 d}+\frac{(3 b f) \int (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 d}+\frac{\left (6 f^3\right ) \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a d^3}-\frac{\left (6 f^3\right ) \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a d^3}\\ &=-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f^2 (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac{\left (3 b f^2\right ) \int (e+f x) \text{Li}_2\left (e^{2 (c+d x)}\right ) \, dx}{a^2 d^2}-\frac{\left (6 b 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^2 d^2}-\frac{\left (6 b 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^2 d^2}+\frac{\left (6 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}-\frac{\left (6 f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a d^4}\\ &=-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f^2 (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac{6 f^3 \text{Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^3}+\frac{3 b f^2 (e+f x) \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{\left (3 b f^3\right ) \int \text{Li}_3\left (e^{2 (c+d x)}\right ) \, dx}{2 a^2 d^3}+\frac{\left (6 b f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a^2 d^3}+\frac{\left (6 b f^3\right ) \int \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a^2 d^3}\\ &=-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f^2 (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac{6 f^3 \text{Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^3}+\frac{3 b f^2 (e+f x) \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{\left (3 b f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 a^2 d^4}+\frac{\left (6 b 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^2 d^4}+\frac{\left (6 b 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^2 d^4}\\ &=-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d}+\frac{b (e+f x)^3 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{6 f^2 (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a d^3}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{3 b f (e+f x)^2 \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^2}-\frac{3 b f (e+f x)^2 \text{Li}_2\left (e^{2 (c+d x)}\right )}{2 a^2 d^2}+\frac{6 f^3 \text{Li}_3\left (-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{Li}_3\left (e^{c+d x}\right )}{a d^4}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{6 b f^2 (e+f x) \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^3}+\frac{3 b f^2 (e+f x) \text{Li}_3\left (e^{2 (c+d x)}\right )}{2 a^2 d^3}+\frac{6 b f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^4}+\frac{6 b f^3 \text{Li}_4\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 d^4}-\frac{3 b f^3 \text{Li}_4\left (e^{2 (c+d x)}\right )}{4 a^2 d^4}\\ \end{align*}

Mathematica [C]  time = 48.7021, size = 5638, normalized size = 9.38 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.909, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}{\rm coth} \left (dx+c\right ){\rm csch} \left (dx+c\right )}{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)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

