3.440 \(\int \frac{(e+f x)^3 \text{csch}(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=1164 \[ \text{result too large to display} \]

[Out]

-((b*(e + f*x)^3)/((a^2 + b^2)*d)) - (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (6*b^2*f*(e + f*x)^2*ArcT
an[E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (b^3*(e + f*x)^3*Log[1 + (
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (3*b*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) -
(3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((6*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^
3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog
[2, I*E^(c + d*x)])/(a*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (3*f*(
e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) + (6*f^2*(e
+ f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - ((6*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*b^2*f^3*
PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) + ((6*I)*f^3*PolyLog[3, I*E^(c + d*x)])/(a*d^4) - ((6*I)*b^2
*f^3*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*b
^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (6*b^3*f^2*
(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (3*b*f^3*PolyLog[3
, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^4) - (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*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) +
(6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) + ((e + f*x)^3*Sec
h[c + d*x])/(a*d) - (b^2*(e + f*x)^3*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)^3*Tanh[c + d*x])/((a^2 +
b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 2.19887, antiderivative size = 1164, normalized size of antiderivative = 1., number of steps used = 53, number of rules used = 22, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647, Rules used = {5589, 2622, 321, 207, 5462, 6741, 12, 6742, 6273, 4182, 2531, 6609, 2282, 6589, 4180, 5573, 3322, 2264, 2190, 4184, 3718, 5451} \[ -\frac{(e+f x)^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}+\frac{(e+f x)^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^3}-\frac{6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac{6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^3}{a \left (a^2+b^2\right )^{3/2} d^4}+\frac{6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac{6 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}-\frac{6 i f^3 \text{PolyLog}\left (3,i e^{c+d x}\right ) b^2}{a \left (a^2+b^2\right ) d^4}-\frac{(e+f x)^3 \text{sech}(c+d x) b^2}{a \left (a^2+b^2\right ) d}-\frac{(e+f x)^3 b}{\left (a^2+b^2\right ) d}+\frac{3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^2}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac{3 f^3 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) b}{2 \left (a^2+b^2\right ) d^4}-\frac{(e+f x)^3 \tanh (c+d x) b}{\left (a^2+b^2\right ) d}-\frac{6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{2 (e+f x)^3 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,i 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{6 f^2 (e+f x) \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{6 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac{6 i f^3 \text{PolyLog}\left (3,i e^{c+d x}\right )}{a d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\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{(e+f x)^3 \text{sech}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(e + f*x)^3)/((a^2 + b^2)*d)) - (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(a*d^2) + (6*b^2*f*(e + f*x)^2*ArcT
an[E^(c + d*x)])/(a*(a^2 + b^2)*d^2) - (2*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a*d) - (b^3*(e + f*x)^3*Log[1 + (
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (b^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a +
Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^(3/2)*d) + (3*b*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)*d^2) -
(3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a*d^2) + ((6*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*d^
3) - ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog
[2, I*E^(c + d*x)])/(a*d^3) + ((6*I)*b^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^3) + (3*f*(
e + f*x)^2*PolyLog[2, E^(c + d*x)])/(a*d^2) - (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2]))])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])
)])/(a*(a^2 + b^2)^(3/2)*d^2) + (3*b*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)*d^3) + (6*f^2*(e
+ f*x)*PolyLog[3, -E^(c + d*x)])/(a*d^3) - ((6*I)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(a*d^4) + ((6*I)*b^2*f^3*
PolyLog[3, (-I)*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) + ((6*I)*f^3*PolyLog[3, I*E^(c + d*x)])/(a*d^4) - ((6*I)*b^2
*f^3*PolyLog[3, I*E^(c + d*x)])/(a*(a^2 + b^2)*d^4) - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a*d^3) + (6*b
^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (6*b^3*f^2*
(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^3) - (3*b*f^3*PolyLog[3
, -E^(2*(c + d*x))])/(2*(a^2 + b^2)*d^4) - (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*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) +
(6*b^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^(3/2)*d^4) + ((e + f*x)^3*Sec
h[c + d*x])/(a*d) - (b^2*(e + f*x)^3*Sech[c + d*x])/(a*(a^2 + b^2)*d) - (b*(e + f*x)^3*Tanh[c + d*x])/((a^2 +
b^2)*d)

Rule 5589

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c
, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 5462

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 6273

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 - u^2), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, 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 4180

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

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

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]

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Sech[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rubi steps

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

Mathematica [A]  time = 16.9409, size = 1467, normalized size = 1.26 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

