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

Optimal. Leaf size=864 \[ \frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{b \sinh ^2(c+d x) (e+f x)^3}{2 a^2 d}-\frac{b \left (a^2+b^2\right ) \log \left (\frac{e^{c+d x} a}{b-\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}-\frac{b \left (a^2+b^2\right ) \log \left (\frac{e^{c+d x} a}{b+\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}+\frac{\cosh ^2(c+d x) \sinh (c+d x) (e+f x)^3}{3 a d}+\frac{b^2 \sinh (c+d x) (e+f x)^3}{a^3 d}+\frac{2 \sinh (c+d x) (e+f x)^3}{3 a d}-\frac{b (e+f x)^3}{4 a^2 d}-\frac{f \cosh ^3(c+d x) (e+f x)^2}{3 a d^2}-\frac{3 b^2 f \cosh (c+d x) (e+f x)^2}{a^3 d^2}-\frac{2 f \cosh (c+d x) (e+f x)^2}{a d^2}-\frac{3 b \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}-\frac{3 b \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}+\frac{3 b f \cosh (c+d x) \sinh (c+d x) (e+f x)^2}{4 a^2 d^2}-\frac{3 b f^2 \sinh ^2(c+d x) (e+f x)}{4 a^2 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac{6 b^2 f^2 \sinh (c+d x) (e+f x)}{a^3 d^3}+\frac{40 f^2 \sinh (c+d x) (e+f x)}{9 a d^3}+\frac{2 f^2 \cosh ^2(c+d x) \sinh (c+d x) (e+f x)}{9 a d^3}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{3 b f^3 x}{8 a^2 d^3}-\frac{6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac{40 f^3 \cosh (c+d x)}{9 a d^4}-\frac{6 b \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^4}-\frac{6 b \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^4}+\frac{3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4} \]

[Out]

(-3*b*f^3*x)/(8*a^2*d^3) - (b*(e + f*x)^3)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^4)/(4*a^4*f) - (40*f^3*Cosh[c
+ d*x])/(9*a*d^4) - (6*b^2*f^3*Cosh[c + d*x])/(a^3*d^4) - (2*f*(e + f*x)^2*Cosh[c + d*x])/(a*d^2) - (3*b^2*f*(
e + f*x)^2*Cosh[c + d*x])/(a^3*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*a*d^4) - (f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*
a*d^2) - (b*(a^2 + b^2)*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(
e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2,
-((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((a*E^(c +
d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^2) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sq
rt[a^2 + b^2]))])/(a^4*d^3) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]
))])/(a^4*d^3) - (6*b*(a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^4) - (6*b*(
a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^4) + (40*f^2*(e + f*x)*Sinh[c + d*
x])/(9*a*d^3) + (6*b^2*f^2*(e + f*x)*Sinh[c + d*x])/(a^3*d^3) + (2*(e + f*x)^3*Sinh[c + d*x])/(3*a*d) + (b^2*(
e + f*x)^3*Sinh[c + d*x])/(a^3*d) + (3*b*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d^4) + (3*b*f*(e + f*x)^2*Cos
h[c + d*x]*Sinh[c + d*x])/(4*a^2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*a*d^3) + ((e + f*x)
^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*a*d) - (3*b*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*a^2*d^3) - (b*(e + f*x)^3*S
inh[c + d*x]^2)/(2*a^2*d)

