3.459 \(\int \frac{(e+f x)^3 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=718 \[ -\frac{6 f^2 \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 b d^3}-\frac{6 f^2 \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 b d^3}+\frac{3 f \left (a^2+b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 b d^2}+\frac{3 f \left (a^2+b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 b d^2}+\frac{6 f^3 \left (a^2+b^2\right ) \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 b d^4}+\frac{6 f^3 \left (a^2+b^2\right ) \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 b d^4}+\frac{3 b f^2 (e+f x) \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac{3 b f^3 \text{PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^2 d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac{6 f^3 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^4}+\frac{\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 b d}+\frac{\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 b d}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac{b (e+f x)^4}{4 a^2 f}-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.64475, antiderivative size = 718, normalized size of antiderivative = 1., number of steps used = 48, number of rules used = 19, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.559, Rules used = {5585, 5450, 3296, 2638, 5452, 4182, 2531, 2282, 6589, 5446, 3311, 32, 2635, 8, 3716, 2190, 6609, 5565, 5561} \[ -\frac{6 f^2 \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 b d^3}-\frac{6 f^2 \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 b d^3}+\frac{3 f \left (a^2+b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 b d^2}+\frac{3 f \left (a^2+b^2\right ) (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 b d^2}+\frac{6 f^3 \left (a^2+b^2\right ) \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 b d^4}+\frac{6 f^3 \left (a^2+b^2\right ) \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 b d^4}+\frac{3 b f^2 (e+f x) \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{3 b f (e+f x)^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{2 a^2 d^2}-\frac{3 b f^3 \text{PolyLog}\left (4,e^{2 (c+d x)}\right )}{4 a^2 d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac{6 f^3 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^4}-\frac{6 f^3 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^4}+\frac{\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 b d}+\frac{\left (a^2+b^2\right ) (e+f x)^3 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 b d}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac{b (e+f x)^3 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}+\frac{b (e+f x)^4}{4 a^2 f}-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

(b*(e + f*x)^4)/(4*a^2*f) - ((a^2 + b^2)*(e + f*x)^4)/(4*a^2*b*f) - (6*f*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*
d^2) - ((e + f*x)^3*Csch[c + d*x])/(a*d) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^
2])])/(a^2*b*d) + ((a^2 + b^2)*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*b*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*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2]))])/(a^2*b*d^2) + (3*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(
a^2*b*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*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2]))])/(a^2*b*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^2*b*d^3) + (3*b*f^2*(e + f*x)*PolyLog[3, E^(2*(c + d*x))])/(2*a^2*d^3) + (6*(a^2 + b^2)*f^3*PolyL
og[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*b*d^4) + (6*(a^2 + b^2)*f^3*PolyLog[4, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))])/(a^2*b*d^4) - (3*b*f^3*PolyLog[4, E^(2*(c + d*x))])/(4*a^2*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 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 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 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 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 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 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 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 \cosh (c+d x) \coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^3 \cosh (c+d x) \coth ^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \cosh ^2(c+d x) \coth (c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac{\int (e+f x)^3 \cosh (c+d x) \, dx}{a}+\frac{\int (e+f x)^3 \coth (c+d x) \text{csch}(c+d x) \, dx}{a}-\frac{b \int (e+f x)^3 \cosh ^2(c+d x) \coth (c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x)^3 \cosh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=-\frac{(e+f x)^3 \text{csch}(c+d x)}{a d}+\frac{(e+f x)^3 \sinh (c+d x)}{a d}-\frac{\int (e+f x)^3 \cosh (c+d x) \, dx}{a}-\frac{b \int (e+f x)^3 \coth (c+d x) \, dx}{a^2}+\frac{\left (a^2+b^2\right ) \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{(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}\\ &=\frac{b (e+f x)^4}{4 a^2 f}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{3 f (e+f x)^2 \cosh (c+d x)}{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{\left (a^2+b^2\right ) \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 (a^2+b^2\right ) \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{(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}+\frac{\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, 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{\left (6 f^2\right ) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a d^2}\\ &=\frac{b (e+f x)^4}{4 a^2 f}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\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{\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^2 b 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^2 b 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{6 f^2 (e+f x) \sinh (c+d x)}{a d^3}+\frac{(3 b f) \int (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a^2 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^2 b 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^2 b d}-\frac{\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\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{\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}\\ &=\frac{b (e+f x)^4}{4 a^2 f}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\frac{6 f (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{6 f^3 \cosh (c+d x)}{a d^4}-\frac{(e+f x)^3 \text{csch}(c+d x)}{a 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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b 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{\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}\\ &=\frac{b (e+f x)^4}{4 a^2 f}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\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{\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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b d^3}\\ &=\frac{b (e+f x)^4}{4 a^2 f}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\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{\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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b d^4}\\ &=\frac{b (e+f x)^4}{4 a^2 f}-\frac{\left (a^2+b^2\right ) (e+f x)^4}{4 a^2 b f}-\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{\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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b 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 \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^2 b 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^2 b 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 = 13.512, size = 10534, normalized size = 14.67 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^3*Cosh[c + d*x]*Coth[c + d*x]^2)/(a + b*Sinh[c + d*x]),x]