4*(-(f*Csch[c + d*x]*(12*b*d^3*e^2*E^(2*c)*x - 12*b*d^3*e^2*(1 + E^(2*c))*x - 12*b*d^3*e*f*x^2 - 4*b*d^3*f^2*x
^3 + 12*a*d^2*e^2*(1 + E^(2*c))*ArcTan[E^(c + d*x)] + 6*b*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x
))]) + (12*I)*a*d*e*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E
^(c + d*x)] + PolyLog[2, I*E^(c + d*x)]) + 6*b*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - P
olyLog[2, -E^(2*(c + d*x))]) + (6*I)*a*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E
^(c + d*x)] - 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c +
d*x)] - 2*PolyLog[3, I*E^(c + d*x)]) + b*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6
*d*x*PolyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))]))*(a + b*Sinh[c + d*x]))/(8*(a^2 + b^2)*d^4
*(1 + E^(2*c))*(b + a*Csch[c + d*x])) + (b^3*Csch[c + d*x]*(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 + Sq
rt[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 + Sq
rt[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]))])*(a + b*Sinh[c + d*x]))/(4*a*(a^2 + b^2)^(3/2)*d^4*(b + a*Csch[c + d*x])) + (Csch[c
+ d*x]*(-2*(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] - Sinh[c + d*x]] + 2*f^2*PolyLog[4, -Cosh[c +
d*x] - Sinh[c + d*x]]))/d^3 + (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]]))/d^3)*(a + b*Si
nh[c + d*x]))/(4*a*d*(b + a*Csch[c + d*x])) + (Csch[c + d*x]*Sech[c]*Sech[c + d*x]*(a*e^3*Cosh[c] + 3*a*e^2*f*
x*Cosh[c] + 3*a*e*f^2*x^2*Cosh[c] + a*f^3*x^3*Cosh[c] - b*e^3*Sinh[d*x] - 3*b*e^2*f*x*Sinh[d*x] - 3*b*e*f^2*x^
2*Sinh[d*x] - b*f^3*x^3*Sinh[d*x])*(a + b*Sinh[c + d*x]))/(4*(a^2 + b^2)*d*(b + a*Csch[c + d*x])))

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Maple [F]  time = 1.674, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3}{\rm csch} \left (dx+c\right ) \left ({\rm sech} \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*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [C]  time = 5.446, size = 21639, normalized size = 18.59 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