________________________________________________________________________________________

Rubi [A]  time = 1.18846, antiderivative size = 864, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 17, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.607, Rules used = {5594, 5579, 3311, 3296, 2638, 3310, 5565, 5446, 32, 2635, 8, 5561, 2190, 2531, 6609, 2282, 6589} \[ \frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{b \sinh ^2(c+d x) (e+f x)^3}{2 a^2 d}-\frac{b \left (a^2+b^2\right ) \log \left (\frac{e^{c+d x} a}{b-\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}-\frac{b \left (a^2+b^2\right ) \log \left (\frac{e^{c+d x} a}{b+\sqrt{a^2+b^2}}+1\right ) (e+f x)^3}{a^4 d}+\frac{\cosh ^2(c+d x) \sinh (c+d x) (e+f x)^3}{3 a d}+\frac{b^2 \sinh (c+d x) (e+f x)^3}{a^3 d}+\frac{2 \sinh (c+d x) (e+f x)^3}{3 a d}-\frac{b (e+f x)^3}{4 a^2 d}-\frac{f \cosh ^3(c+d x) (e+f x)^2}{3 a d^2}-\frac{3 b^2 f \cosh (c+d x) (e+f x)^2}{a^3 d^2}-\frac{2 f \cosh (c+d x) (e+f x)^2}{a d^2}-\frac{3 b \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}-\frac{3 b \left (a^2+b^2\right ) f \text{PolyLog}\left (2,-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) (e+f x)^2}{a^4 d^2}+\frac{3 b f \cosh (c+d x) \sinh (c+d x) (e+f x)^2}{4 a^2 d^2}-\frac{3 b f^2 \sinh ^2(c+d x) (e+f x)}{4 a^2 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 \text{PolyLog}\left (3,-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) (e+f x)}{a^4 d^3}+\frac{6 b^2 f^2 \sinh (c+d x) (e+f x)}{a^3 d^3}+\frac{40 f^2 \sinh (c+d x) (e+f x)}{9 a d^3}+\frac{2 f^2 \cosh ^2(c+d x) \sinh (c+d x) (e+f x)}{9 a d^3}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{3 b f^3 x}{8 a^2 d^3}-\frac{6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac{40 f^3 \cosh (c+d x)}{9 a d^4}-\frac{6 b \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^4}-\frac{6 b \left (a^2+b^2\right ) f^3 \text{PolyLog}\left (4,-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^4}+\frac{3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*f^3*x)/(8*a^2*d^3) - (b*(e + f*x)^3)/(4*a^2*d) + (b*(a^2 + b^2)*(e + f*x)^4)/(4*a^4*f) - (40*f^3*Cosh[c
+ d*x])/(9*a*d^4) - (6*b^2*f^3*Cosh[c + d*x])/(a^3*d^4) - (2*f*(e + f*x)^2*Cosh[c + d*x])/(a*d^2) - (3*b^2*f*(
e + f*x)^2*Cosh[c + d*x])/(a^3*d^2) - (2*f^3*Cosh[c + d*x]^3)/(27*a*d^4) - (f*(e + f*x)^2*Cosh[c + d*x]^3)/(3*
a*d^2) - (b*(a^2 + b^2)*(e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b - Sqrt[a^2 + b^2])])/(a^4*d) - (b*(a^2 + b^2)*(
e + f*x)^3*Log[1 + (a*E^(c + d*x))/(b + Sqrt[a^2 + b^2])])/(a^4*d) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2,
-((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^2) - (3*b*(a^2 + b^2)*f*(e + f*x)^2*PolyLog[2, -((a*E^(c +
d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^2) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b - Sq
rt[a^2 + b^2]))])/(a^4*d^3) + (6*b*(a^2 + b^2)*f^2*(e + f*x)*PolyLog[3, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]
))])/(a^4*d^3) - (6*b*(a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b - Sqrt[a^2 + b^2]))])/(a^4*d^4) - (6*b*(
a^2 + b^2)*f^3*PolyLog[4, -((a*E^(c + d*x))/(b + Sqrt[a^2 + b^2]))])/(a^4*d^4) + (40*f^2*(e + f*x)*Sinh[c + d*
x])/(9*a*d^3) + (6*b^2*f^2*(e + f*x)*Sinh[c + d*x])/(a^3*d^3) + (2*(e + f*x)^3*Sinh[c + d*x])/(3*a*d) + (b^2*(
e + f*x)^3*Sinh[c + d*x])/(a^3*d) + (3*b*f^3*Cosh[c + d*x]*Sinh[c + d*x])/(8*a^2*d^4) + (3*b*f*(e + f*x)^2*Cos
h[c + d*x]*Sinh[c + d*x])/(4*a^2*d^2) + (2*f^2*(e + f*x)*Cosh[c + d*x]^2*Sinh[c + d*x])/(9*a*d^3) + ((e + f*x)
^3*Cosh[c + d*x]^2*Sinh[c + d*x])/(3*a*d) - (3*b*f^2*(e + f*x)*Sinh[c + d*x]^2)/(4*a^2*d^3) - (b*(e + f*x)^3*S
inh[c + d*x]^2)/(2*a^2*d)

Rule 5594

Int[(((e_.) + (f_.)*(x_))^(m_.)*(F_)[(c_.) + (d_.)*(x_)]^(n_.))/(Csch[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Sym
bol] :> Int[((e + f*x)^m*Sinh[c + d*x]*F[c + d*x]^n)/(b + a*Sinh[c + d*x]), x] /; FreeQ[{a, b, c, d, e, f}, x]
&& HyperbolicQ[F] && IntegersQ[m, n]

Rule 5579

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^p*Sinh[c + d*x]^(n - 1), x], x]
- Dist[a/b, Int[((e + f*x)^m*Cosh[c + d*x]^p*Sinh[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 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 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 3310

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

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 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 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 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]