[Out]

Result too large to show

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Maple [F]  time = 0.986, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}\cosh \left ( dx+c \right ) \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}}{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)^2/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*cosh(d*x+c)*coth(d*x+c)^2/(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*cosh(d*x+c)*coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

e^3*((d*x + c)/(b*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e
^(-d*x - c) - 1)/(a^2*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*b*d)) - 3*e^2*f*lo
g(e^(d*x + c) + 1)/(a*d^2) + 3*e^2*f*log(e^(d*x + c) - 1)/(a*d^2) - 1/4*(a*d*f^3*x^4 + 4*a*d*e*f^2*x^3 + 6*a*d
*e^2*f*x^2 - (a*d*f^3*x^4*e^(2*c) + 4*a*d*e*f^2*x^3*e^(2*c) + 6*a*d*e^2*f*x^2*e^(2*c))*e^(2*d*x) + 8*(b*f^3*x^
3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^2*f*x*e^c)*e^(d*x))/(a*b*d*e^(2*d*x + 2*c) - a*b*d) - (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*p
olylog(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*dilog(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*((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^2*b^2*e^(2*d*x + 2*c) + 2*a^3*b*e^(d*x + c) - a^2*b^2), x)

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Fricas [C]  time = 3.42364, size = 13009, normalized size = 18.12 \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)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/4*(a^2*d^4*f^3*x^4 + 4*a^2*d^4*e*f^2*x^3 + 6*a^2*d^4*e^2*f*x^2 + 4*a^2*d^4*e^3*x + 8*a^2*c*d^3*e^3 - 12*a^2*
c^2*d^2*e^2*f + 8*a^2*c^3*d*e*f^2 - 2*a^2*c^4*f^3 - (a^2*d^4*f^3*x^4 + 4*a^2*d^4*e*f^2*x^3 + 6*a^2*d^4*e^2*f*x
^2 + 4*a^2*d^4*e^3*x + 8*a^2*c*d^3*e^3 - 12*a^2*c^2*d^2*e^2*f + 8*a^2*c^3*d*e*f^2 - 2*a^2*c^4*f^3)*cosh(d*x +
c)^2 - (a^2*d^4*f^3*x^4 + 4*a^2*d^4*e*f^2*x^3 + 6*a^2*d^4*e^2*f*x^2 + 4*a^2*d^4*e^3*x + 8*a^2*c*d^3*e^3 - 12*a
^2*c^2*d^2*e^2*f + 8*a^2*c^3*d*e*f^2 - 2*a^2*c^4*f^3)*sinh(d*x + c)^2 - 8*(a*b*d^3*f^3*x^3 + 3*a*b*d^3*e*f^2*x
^2 + 3*a*b*d^3*e^2*f*x + a*b*d^3*e^3)*cosh(d*x + c) - 12*((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 - ((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)*cosh(d
*x + c)^2 - 2*((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)*cosh(d*x + c)*sinh
(d*x + c) - ((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)*sinh(d*x + c)^2)*dil
og((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*((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 - ((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)*cosh(d*x + c)^2 - 2*((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)*cosh(d*x + c)*sinh(d*x + c) - ((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)*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) + 12*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^
2 - (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*cosh(d*x + c)^2 - 2*(b
^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*cosh(d*x + c)*sinh(d*x + c)
- (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f - 2*a*b*d*e*f^2 + 2*(b^2*d^2*e*f^2 - a*b*d*f^3)*x)*sinh(d*x + c)^2 + 2*(b^2
*d^2*e*f^2 - a*b*d*f^3)*x)*dilog(cosh(d*x + c) + sinh(d*x + c)) + 12*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*
d*e*f^2 - (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d*x + c)^2
- 2*(b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*cosh(d*x + c)*sinh(d*x
+ c) - (b^2*d^2*f^3*x^2 + b^2*d^2*e^2*f + 2*a*b*d*e*f^2 + 2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*sinh(d*x + c)^2 +
2*(b^2*d^2*e*f^2 + a*b*d*f^3)*x)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - 4*((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 - ((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)*cosh(d*x + c)^2 - 2*((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)*cosh(d*x + c)*sinh(d*x + c) - ((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)*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) - 4*((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 - ((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)*cosh(d*x + c)^2 - 2*((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)*cosh(d*x + c)*sinh(d*x + c) - ((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)*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) - 4*((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 - ((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)*cosh(d*x + c)^2 - 2*((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)*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 + 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