1/2*(4*(a^3*b + a*b^3)*d^3*e^3 - 12*(a^3*b + a*b^3)*c*d^2*e^2*f + 12*(a^3*b + a*b^3)*c^2*d*e*f^2 - 4*(a^3*b +
a*b^3)*c^3*f^3 - 4*((a^3*b + a*b^3)*d^3*f^3*x^3 + 3*(a^3*b + a*b^3)*d^3*e*f^2*x^2 + 3*(a^3*b + a*b^3)*d^3*e^2*
f*x + 3*(a^3*b + a*b^3)*c*d^2*e^2*f - 3*(a^3*b + a*b^3)*c^2*d*e*f^2 + (a^3*b + a*b^3)*c^3*f^3)*cosh(d*x + c)^2
- 4*((a^3*b + a*b^3)*d^3*f^3*x^3 + 3*(a^3*b + a*b^3)*d^3*e*f^2*x^2 + 3*(a^3*b + a*b^3)*d^3*e^2*f*x + 3*(a^3*b
+ a*b^3)*c*d^2*e^2*f - 3*(a^3*b + a*b^3)*c^2*d*e*f^2 + (a^3*b + a*b^3)*c^3*f^3)*sinh(d*x + c)^2 - 6*(b^4*d^2*
f^3*x^2 + 2*b^4*d^2*e*f^2*x + b^4*d^2*e^2*f + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*e*f^2*x + b^4*d^2*e^2*f)*cosh(d*x +
c)^2 + 2*(b^4*d^2*f^3*x^2 + 2*b^4*d^2*e*f^2*x + b^4*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d^2*f^3*x^2
+ 2*b^4*d^2*e*f^2*x + b^4*d^2*e^2*f)*sinh(d*x + c)^2)*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) + 6*(b^4*d^2*f^3*x^2 + 2*b^4*d
^2*e*f^2*x + b^4*d^2*e^2*f + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*e*f^2*x + b^4*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(b^4*d^
2*f^3*x^2 + 2*b^4*d^2*e*f^2*x + b^4*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d^2*f^3*x^2 + 2*b^4*d^2*e*f^
2*x + b^4*d^2*e^2*f)*sinh(d*x + c)^2)*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) + 2*(b^4*d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^
2*d*e*f^2 - b^4*c^3*f^3 + (b^4*d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^2*d*e*f^2 - b^4*c^3*f^3)*cosh(d*x + c)^2
+ 2*(b^4*d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^2*d*e*f^2 - b^4*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d^3
*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^2*d*e*f^2 - b^4*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cos
h(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(b^4*d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4
*c^2*d*e*f^2 - b^4*c^3*f^3 + (b^4*d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^2*d*e*f^2 - b^4*c^3*f^3)*cosh(d*x + c)
^2 + 2*(b^4*d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^2*d*e*f^2 - b^4*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (b^4*
d^3*e^3 - 3*b^4*c*d^2*e^2*f + 3*b^4*c^2*d*e*f^2 - b^4*c^3*f^3)*sinh(d*x + c)^2)*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) - 2*(b^4*d^3*f^3*x^3 + 3*b^4*d^3*e*f^2*x^
2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4*c^2*d*e*f^2 + b^4*c^3*f^3 + (b^4*d^3*f^3*x^3 + 3*b^4*d^3*e*f
^2*x^2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4*c^2*d*e*f^2 + b^4*c^3*f^3)*cosh(d*x + c)^2 + 2*(b^4*d^3
*f^3*x^3 + 3*b^4*d^3*e*f^2*x^2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4*c^2*d*e*f^2 + b^4*c^3*f^3)*cosh
(d*x + c)*sinh(d*x + c) + (b^4*d^3*f^3*x^3 + 3*b^4*d^3*e*f^2*x^2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b
^4*c^2*d*e*f^2 + b^4*c^3*f^3)*sinh(d*x + c)^2)*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) + 2*(b^4*d^3*f^3*x^3 + 3*b^4*d^3*e*f^2*x^2
+ 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4*c^2*d*e*f^2 + b^4*c^3*f^3 + (b^4*d^3*f^3*x^3 + 3*b^4*d^3*e*f^2
*x^2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4*c^2*d*e*f^2 + b^4*c^3*f^3)*cosh(d*x + c)^2 + 2*(b^4*d^3*f
^3*x^3 + 3*b^4*d^3*e*f^2*x^2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4*c^2*d*e*f^2 + b^4*c^3*f^3)*cosh(d
*x + c)*sinh(d*x + c) + (b^4*d^3*f^3*x^3 + 3*b^4*d^3*e*f^2*x^2 + 3*b^4*d^3*e^2*f*x + 3*b^4*c*d^2*e^2*f - 3*b^4
*c^2*d*e*f^2 + b^4*c^3*f^3)*sinh(d*x + c)^2)*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) - 12*(b^4*f^3*cosh(d*x + c)^2 + 2*b^4*f^3*cos
h(d*x + c)*sinh(d*x + c) + b^4*f^3*sinh(d*x + c)^2 + b^4*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) + 12*(b^4*f^3*cosh(d*x + c
)^2 + 2*b^4*f^3*cosh(d*x + c)*sinh(d*x + c) + b^4*f^3*sinh(d*x + c)^2 + b^4*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) + 12*(b
^4*d*f^3*x + b^4*d*e*f^2 + (b^4*d*f^3*x + b^4*d*e*f^2)*cosh(d*x + c)^2 + 2*(b^4*d*f^3*x + b^4*d*e*f^2)*cosh(d*