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 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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^3 \cosh ^3(c+d x)}{a+b \text{csch}(c+d x)} \, dx &=\int \frac{(e+f x)^3 \cosh ^3(c+d x) \sinh (c+d x)}{b+a \sinh (c+d x)} \, dx\\ &=\frac{\int (e+f x)^3 \cosh ^3(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^3 \cosh ^3(c+d x)}{b+a \sinh (c+d x)} \, dx}{a}\\ &=-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}+\frac{2 \int (e+f x)^3 \cosh (c+d x) \, dx}{3 a}-\frac{b \int (e+f x)^3 \cosh (c+d x) \sinh (c+d x) \, dx}{a^2}+\frac{b^2 \int (e+f x)^3 \cosh (c+d x) \, dx}{a^3}-\frac{\left (b \left (a^2+b^2\right )\right ) \int \frac{(e+f x)^3 \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx}{a^3}+\frac{\left (2 f^2\right ) \int (e+f x) \cosh ^3(c+d x) \, dx}{3 a d^2}\\ &=\frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}+\frac{2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac{b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac{b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac{\left (b \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{b-\sqrt{a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac{\left (b \left (a^2+b^2\right )\right ) \int \frac{e^{c+d x} (e+f x)^3}{b+\sqrt{a^2+b^2}+a e^{c+d x}} \, dx}{a^3}-\frac{(2 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{a d}+\frac{(3 b f) \int (e+f x)^2 \sinh ^2(c+d x) \, dx}{2 a^2 d}-\frac{\left (3 b^2 f\right ) \int (e+f x)^2 \sinh (c+d x) \, dx}{a^3 d}+\frac{\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{9 a d^2}\\ &=\frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac{3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d}+\frac{4 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac{2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac{b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac{3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac{3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac{b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac{(3 b f) \int (e+f x)^2 \, dx}{4 a^2 d}+\frac{\left (3 b \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) \, dx}{a^4 d}+\frac{\left (3 b \left (a^2+b^2\right ) f\right ) \int (e+f x)^2 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) \, dx}{a^4 d}+\frac{\left (4 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a d^2}+\frac{\left (6 b^2 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{a^3 d^2}-\frac{\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{9 a d^3}+\frac{\left (3 b f^3\right ) \int \sinh ^2(c+d x) \, dx}{4 a^2 d^3}\\ &=-\frac{b (e+f x)^3}{4 a^2 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{4 f^3 \cosh (c+d x)}{9 a d^4}-\frac{2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac{3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^2}+\frac{40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac{6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac{2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac{b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac{3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac{3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac{3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac{b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}+\frac{\left (6 b \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) \, dx}{a^4 d^2}+\frac{\left (6 b \left (a^2+b^2\right ) f^2\right ) \int (e+f x) \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) \, dx}{a^4 d^2}-\frac{\left (4 f^3\right ) \int \sinh (c+d x) \, dx}{a d^3}-\frac{\left (3 b f^3\right ) \int 1 \, dx}{8 a^2 d^3}-\frac{\left (6 b^2 f^3\right ) \int \sinh (c+d x) \, dx}{a^3 d^3}\\ &=-\frac{3 b f^3 x}{8 a^2 d^3}-\frac{b (e+f x)^3}{4 a^2 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{40 f^3 \cosh (c+d x)}{9 a d^4}-\frac{6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac{2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac{3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^2}+\frac{6 b \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^3}+\frac{40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac{6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac{2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac{b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac{3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac{3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac{3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac{b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac{\left (6 b \left (a^2+b^2\right ) f^3\right ) \int \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) \, dx}{a^4 d^3}-\frac{\left (6 b \left (a^2+b^2\right ) f^3\right ) \int \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) \, dx}{a^4 d^3}\\ &=-\frac{3 b f^3 x}{8 a^2 d^3}-\frac{b (e+f x)^3}{4 a^2 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{40 f^3 \cosh (c+d x)}{9 a d^4}-\frac{6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac{2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac{3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^2}+\frac{6 b \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^3}+\frac{40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac{6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac{2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac{b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac{3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac{3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac{3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac{b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}-\frac{\left (6 b \left (a^2+b^2\right ) f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{a x}{-b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^4}-\frac{\left (6 b \left (a^2+b^2\right ) f^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-\frac{a x}{b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^4 d^4}\\ &=-\frac{3 b f^3 x}{8 a^2 d^3}-\frac{b (e+f x)^3}{4 a^2 d}+\frac{b \left (a^2+b^2\right ) (e+f x)^4}{4 a^4 f}-\frac{40 f^3 \cosh (c+d x)}{9 a d^4}-\frac{6 b^2 f^3 \cosh (c+d x)}{a^3 d^4}-\frac{2 f (e+f x)^2 \cosh (c+d x)}{a d^2}-\frac{3 b^2 f (e+f x)^2 \cosh (c+d x)}{a^3 d^2}-\frac{2 f^3 \cosh ^3(c+d x)}{27 a d^4}-\frac{f (e+f x)^2 \cosh ^3(c+d x)}{3 a d^2}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{b \left (a^2+b^2\right ) (e+f x)^3 \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^2}-\frac{3 b \left (a^2+b^2\right ) f (e+f x)^2 \text{Li}_2\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^2}+\frac{6 b \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^3}+\frac{6 b \left (a^2+b^2\right ) f^2 (e+f x) \text{Li}_3\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^3}-\frac{6 b \left (a^2+b^2\right ) f^3 \text{Li}_4\left (-\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right )}{a^4 d^4}-\frac{6 b \left (a^2+b^2\right ) f^3 \text{Li}_4\left (-\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right )}{a^4 d^4}+\frac{40 f^2 (e+f x) \sinh (c+d x)}{9 a d^3}+\frac{6 b^2 f^2 (e+f x) \sinh (c+d x)}{a^3 d^3}+\frac{2 (e+f x)^3 \sinh (c+d x)}{3 a d}+\frac{b^2 (e+f x)^3 \sinh (c+d x)}{a^3 d}+\frac{3 b f^3 \cosh (c+d x) \sinh (c+d x)}{8 a^2 d^4}+\frac{3 b f (e+f x)^2 \cosh (c+d x) \sinh (c+d x)}{4 a^2 d^2}+\frac{2 f^2 (e+f x) \cosh ^2(c+d x) \sinh (c+d x)}{9 a d^3}+\frac{(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{3 a d}-\frac{3 b f^2 (e+f x) \sinh ^2(c+d x)}{4 a^2 d^3}-\frac{b (e+f x)^3 \sinh ^2(c+d x)}{2 a^2 d}\\ \end{align*}