)*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) - 4*((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 - ((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)*cosh(d*x + c)^2 - 2*((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)*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 + 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)*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) + 4*(b^2*d^3*f^3*x^3 + b^2*d^3*e^3
+ 3*a*b*d^2*e^2*f + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 - (b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3
*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2)*x)*cosh(d*x + c)^2 - 2*(b^2*d^3*f^3*x
^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*(b^2*d^3*e*f^2 + a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2)
*x)*cosh(d*x + c)*sinh(d*x + c) - (b^2*d^3*f^3*x^3 + b^2*d^3*e^3 + 3*a*b*d^2*e^2*f + 3*(b^2*d^3*e*f^2 + a*b*d^
2*f^3)*x^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2)*x)*sinh(d*x + c)^2 + 3*(b^2*d^3*e^2*f + 2*a*b*d^2*e*f^2)*x)*l
og(cosh(d*x + c) + sinh(d*x + c) + 1) + 4*(b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c)*d*e
*f^2 - (b^2*c^3 + 3*a*b*c^2)*f^3 - (b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 -
(b^2*c^3 + 3*a*b*c^2)*f^3)*cosh(d*x + c)^2 - 2*(b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f + 3*(b^2*c^2 + 2*a*b*c
)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2)*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b^2*d^3*e^3 - 3*(b^2*c + a*b)*d^2*e^2*f
+ 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 - (b^2*c^3 + 3*a*b*c^2)*f^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c
) - 1) + 4*(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2)*f^3 +
3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 - (b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (
b^2*c^3 + 3*a*b*c^2)*f^3 + 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2)*x)*cosh(d
*x + c)^2 - 2*(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2)*f^3
+ 3*(b^2*d^3*e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) -
(b^2*d^3*f^3*x^3 + 3*b^2*c*d^2*e^2*f - 3*(b^2*c^2 + 2*a*b*c)*d*e*f^2 + (b^2*c^3 + 3*a*b*c^2)*f^3 + 3*(b^2*d^3*
e*f^2 - a*b*d^2*f^3)*x^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d^2*e*f^2)*x)*sinh(d*x + c)^2 + 3*(b^2*d^3*e^2*f - 2*a*b*d
^2*e*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 24*((a^2 + b^2)*f^3*cosh(d*x + c)^2 + 2*(a^2 + b^2)*f^3
*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f^3*sinh(d*x + c)^2 - (a^2 + b^2)*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) + 24*((a^2 + b^2)*f^3*cosh(d
*x + c)^2 + 2*(a^2 + b^2)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a^2 + b^2)*f^3*sinh(d*x + c)^2 - (a^2 + b^2)*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)
- 24*(b^2*f^3*cosh(d*x + c)^2 + 2*b^2*f^3*cosh(d*x + c)*sinh(d*x + c) + b^2*f^3*sinh(d*x + c)^2 - b^2*f^3)*po
lylog(4, cosh(d*x + c) + sinh(d*x + c)) - 24*(b^2*f^3*cosh(d*x + c)^2 + 2*b^2*f^3*cosh(d*x + c)*sinh(d*x + c)
+ b^2*f^3*sinh(d*x + c)^2 - b^2*f^3)*polylog(4, -cosh(d*x + c) - sinh(d*x + c)) + 24*((a^2 + b^2)*d*f^3*x + (a
^2 + b^2)*d*e*f^2 - ((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d*f^3*x + (a^
2 + b^2)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*sinh(d*x + c)^2)*p
olylog(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*((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2 - ((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*cosh(d*x + c)^2 -
2*((a^2 + b^2)*d*f^3*x + (a^2 + b^2)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2 + b^2)*d*f^3*x + (a^2 + b^2
)*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) - 24*(b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3 - (b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*
cosh(d*x + c)^2 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 - a*b*f^3)*cosh(d*x + c)*sinh(d*x + c) - (b^2*d*f^3*x + b^2*d*e
*f^2 - a*b*f^3)*sinh(d*x + c)^2)*polylog(3, cosh(d*x + c) + sinh(d*x + c)) - 24*(b^2*d*f^3*x + b^2*d*e*f^2 + a
*b*f^3 - (b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(d*x + c)^2 - 2*(b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*cosh(
d*x + c)*sinh(d*x + c) - (b^2*d*f^3*x + b^2*d*e*f^2 + a*b*f^3)*sinh(d*x + c)^2)*polylog(3, -cosh(d*x + c) - si
nh(d*x + c)) - 2*(4*a*b*d^3*f^3*x^3 + 12*a*b*d^3*e*f^2*x^2 + 12*a*b*d^3*e^2*f*x + 4*a*b*d^3*e^3 + (a^2*d^4*f^3
*x^4 + 4*a^2*d^4*e*f^2*x^3 + 6*a^2*d^4*e^2*f*x^2 + 4*a^2*d^4*e^3*x + 8*a^2*c*d^3*e^3 - 12*a^2*c^2*d^2*e^2*f +
8*a^2*c^3*d*e*f^2 - 2*a^2*c^4*f^3)*cosh(d*x + c))*sinh(d*x + c))/(a^2*b*d^4*cosh(d*x + c)^2 + 2*a^2*b*d^4*cosh
(d*x + c)*sinh(d*x + c) + a^2*b*d^4*sinh(d*x + c)^2 - a^2*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)**2/(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)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out