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

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Fricas [C]  time = 3.62349, size = 9829, normalized size = 16.35 \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)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x + a*d^3*e^3)*cosh(d*x + c) + 3*(b*d^2*f^3*x^2 + 2*b*d
^2*e*f^2*x + b*d^2*e^2*f - (b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh(d*x + c)^2 - 2*(b*d^2*f^3*x^2
+ 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)
*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b
^2)/b^2) - b)/b + 1) + 3*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f - (b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b
*d^2*e^2*f)*cosh(d*x + c)^2 - 2*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) -
(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b
*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 3*(b*d^2*f^3*x^2 + b*d^2*e^2*f - 2*a*d*e
*f^2 - (b*d^2*f^3*x^2 + b*d^2*e^2*f - 2*a*d*e*f^2 + 2*(b*d^2*e*f^2 - a*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b*d^2*f^
3*x^2 + b*d^2*e^2*f - 2*a*d*e*f^2 + 2*(b*d^2*e*f^2 - a*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^3*x^2
+ b*d^2*e^2*f - 2*a*d*e*f^2 + 2*(b*d^2*e*f^2 - a*d*f^3)*x)*sinh(d*x + c)^2 + 2*(b*d^2*e*f^2 - a*d*f^3)*x)*dilo
g(cosh(d*x + c) + sinh(d*x + c)) - 3*(b*d^2*f^3*x^2 + b*d^2*e^2*f + 2*a*d*e*f^2 - (b*d^2*f^3*x^2 + b*d^2*e^2*f
+ 2*a*d*e*f^2 + 2*(b*d^2*e*f^2 + a*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b*d^2*f^3*x^2 + b*d^2*e^2*f + 2*a*d*e*f^2 +
2*(b*d^2*e*f^2 + a*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^3*x^2 + b*d^2*e^2*f + 2*a*d*e*f^2 + 2*(b*
d^2*e*f^2 + a*d*f^3)*x)*sinh(d*x + c)^2 + 2*(b*d^2*e*f^2 + a*d*f^3)*x)*dilog(-cosh(d*x + c) - sinh(d*x + c)) +
(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3 - (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 -
b*c^3*f^3)*cosh(d*x + c)^2 - 2*(b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cosh(d*x + c)*sinh
(d*x + c) - (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c)
+ 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c
^3*f^3 - (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cosh(d*x + c)^2 - 2*(b*d^3*e^3 - 3*b*c*d^
2*e^2*f + 3*b*c^2*d*e*f^2 - b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b*d^3*e^3 - 3*b*c*d^2*e^2*f + 3*b*c^2*d*
e*f^2 - b*c^3*f^3)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*
a) + (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3 - (b
*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*cosh(d*x +
c)^2 - 2*(b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3
)*cosh(d*x + c)*sinh(d*x + c) - (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c
^2*d*e*f^2 + b*c^3*f^3)*sinh(d*x + c)^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d
*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*
f - 3*b*c^2*d*e*f^2 + b*c^3*f^3 - (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b
*c^2*d*e*f^2 + b*c^3*f^3)*cosh(d*x + c)^2 - 2*(b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*x + 3*b*c*d^2
*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d
^3*e^2*f*x + 3*b*c*d^2*e^2*f - 3*b*c^2*d*e*f^2 + b*c^3*f^3)*sinh(d*x + c)^2)*log(-(a*cosh(d*x + c) + a*sinh(d*
x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - (b*d^3*f^3*x^3 + b*d^3*e^3 + 3*a*
d^2*e^2*f + 3*(b*d^3*e*f^2 + a*d^2*f^3)*x^2 - (b*d^3*f^3*x^3 + b*d^3*e^3 + 3*a*d^2*e^2*f + 3*(b*d^3*e*f^2 + a*
d^2*f^3)*x^2 + 3*(b*d^3*e^2*f + 2*a*d^2*e*f^2)*x)*cosh(d*x + c)^2 - 2*(b*d^3*f^3*x^3 + b*d^3*e^3 + 3*a*d^2*e^2