Mathematica [C]  time = 40.1403, size = 7881, normalized size = 9.12 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \cosh \left ( dx+c \right ) \right ) ^{3}}{a+b{\rm csch} \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)^3/(a+b*csch(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

-1/24*e^3*((3*a*b*e^(-d*x - c) - a^2 - 3*(3*a^2 + 4*b^2)*e^(-2*d*x - 2*c))*e^(3*d*x + 3*c)/(a^3*d) + 24*(a^2*b
+ b^3)*(d*x + c)/(a^4*d) + (3*a*b*e^(-2*d*x - 2*c) + a^2*e^(-3*d*x - 3*c) + 3*(3*a^2 + 4*b^2)*e^(-d*x - c))/(
a^3*d) + 24*(a^2*b + b^3)*log(-2*b*e^(-d*x - c) + a*e^(-2*d*x - 2*c) - a)/(a^4*d)) - 1/864*(216*(a^2*b*d^4*f^3
*e^(3*c) + b^3*d^4*f^3*e^(3*c))*x^4 + 864*(a^2*b*d^4*e*f^2*e^(3*c) + b^3*d^4*e*f^2*e^(3*c))*x^3 + 1296*(a^2*b*
d^4*e^2*f*e^(3*c) + b^3*d^4*e^2*f*e^(3*c))*x^2 - 4*(9*a^3*d^3*f^3*x^3*e^(6*c) + 9*(3*d^3*e*f^2 - d^2*f^3)*a^3*
x^2*e^(6*c) + 3*(9*d^3*e^2*f - 6*d^2*e*f^2 + 2*d*f^3)*a^3*x*e^(6*c) - (9*d^2*e^2*f - 6*d*e*f^2 + 2*f^3)*a^3*e^
(6*c))*e^(3*d*x) + 27*(4*a^2*b*d^3*f^3*x^3*e^(5*c) + 6*(2*d^3*e*f^2 - d^2*f^3)*a^2*b*x^2*e^(5*c) + 6*(2*d^3*e^
2*f - 2*d^2*e*f^2 + d*f^3)*a^2*b*x*e^(5*c) - 3*(2*d^2*e^2*f - 2*d*e*f^2 + f^3)*a^2*b*e^(5*c))*e^(2*d*x) + 108*
(9*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a^3*e^(4*c) + 12*(d^2*e^2*f - 2*d*e*f^2 + 2*f^3)*a*b^2*e^(4*c) - (3*a^3*d^3
*f^3*e^(4*c) + 4*a*b^2*d^3*f^3*e^(4*c))*x^3 - 3*(3*(d^3*e*f^2 - d^2*f^3)*a^3*e^(4*c) + 4*(d^3*e*f^2 - d^2*f^3)
*a*b^2*e^(4*c))*x^2 - 3*(3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*a^3*e^(4*c) + 4*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*
f^3)*a*b^2*e^(4*c))*x)*e^(d*x) + 108*(9*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*a^3*e^(2*c) + 12*(d^2*e^2*f + 2*d*e*f^
2 + 2*f^3)*a*b^2*e^(2*c) + (3*a^3*d^3*f^3*e^(2*c) + 4*a*b^2*d^3*f^3*e^(2*c))*x^3 + 3*(3*(d^3*e*f^2 + d^2*f^3)*
a^3*e^(2*c) + 4*(d^3*e*f^2 + d^2*f^3)*a*b^2*e^(2*c))*x^2 + 3*(3*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a^3*e^(2*c
) + 4*(d^3*e^2*f + 2*d^2*e*f^2 + 2*d*f^3)*a*b^2*e^(2*c))*x)*e^(-d*x) + 27*(4*a^2*b*d^3*f^3*x^3*e^c + 6*(2*d^3*
e*f^2 + d^2*f^3)*a^2*b*x^2*e^c + 6*(2*d^3*e^2*f + 2*d^2*e*f^2 + d*f^3)*a^2*b*x*e^c + 3*(2*d^2*e^2*f + 2*d*e*f^
2 + f^3)*a^2*b*e^c)*e^(-2*d*x) + 4*(9*a^3*d^3*f^3*x^3 + 9*(3*d^3*e*f^2 + d^2*f^3)*a^3*x^2 + 3*(9*d^3*e^2*f + 6
*d^2*e*f^2 + 2*d*f^3)*a^3*x + (9*d^2*e^2*f + 6*d*e*f^2 + 2*f^3)*a^3)*e^(-3*d*x))*e^(-3*c)/(a^4*d^4) + integrat
e(-2*((a^3*b*f^3 + a*b^3*f^3)*x^3 + 3*(a^3*b*e*f^2 + a*b^3*e*f^2)*x^2 + 3*(a^3*b*e^2*f + a*b^3*e^2*f)*x - ((a^
2*b^2*f^3*e^c + b^4*f^3*e^c)*x^3 + 3*(a^2*b^2*e*f^2*e^c + b^4*e*f^2*e^c)*x^2 + 3*(a^2*b^2*e^2*f*e^c + b^4*e^2*
f*e^c)*x)*e^(d*x))/(a^5*e^(2*d*x + 2*c) + 2*a^4*b*e^(d*x + c) - a^5), x)

