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

Optimal. Leaf size=638 \[ \frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b d^3}-\frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b d^3}-\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b d^2}+\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b d^2}-\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b d^4}+\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b d^4}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a b d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{(e+f x)^4}{4 b f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.27521, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 14, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5585, 5450, 3296, 2637, 4182, 2531, 6609, 2282, 6589, 5565, 32, 3322, 2264, 2190} \[ \frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b d^3}-\frac{6 f^2 \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b d^3}-\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b d^2}+\frac{3 f \sqrt{a^2+b^2} (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b d^2}-\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a b d^4}+\frac{6 f^3 \sqrt{a^2+b^2} \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a b d^4}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \text{PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac{6 f^3 \text{PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a b d}+\frac{\sqrt{a^2+b^2} (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a b d}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{(e+f x)^4}{4 b f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A]  time = 2.47645, size = 802, normalized size = 1.26 \[ \frac{a x \left (4 e^3+6 f x e^2+4 f^2 x^2 e+f^3 x^3\right ) d^4+4 \sqrt{a^2+b^2} \left (2 e^3 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^3-f^3 x^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3-3 e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3-3 e^2 f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3+f^3 x^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3+3 e^2 f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3-3 f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d^2+3 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d^2+6 e f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d+6 f^3 x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d-6 e f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d-6 f^3 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d-6 f^3 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right )-4 b \left (2 d^3 \tanh ^{-1}(\cosh (c+d x)+\sinh (c+d x)) (e+f x)^3+3 f \left (2 \text{PolyLog}(4,-\cosh (c+d x)-\sinh (c+d x)) f^2-2 d (e+f x) \text{PolyLog}(3,-\cosh (c+d x)-\sinh (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,-\cosh (c+d x)-\sinh (c+d x))\right )-3 f \left (2 \text{PolyLog}(4,\cosh (c+d x)+\sinh (c+d x)) f^2-2 d (e+f x) \text{PolyLog}(3,\cosh (c+d x)+\sinh (c+d x)) f+d^2 (e+f x)^2 \text{PolyLog}(2,\cosh (c+d x)+\sinh (c+d x))\right )\right )}{4 a b d^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) + 4*Sqrt[a^2 + b^2]*(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]))]) - 4*b*(2*d^3*(e + f*x)^3*ArcTanh[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] - Si
nh[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]])))/(4*a*b*d^4)

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Maple [F]  time = 0.734, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\cosh \left ( dx+c \right ){\rm coth} \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*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.41837, size = 3588, normalized size = 5.62 \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*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(a*d^4*f^3*x^4 + 4*a*d^4*e*f^2*x^3 + 6*a*d^4*e^2*f*x^2 + 4*a*d^4*e^3*x - 24*b*f^3*sqrt((a^2 + b^2)/b^2)*po
lylog(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) +
24*b*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) + 24*b*f^3*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - 24*b*f^3*polylog(4, -cos
h(d*x + c) - sinh(d*x + c)) - 12*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*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) + 12
*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*x + b*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) + 4*(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)*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) - 4*(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)*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) - 4*(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)*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) + 4*(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)*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) + 24*(b*
d*f^3*x + b*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) - 24*(b*d*f^3*x + b*d*e*f^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*co
sh(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) + 12*(b*d^2*f^3*
x^2 + 2*b*d^2*e*f^2*x + b*d^2*e^2*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) - 12*(b*d^2*f^3*x^2 + 2*b*d^2*e*f^2*
x + b*d^2*e^2*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - 4*(b*d^3*f^3*x^3 + 3*b*d^3*e*f^2*x^2 + 3*b*d^3*e^2*f*
x + b*d^3*e^3)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + 4*(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)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 4*(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)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 24*(b*d*f^3*x + b*d*e*f^2
)*polylog(3, cosh(d*x + c) + sinh(d*x + c)) + 24*(b*d*f^3*x + b*d*e*f^2)*polylog(3, -cosh(d*x + c) - sinh(d*x
+ c)))/(a*b*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*cosh(d*x+c)*coth(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*cosh(d*x+c)*coth(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out