*f + 3*(b*d^3*e*f^2 + a*d^2*f^3)*x^2 + 3*(b*d^3*e^2*f + 2*a*d^2*e*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) - (b*d^3
*f^3*x^3 + b*d^3*e^3 + 3*a*d^2*e^2*f + 3*(b*d^3*e*f^2 + a*d^2*f^3)*x^2 + 3*(b*d^3*e^2*f + 2*a*d^2*e*f^2)*x)*si
nh(d*x + c)^2 + 3*(b*d^3*e^2*f + 2*a*d^2*e*f^2)*x)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (b*d^3*e^3 - 3*(b*
c + a)*d^2*e^2*f + 3*(b*c^2 + 2*a*c)*d*e*f^2 - (b*c^3 + 3*a*c^2)*f^3 - (b*d^3*e^3 - 3*(b*c + a)*d^2*e^2*f + 3*
(b*c^2 + 2*a*c)*d*e*f^2 - (b*c^3 + 3*a*c^2)*f^3)*cosh(d*x + c)^2 - 2*(b*d^3*e^3 - 3*(b*c + a)*d^2*e^2*f + 3*(b
*c^2 + 2*a*c)*d*e*f^2 - (b*c^3 + 3*a*c^2)*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b*d^3*e^3 - 3*(b*c + a)*d^2*e^2*
f + 3*(b*c^2 + 2*a*c)*d*e*f^2 - (b*c^3 + 3*a*c^2)*f^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1)
- (b*d^3*f^3*x^3 + 3*b*c*d^2*e^2*f - 3*(b*c^2 + 2*a*c)*d*e*f^2 + (b*c^3 + 3*a*c^2)*f^3 + 3*(b*d^3*e*f^2 - a*d
^2*f^3)*x^2 - (b*d^3*f^3*x^3 + 3*b*c*d^2*e^2*f - 3*(b*c^2 + 2*a*c)*d*e*f^2 + (b*c^3 + 3*a*c^2)*f^3 + 3*(b*d^3*
e*f^2 - a*d^2*f^3)*x^2 + 3*(b*d^3*e^2*f - 2*a*d^2*e*f^2)*x)*cosh(d*x + c)^2 - 2*(b*d^3*f^3*x^3 + 3*b*c*d^2*e^2
*f - 3*(b*c^2 + 2*a*c)*d*e*f^2 + (b*c^3 + 3*a*c^2)*f^3 + 3*(b*d^3*e*f^2 - a*d^2*f^3)*x^2 + 3*(b*d^3*e^2*f - 2*
a*d^2*e*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) - (b*d^3*f^3*x^3 + 3*b*c*d^2*e^2*f - 3*(b*c^2 + 2*a*c)*d*e*f^2 + (
b*c^3 + 3*a*c^2)*f^3 + 3*(b*d^3*e*f^2 - a*d^2*f^3)*x^2 + 3*(b*d^3*e^2*f - 2*a*d^2*e*f^2)*x)*sinh(d*x + c)^2 +
3*(b*d^3*e^2*f - 2*a*d^2*e*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 6*(b*f^3*cosh(d*x + c)^2 + 2*b*f^
3*cosh(d*x + c)*sinh(d*x + c) + b*f^3*sinh(d*x + c)^2 - b*f^3)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) +
(b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*(b*f^3*cosh(d*x + c)^2 + 2*b*f^3*cosh(d*x +
c)*sinh(d*x + c) + b*f^3*sinh(d*x + c)^2 - b*f^3)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x
+ c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 6*(b*f^3*cosh(d*x + c)^2 + 2*b*f^3*cosh(d*x + c)*sinh(d*x
+ c) + b*f^3*sinh(d*x + c)^2 - b*f^3)*polylog(4, cosh(d*x + c) + sinh(d*x + c)) + 6*(b*f^3*cosh(d*x + c)^2 +
2*b*f^3*cosh(d*x + c)*sinh(d*x + c) + b*f^3*sinh(d*x + c)^2 - b*f^3)*polylog(4, -cosh(d*x + c) - sinh(d*x + c)
) - 6*(b*d*f^3*x + b*d*e*f^2 - (b*d*f^3*x + b*d*e*f^2)*cosh(d*x + c)^2 - 2*(b*d*f^3*x + b*d*e*f^2)*cosh(d*x +
c)*sinh(d*x + c) - (b*d*f^3*x + b*d*e*f^2)*sinh(d*x + c)^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b
*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 6*(b*d*f^3*x + b*d*e*f^2 - (b*d*f^3*x + b*d*e*f^
2)*cosh(d*x + c)^2 - 2*(b*d*f^3*x + b*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^3*x + b*d*e*f^2)*sinh(d*x
+ c)^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b
^2))/b) + 6*(b*d*f^3*x + b*d*e*f^2 - a*f^3 - (b*d*f^3*x + b*d*e*f^2 - a*f^3)*cosh(d*x + c)^2 - 2*(b*d*f^3*x +
b*d*e*f^2 - a*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^3*x + b*d*e*f^2 - a*f^3)*sinh(d*x + c)^2)*polylog(3, c
osh(d*x + c) + sinh(d*x + c)) + 6*(b*d*f^3*x + b*d*e*f^2 + a*f^3 - (b*d*f^3*x + b*d*e*f^2 + a*f^3)*cosh(d*x +
c)^2 - 2*(b*d*f^3*x + b*d*e*f^2 + a*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^3*x + b*d*e*f^2 + a*f^3)*sinh(d*
x + c)^2)*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) + 2*(a*d^3*f^3*x^3 + 3*a*d^3*e*f^2*x^2 + 3*a*d^3*e^2*f*x
+ a*d^3*e^3)*sinh(d*x + c))/(a^2*d^4*cosh(d*x + c)^2 + 2*a^2*d^4*cosh(d*x + c)*sinh(d*x + c) + a^2*d^4*sinh(d*
x + c)^2 - a^2*d^4)

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

[Out]

Timed out