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Fricas [C]  time = 2.72124, size = 17256, normalized size = 19.97 \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)^3/(a+b*csch(d*x+c)),x, algorithm="fricas")

[Out]

-1/864*(36*a^3*d^3*f^3*x^3 + 36*a^3*d^3*e^3 + 36*a^3*d^2*e^2*f + 24*a^3*d*e*f^2 - 4*(9*a^3*d^3*f^3*x^3 + 9*a^3
*d^3*e^3 - 9*a^3*d^2*e^2*f + 6*a^3*d*e*f^2 - 2*a^3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*
e^2*f - 6*a^3*d^2*e*f^2 + 2*a^3*d*f^3)*x)*cosh(d*x + c)^6 - 4*(9*a^3*d^3*f^3*x^3 + 9*a^3*d^3*e^3 - 9*a^3*d^2*e
^2*f + 6*a^3*d*e*f^2 - 2*a^3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*e^2*f - 6*a^3*d^2*e*f^
2 + 2*a^3*d*f^3)*x)*sinh(d*x + c)^6 + 8*a^3*f^3 + 27*(4*a^2*b*d^3*f^3*x^3 + 4*a^2*b*d^3*e^3 - 6*a^2*b*d^2*e^2*
f + 6*a^2*b*d*e*f^2 - 3*a^2*b*f^3 + 6*(2*a^2*b*d^3*e*f^2 - a^2*b*d^2*f^3)*x^2 + 6*(2*a^2*b*d^3*e^2*f - 2*a^2*b
*d^2*e*f^2 + a^2*b*d*f^3)*x)*cosh(d*x + c)^5 + 3*(36*a^2*b*d^3*f^3*x^3 + 36*a^2*b*d^3*e^3 - 54*a^2*b*d^2*e^2*f
+ 54*a^2*b*d*e*f^2 - 27*a^2*b*f^3 + 54*(2*a^2*b*d^3*e*f^2 - a^2*b*d^2*f^3)*x^2 + 54*(2*a^2*b*d^3*e^2*f - 2*a^
2*b*d^2*e*f^2 + a^2*b*d*f^3)*x - 8*(9*a^3*d^3*f^3*x^3 + 9*a^3*d^3*e^3 - 9*a^3*d^2*e^2*f + 6*a^3*d*e*f^2 - 2*a^
3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*e^2*f - 6*a^3*d^2*e*f^2 + 2*a^3*d*f^3)*x)*cosh(d*
x + c))*sinh(d*x + c)^5 - 108*((3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + (3*a^3 + 4*a*b^2)*d^3*e^3 - 3*(3*a^3 + 4*a*b^2)
*d^2*e^2*f + 6*(3*a^3 + 4*a*b^2)*d*e*f^2 - 6*(3*a^3 + 4*a*b^2)*f^3 + 3*((3*a^3 + 4*a*b^2)*d^3*e*f^2 - (3*a^3 +
4*a*b^2)*d^2*f^3)*x^2 + 3*((3*a^3 + 4*a*b^2)*d^3*e^2*f - 2*(3*a^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*
d*f^3)*x)*cosh(d*x + c)^4 - 3*(36*(3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + 36*(3*a^3 + 4*a*b^2)*d^3*e^3 - 108*(3*a^3 +
4*a*b^2)*d^2*e^2*f + 216*(3*a^3 + 4*a*b^2)*d*e*f^2 - 216*(3*a^3 + 4*a*b^2)*f^3 + 108*((3*a^3 + 4*a*b^2)*d^3*e*
f^2 - (3*a^3 + 4*a*b^2)*d^2*f^3)*x^2 + 20*(9*a^3*d^3*f^3*x^3 + 9*a^3*d^3*e^3 - 9*a^3*d^2*e^2*f + 6*a^3*d*e*f^2
- 2*a^3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*e^2*f - 6*a^3*d^2*e*f^2 + 2*a^3*d*f^3)*x)*
cosh(d*x + c)^2 + 108*((3*a^3 + 4*a*b^2)*d^3*e^2*f - 2*(3*a^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3
)*x - 45*(4*a^2*b*d^3*f^3*x^3 + 4*a^2*b*d^3*e^3 - 6*a^2*b*d^2*e^2*f + 6*a^2*b*d*e*f^2 - 3*a^2*b*f^3 + 6*(2*a^2
*b*d^3*e*f^2 - a^2*b*d^2*f^3)*x^2 + 6*(2*a^2*b*d^3*e^2*f - 2*a^2*b*d^2*e*f^2 + a^2*b*d*f^3)*x)*cosh(d*x + c))*
sinh(d*x + c)^4 - 216*((a^2*b + b^3)*d^4*f^3*x^4 + 4*(a^2*b + b^3)*d^4*e*f^2*x^3 + 6*(a^2*b + b^3)*d^4*e^2*f*x
^2 + 4*(a^2*b + b^3)*d^4*e^3*x + 8*(a^2*b + b^3)*c*d^3*e^3 - 12*(a^2*b + b^3)*c^2*d^2*e^2*f + 8*(a^2*b + b^3)*
c^3*d*e*f^2 - 2*(a^2*b + b^3)*c^4*f^3)*cosh(d*x + c)^3 - 2*(108*(a^2*b + b^3)*d^4*f^3*x^4 + 432*(a^2*b + b^3)*
d^4*e*f^2*x^3 + 648*(a^2*b + b^3)*d^4*e^2*f*x^2 + 432*(a^2*b + b^3)*d^4*e^3*x + 864*(a^2*b + b^3)*c*d^3*e^3 -
1296*(a^2*b + b^3)*c^2*d^2*e^2*f + 864*(a^2*b + b^3)*c^3*d*e*f^2 - 216*(a^2*b + b^3)*c^4*f^3 + 40*(9*a^3*d^3*f
^3*x^3 + 9*a^3*d^3*e^3 - 9*a^3*d^2*e^2*f + 6*a^3*d*e*f^2 - 2*a^3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 +
