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

Optimal. Leaf size=605 \[ \frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \sqrt{a^2+b^2}}-\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \sqrt{a^2+b^2}}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.964937, antiderivative size = 605, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5575, 4182, 2531, 6609, 2282, 6589, 3322, 2264, 2190} \[ \frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{6 b f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \sqrt{a^2+b^2}}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \sqrt{a^2+b^2}}+\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \sqrt{a^2+b^2}}-\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 b f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^4 \sqrt{a^2+b^2}}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \sqrt{a^2+b^2}}+\frac{b (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \sqrt{a^2+b^2}}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(
a*Sqrt[a^2 + b^2]*d) + (b*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d) -
(3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + (3*f*(e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*b*
f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (3*b*f*(e + f*x)
^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^2) + (6*f^2*(e + f*x)*PolyLog[3,
-E^(c + d*x)])/(a*d^3) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog[3, -(
(b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d
*x))/(a + Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^2]*d^3) - (6*f^3*PolyLog[4, -E^(c + d*x)])/(a*d^4) + (6*f^3*Poly
Log[4, E^(c + d*x)])/(a*d^4) - (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*Sqrt[a^2 + b^
2]*d^4) + (6*b*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*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 4182

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 6609

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 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]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rubi steps

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

Mathematica [A]  time = 3.11956, size = 757, normalized size = 1.25 \[ \frac{\frac{b \left (-3 d^2 f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+3 d^2 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+6 d e f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-6 d e f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+6 d f^3 x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-6 d f^3 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-6 f^3 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )-3 d^3 e^2 f x \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+3 d^3 e^2 f x \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+2 d^3 e^3 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-3 d^3 e f^2 x^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+3 d^3 e f^2 x^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-d^3 f^3 x^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+d^3 f^3 x^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )\right )}{\sqrt{a^2+b^2}}-3 f \left (d^2 (e+f x)^2 \text{PolyLog}(2,-\sinh (c+d x)-\cosh (c+d x))-2 d f (e+f x) \text{PolyLog}(3,-\sinh (c+d x)-\cosh (c+d x))+2 f^2 \text{PolyLog}(4,-\sinh (c+d x)-\cosh (c+d x))\right )+3 f \left (d^2 (e+f x)^2 \text{PolyLog}(2,\sinh (c+d x)+\cosh (c+d x))-2 d f (e+f x) \text{PolyLog}(3,\sinh (c+d x)+\cosh (c+d x))+2 f^2 \text{PolyLog}(4,\sinh (c+d x)+\cosh (c+d x))\right )-2 d^3 (e+f x)^3 \tanh ^{-1}(\sinh (c+d x)+\cosh (c+d x))}{a d^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*d^3*(e + f*x)^3*ArcTanh[Cosh[c + d*x] + Sinh[c + d*x]] + (b*(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*Po
lyLog[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]))]))/Sqrt[a^2 + b^2] - 3*f*(d^2*(e + f*x)^2*PolyLog[2, -Cosh[c + d*x] - Sinh[c +
d*x]] - 2*d*f*(e + f*x)*PolyLog[3, -Cosh[c + d*x] - Sinh[c + d*x]] + 2*f^2*PolyLog[4, -Cosh[c + d*x] - Sinh[c
+ d*x]]) + 3*f*(d^2*(e + f*x)^2*PolyLog[2, Cosh[c + d*x] + Sinh[c + d*x]] - 2*d*f*(e + f*x)*PolyLog[3, Cosh[c
+ d*x] + Sinh[c + d*x]] + 2*f^2*PolyLog[4, Cosh[c + d*x] + Sinh[c + d*x]]))/(a*d^4)

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Maple [F]  time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}{\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*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.05163, size = 3903, normalized size = 6.45 \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*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(6*b^2*f^3*sqrt((a^2 + b^2)/b^2)*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^2*f^3*sqrt((a^2 + b^2)/b^2)*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*(a^2 + b^2)*f^3*polylog(4, cosh(d*x +
c) + sinh(d*x + c)) + 6*(a^2 + b^2)*f^3*polylog(4, -cosh(d*x + c) - sinh(d*x + c)) + 3*(b^2*d^2*f^3*x^2 + 2*b^
2*d^2*e*f^2*x + b^2*d^2*e^2*f)*sqrt((a^2 + b^2)/b^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^2*d^2*f^3*x^2 + 2*b^2*d^2*e*f^2*x + b^2*d^2*e^2
*f)*sqrt((a^2 + b^2)/b^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) - (b^2*d^3*e^3 - 3*b^2*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*sqrt((a^2 +
b^2)/b^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^2*d^3*e^3 - 3*b^2
*c*d^2*e^2*f + 3*b^2*c^2*d*e*f^2 - b^2*c^3*f^3)*sqrt((a^2 + b^2)/b^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^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^2*d^3*e^2*f*x + 3*b^2*c*d^
2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*sqrt((a^2 + b^2)/b^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^2*d^3*f^3*x^3 + 3*b^2*d^3*e*f^2*x^2 + 3*b^
2*d^3*e^2*f*x + 3*b^2*c*d^2*e^2*f - 3*b^2*c^2*d*e*f^2 + b^2*c^3*f^3)*sqrt((a^2 + b^2)/b^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) - 6*(b^2*d*f^3*x + b^
2*d*e*f^2)*sqrt((a^2 + b^2)/b^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^2*d*f^3*x + b^2*d*e*f^2)*sqrt((a^2 + b^2)/b^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) - 3*((a^2 + b^2)*d^2*f^
3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) + 3*((a^2 + b^
2)*d^2*f^3*x^2 + 2*(a^2 + b^2)*d^2*e*f^2*x + (a^2 + b^2)*d^2*e^2*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)) + ((
a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^2*f*x + (a^2 + b^2)*d^3*e^3)*log(co
sh(d*x + c) + sinh(d*x + c) + 1) - ((a^2 + b^2)*d^3*e^3 - 3*(a^2 + b^2)*c*d^2*e^2*f + 3*(a^2 + b^2)*c^2*d*e*f^
2 - (a^2 + b^2)*c^3*f^3)*log(cosh(d*x + c) + sinh(d*x + c) - 1) - ((a^2 + b^2)*d^3*f^3*x^3 + 3*(a^2 + b^2)*d^3
*e*f^2*x^2 + 3*(a^2 + b^2)*d^3*e^2*f*x + 3*(a^2 + b^2)*c*d^2*e^2*f - 3*(a^2 + b^2)*c^2*d*e*f^2 + (a^2 + b^2)*c
^3*f^3)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 6*((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*polylog(3, cos
h(d*x + c) + sinh(d*x + c)) - 6*((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*polylog(3, -cosh(d*x + c) - sinh(d
*x + c)))/((a^3 + a*b^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*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*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out