3*(9*a^3*d^3*e^2*f - 6*a^3*d^2*e*f^2 + 2*a^3*d*f^3)*x)*cosh(d*x + c)^3 - 135*(4*a^2*b*d^3*f^3*x^3 + 4*a^2*b*d
^3*e^3 - 6*a^2*b*d^2*e^2*f + 6*a^2*b*d*e*f^2 - 3*a^2*b*f^3 + 6*(2*a^2*b*d^3*e*f^2 - a^2*b*d^2*f^3)*x^2 + 6*(2*
a^2*b*d^3*e^2*f - 2*a^2*b*d^2*e*f^2 + a^2*b*d*f^3)*x)*cosh(d*x + c)^2 + 216*((3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + (
3*a^3 + 4*a*b^2)*d^3*e^3 - 3*(3*a^3 + 4*a*b^2)*d^2*e^2*f + 6*(3*a^3 + 4*a*b^2)*d*e*f^2 - 6*(3*a^3 + 4*a*b^2)*f
^3 + 3*((3*a^3 + 4*a*b^2)*d^3*e*f^2 - (3*a^3 + 4*a*b^2)*d^2*f^3)*x^2 + 3*((3*a^3 + 4*a*b^2)*d^3*e^2*f - 2*(3*a
^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c)^3 + 36*(3*a^3*d^3*e*f^2 +
a^3*d^2*f^3)*x^2 + 108*((3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + (3*a^3 + 4*a*b^2)*d^3*e^3 + 3*(3*a^3 + 4*a*b^2)*d^2*e
^2*f + 6*(3*a^3 + 4*a*b^2)*d*e*f^2 + 6*(3*a^3 + 4*a*b^2)*f^3 + 3*((3*a^3 + 4*a*b^2)*d^3*e*f^2 + (3*a^3 + 4*a*b
^2)*d^2*f^3)*x^2 + 3*((3*a^3 + 4*a*b^2)*d^3*e^2*f + 2*(3*a^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3)
*x)*cosh(d*x + c)^2 + 6*(18*(3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + 18*(3*a^3 + 4*a*b^2)*d^3*e^3 + 54*(3*a^3 + 4*a*b^2
)*d^2*e^2*f + 108*(3*a^3 + 4*a*b^2)*d*e*f^2 - 10*(9*a^3*d^3*f^3*x^3 + 9*a^3*d^3*e^3 - 9*a^3*d^2*e^2*f + 6*a^3*
d*e*f^2 - 2*a^3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*e^2*f - 6*a^3*d^2*e*f^2 + 2*a^3*d*f
^3)*x)*cosh(d*x + c)^4 + 108*(3*a^3 + 4*a*b^2)*f^3 + 45*(4*a^2*b*d^3*f^3*x^3 + 4*a^2*b*d^3*e^3 - 6*a^2*b*d^2*e
^2*f + 6*a^2*b*d*e*f^2 - 3*a^2*b*f^3 + 6*(2*a^2*b*d^3*e*f^2 - a^2*b*d^2*f^3)*x^2 + 6*(2*a^2*b*d^3*e^2*f - 2*a^
2*b*d^2*e*f^2 + a^2*b*d*f^3)*x)*cosh(d*x + c)^3 + 54*((3*a^3 + 4*a*b^2)*d^3*e*f^2 + (3*a^3 + 4*a*b^2)*d^2*f^3)
*x^2 - 108*((3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + (3*a^3 + 4*a*b^2)*d^3*e^3 - 3*(3*a^3 + 4*a*b^2)*d^2*e^2*f + 6*(3*a
^3 + 4*a*b^2)*d*e*f^2 - 6*(3*a^3 + 4*a*b^2)*f^3 + 3*((3*a^3 + 4*a*b^2)*d^3*e*f^2 - (3*a^3 + 4*a*b^2)*d^2*f^3)*
x^2 + 3*((3*a^3 + 4*a*b^2)*d^3*e^2*f - 2*(3*a^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3)*x)*cosh(d*x
+ c)^2 + 54*((3*a^3 + 4*a*b^2)*d^3*e^2*f + 2*(3*a^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3)*x - 108*
((a^2*b + b^3)*d^4*f^3*x^4 + 4*(a^2*b + b^3)*d^4*e*f^2*x^3 + 6*(a^2*b + b^3)*d^4*e^2*f*x^2 + 4*(a^2*b + b^3)*d
^4*e^3*x + 8*(a^2*b + b^3)*c*d^3*e^3 - 12*(a^2*b + b^3)*c^2*d^2*e^2*f + 8*(a^2*b + b^3)*c^3*d*e*f^2 - 2*(a^2*b
+ b^3)*c^4*f^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 12*(9*a^3*d^3*e^2*f + 6*a^3*d^2*e*f^2 + 2*a^3*d*f^3)*x + 27*
(4*a^2*b*d^3*f^3*x^3 + 4*a^2*b*d^3*e^3 + 6*a^2*b*d^2*e^2*f + 6*a^2*b*d*e*f^2 + 3*a^2*b*f^3 + 6*(2*a^2*b*d^3*e*
f^2 + a^2*b*d^2*f^3)*x^2 + 6*(2*a^2*b*d^3*e^2*f + 2*a^2*b*d^2*e*f^2 + a^2*b*d*f^3)*x)*cosh(d*x + c) + 2592*(((
a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*cosh(d*x + c)^3 + 3*((a^2*b
+ b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*cosh(d*x + c)^2*sinh(d*x + c) + 3*
((a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c
)^2 + ((a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*sinh(d*x + c)^3)*dil
og((b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a + 1)
+ 2592*(((a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*cosh(d*x + c)^3 +
3*((a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*cosh(d*x + c)^2*sinh(d*x
+ c) + 3*((a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*cosh(d*x + c)*si
nh(d*x + c)^2 + ((a^2*b + b^3)*d^2*f^3*x^2 + 2*(a^2*b + b^3)*d^2*e*f^2*x + (a^2*b + b^3)*d^2*e^2*f)*sinh(d*x +
c)^3)*dilog((b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) -
a)/a + 1) + 864*(((a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*c*d^2*e^2*f + 3*(a^2*b + b^3)*c^2*d*e*f^2 - (a^2*b +
b^3)*c^3*f^3)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*c*d^2*e^2*f + 3*(a^2*b + b^3)*c^2*
d*e*f^2 - (a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*c*
d^2*e^2*f + 3*(a^2*b + b^3)*c^2*d*e*f^2 - (a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3
)*d^3*e^3 - 3*(a^2*b + b^3)*c*d^2*e^2*f + 3*(a^2*b + b^3)*c^2*d*e*f^2 - (a^2*b + b^3)*c^3*f^3)*sinh(d*x + c)^3
)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 864*(((a^2*b + b^3)*d^3*e^3 -
3*(a^2*b + b^3)*c*d^2*e^2*f + 3*(a^2*b + b^3)*c^2*d*e*f^2 - (a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)^3 + 3*((a^2*
b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*c*d^2*e^2*f + 3*(a^2*b + b^3)*c^2*d*e*f^2 - (a^2*b + b^3)*c^3*f^3)*cosh(d*x
+ c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*c*d^2*e^2*f + 3*(a^2*b + b^3)*c^2*d*e*f^2 -
(a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*c*d^2*e^2*f +
3*(a^2*b + b^3)*c^2*d*e*f^2 - (a^2*b + b^3)*c^3*f^3)*sinh(d*x + c)^3)*log(2*a*cosh(d*x + c) + 2*a*sinh(d*x +
c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) + 864*(((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a
^2*b + b^3)*d^3*e^2*f*x + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*c
osh(d*x + c)^3 + 3*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x +
3*(a^2*b + b^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)^2*sinh(d*x +
c) + 3*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + 3*(a^2*b + b
^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b
+ b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + 3*(a^2*b + b^3)*c*d^2*e^2*
f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*sinh(d*x + c)^3)*log(-(b*cosh(d*x + c) + b*sinh(d*x +
c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) + 864*(((a^2*b + b^3)*d^3*f^3*x^3 + 3*
(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*
d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^
2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^
3*f^3)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b
+ b^3)*d^3*e^2*f*x + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*cosh(
d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^
2*f*x + 3*(a^2*b + b^3)*c*d^2*e^2*f - 3*(a^2*b + b^3)*c^2*d*e*f^2 + (a^2*b + b^3)*c^3*f^3)*sinh(d*x + c)^3)*lo
g(-(b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2) - a)/a) + 51
84*((a^2*b + b^3)*f^3*cosh(d*x + c)^3 + 3*(a^2*b + b^3)*f^3*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b + b^3)*f^
3*cosh(d*x + c)*sinh(d*x + c)^2 + (a^2*b + b^3)*f^3*sinh(d*x + c)^3)*polylog(4, (b*cosh(d*x + c) + b*sinh(d*x
+ c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) + 5184*((a^2*b + b^3)*f^3*cosh(d*x + c)^3
+ 3*(a^2*b + b^3)*f^3*cosh(d*x + c)^2*sinh(d*x + c) + 3*(a^2*b + b^3)*f^3*cosh(d*x + c)*sinh(d*x + c)^2 + (a^
2*b + b^3)*f^3*sinh(d*x + c)^3)*polylog(4, (b*cosh(d*x + c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x
+ c))*sqrt((a^2 + b^2)/a^2))/a) - 5184*(((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)^3 + 3*((
a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*f^3*x + (a^2*
b + b^3)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2)*sinh(d*x + c
)^3)*polylog(3, (b*cosh(d*x + c) + b*sinh(d*x + c) + (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2)
)/a) - 5184*(((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)^3 + 3*((a^2*b + b^3)*d*f^3*x + (a^2
*b + b^3)*d*e*f^2)*cosh(d*x + c)^2*sinh(d*x + c) + 3*((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2)*cosh(d*x
+ c)*sinh(d*x + c)^2 + ((a^2*b + b^3)*d*f^3*x + (a^2*b + b^3)*d*e*f^2)*sinh(d*x + c)^3)*polylog(3, (b*cosh(d*x
+ c) + b*sinh(d*x + c) - (a*cosh(d*x + c) + a*sinh(d*x + c))*sqrt((a^2 + b^2)/a^2))/a) + 3*(36*a^2*b*d^3*f^3*
x^3 + 36*a^2*b*d^3*e^3 + 54*a^2*b*d^2*e^2*f + 54*a^2*b*d*e*f^2 + 27*a^2*b*f^3 - 8*(9*a^3*d^3*f^3*x^3 + 9*a^3*d
^3*e^3 - 9*a^3*d^2*e^2*f + 6*a^3*d*e*f^2 - 2*a^3*f^3 + 9*(3*a^3*d^3*e*f^2 - a^3*d^2*f^3)*x^2 + 3*(9*a^3*d^3*e^
2*f - 6*a^3*d^2*e*f^2 + 2*a^3*d*f^3)*x)*cosh(d*x + c)^5 + 45*(4*a^2*b*d^3*f^3*x^3 + 4*a^2*b*d^3*e^3 - 6*a^2*b*
d^2*e^2*f + 6*a^2*b*d*e*f^2 - 3*a^2*b*f^3 + 6*(2*a^2*b*d^3*e*f^2 - a^2*b*d^2*f^3)*x^2 + 6*(2*a^2*b*d^3*e^2*f -
2*a^2*b*d^2*e*f^2 + a^2*b*d*f^3)*x)*cosh(d*x + c)^4 - 144*((3*a^3 + 4*a*b^2)*d^3*f^3*x^3 + (3*a^3 + 4*a*b^2)*
d^3*e^3 - 3*(3*a^3 + 4*a*b^2)*d^2*e^2*f + 6*(3*a^3 + 4*a*b^2)*d*e*f^2 - 6*(3*a^3 + 4*a*b^2)*f^3 + 3*((3*a^3 +
4*a*b^2)*d^3*e*f^2 - (3*a^3 + 4*a*b^2)*d^2*f^3)*x^2 + 3*((3*a^3 + 4*a*b^2)*d^3*e^2*f - 2*(3*a^3 + 4*a*b^2)*d^2
*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3)*x)*cosh(d*x + c)^3 + 54*(2*a^2*b*d^3*e*f^2 + a^2*b*d^2*f^3)*x^2 - 216*((a^
2*b + b^3)*d^4*f^3*x^4 + 4*(a^2*b + b^3)*d^4*e*f^2*x^3 + 6*(a^2*b + b^3)*d^4*e^2*f*x^2 + 4*(a^2*b + b^3)*d^4*e
^3*x + 8*(a^2*b + b^3)*c*d^3*e^3 - 12*(a^2*b + b^3)*c^2*d^2*e^2*f + 8*(a^2*b + b^3)*c^3*d*e*f^2 - 2*(a^2*b + b
^3)*c^4*f^3)*cosh(d*x + c)^2 + 54*(2*a^2*b*d^3*e^2*f + 2*a^2*b*d^2*e*f^2 + a^2*b*d*f^3)*x + 72*((3*a^3 + 4*a*b
^2)*d^3*f^3*x^3 + (3*a^3 + 4*a*b^2)*d^3*e^3 + 3*(3*a^3 + 4*a*b^2)*d^2*e^2*f + 6*(3*a^3 + 4*a*b^2)*d*e*f^2 + 6*
(3*a^3 + 4*a*b^2)*f^3 + 3*((3*a^3 + 4*a*b^2)*d^3*e*f^2 + (3*a^3 + 4*a*b^2)*d^2*f^3)*x^2 + 3*((3*a^3 + 4*a*b^2)
*d^3*e^2*f + 2*(3*a^3 + 4*a*b^2)*d^2*e*f^2 + 2*(3*a^3 + 4*a*b^2)*d*f^3)*x)*cosh(d*x + c))*sinh(d*x + c))/(a^4*
d^4*cosh(d*x + c)^3 + 3*a^4*d^4*cosh(d*x + c)^2*sinh(d*x + c) + 3*a^4*d^4*cosh(d*x + c)*sinh(d*x + c)^2 + a^4*
d^4*sinh(d*x + c)^3)

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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)**3/(a+b*csch(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{3}}{b \operatorname{csch}\left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*cosh(d*x+c)^3/(a+b*csch(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^3*cosh(d*x + c)^3/(b*csch(d*x + c) + a), x)