x + c)*sinh(d*x + c) + (b^4*d*f^3*x + b^4*d*e*f^2)*sinh(d*x + c)^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) - 12*(b^4*d*f^3*x +
b^4*d*e*f^2 + (b^4*d*f^3*x + b^4*d*e*f^2)*cosh(d*x + c)^2 + 2*(b^4*d*f^3*x + b^4*d*e*f^2)*cosh(d*x + c)*sinh(d
*x + c) + (b^4*d*f^3*x + b^4*d*e*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*s
inh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 4*((a^4 + a^2*b^2)*d^3*f^3*x^3
+ 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + (a^4 + a^2*b^2)*d^3*e^3)*cosh(d*x + c) + 6
*((a^4 + 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^
2*f + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d
^2*e^2*f)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (
a^4 + 2*a^2*b^2 + b^4)*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4
+ 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2*f)*sinh(d*x + c)^2)*dilog(cosh(d*x + c) + sin
h(d*x + c)) + (-12*I*(a^4 + a^2*b^2)*d*f^3*x + 12*(a^3*b + a*b^3)*d*f^3*x - 12*I*(a^4 + a^2*b^2)*d*e*f^2 + 12*
(a^3*b + a*b^3)*d*e*f^2 + (-12*I*(a^4 + a^2*b^2)*d*f^3*x + 12*(a^3*b + a*b^3)*d*f^3*x - 12*I*(a^4 + a^2*b^2)*d
*e*f^2 + 12*(a^3*b + a*b^3)*d*e*f^2)*cosh(d*x + c)^2 + (-24*I*(a^4 + a^2*b^2)*d*f^3*x + 24*(a^3*b + a*b^3)*d*f
^3*x - 24*I*(a^4 + a^2*b^2)*d*e*f^2 + 24*(a^3*b + a*b^3)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (-12*I*(a^4 +
a^2*b^2)*d*f^3*x + 12*(a^3*b + a*b^3)*d*f^3*x - 12*I*(a^4 + a^2*b^2)*d*e*f^2 + 12*(a^3*b + a*b^3)*d*e*f^2)*sin
h(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + (12*I*(a^4 + a^2*b^2)*d*f^3*x + 12*(a^3*b + a*b^3)*d*
f^3*x + 12*I*(a^4 + a^2*b^2)*d*e*f^2 + 12*(a^3*b + a*b^3)*d*e*f^2 + (12*I*(a^4 + a^2*b^2)*d*f^3*x + 12*(a^3*b
+ a*b^3)*d*f^3*x + 12*I*(a^4 + a^2*b^2)*d*e*f^2 + 12*(a^3*b + a*b^3)*d*e*f^2)*cosh(d*x + c)^2 + (24*I*(a^4 + a
^2*b^2)*d*f^3*x + 24*(a^3*b + a*b^3)*d*f^3*x + 24*I*(a^4 + a^2*b^2)*d*e*f^2 + 24*(a^3*b + a*b^3)*d*e*f^2)*cosh
(d*x + c)*sinh(d*x + c) + (12*I*(a^4 + a^2*b^2)*d*f^3*x + 12*(a^3*b + a*b^3)*d*f^3*x + 12*I*(a^4 + a^2*b^2)*d*
e*f^2 + 12*(a^3*b + a*b^3)*d*e*f^2)*sinh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) - 6*((a^4 + 2*a
^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2*f + ((a^4
+ 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2*f)*co
sh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*
b^2 + b^4)*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d^2*f^3*x^2 + 2*(a^4 + 2*a^2*b^2
+ b^4)*d^2*e*f^2*x + (a^4 + 2*a^2*b^2 + b^4)*d^2*e^2*f)*sinh(d*x + c)^2)*dilog(-cosh(d*x + c) - sinh(d*x + c))
- 2*((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4
)*d^3*e^2*f*x + (a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 + ((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 3*(a^4 + 2*a^2*b^2 +
b^4)*d^3*e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f*x + (a^4 + 2*a^2*b^2 + b^4)*d^3*e^3)*cosh(d*x + c)^2
+ 2*((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4)
*d^3*e^2*f*x + (a^4 + 2*a^2*b^2 + b^4)*d^3*e^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3
*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f*x + (a^4 + 2*a^2*b^2 + b^
4)*d^3*e^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + (-6*I*(a^4 + a^2*b^2)*d^2*e^2*f + 6*(a^3
*b + a*b^3)*d^2*e^2*f + 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 - 12*(a^3*b + a*b^3)*c*d*e*f^2 - 6*I*(a^4 + a^2*b^2)*c^
2*f^3 + 6*(a^3*b + a*b^3)*c^2*f^3 + (-6*I*(a^4 + a^2*b^2)*d^2*e^2*f + 6*(a^3*b + a*b^3)*d^2*e^2*f + 12*I*(a^4
+ a^2*b^2)*c*d*e*f^2 - 12*(a^3*b + a*b^3)*c*d*e*f^2 - 6*I*(a^4 + a^2*b^2)*c^2*f^3 + 6*(a^3*b + a*b^3)*c^2*f^3)
*cosh(d*x + c)^2 + (-12*I*(a^4 + a^2*b^2)*d^2*e^2*f + 12*(a^3*b + a*b^3)*d^2*e^2*f + 24*I*(a^4 + a^2*b^2)*c*d*
e*f^2 - 24*(a^3*b + a*b^3)*c*d*e*f^2 - 12*I*(a^4 + a^2*b^2)*c^2*f^3 + 12*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c
)*sinh(d*x + c) + (-6*I*(a^4 + a^2*b^2)*d^2*e^2*f + 6*(a^3*b + a*b^3)*d^2*e^2*f + 12*I*(a^4 + a^2*b^2)*c*d*e*f
^2 - 12*(a^3*b + a*b^3)*c*d*e*f^2 - 6*I*(a^4 + a^2*b^2)*c^2*f^3 + 6*(a^3*b + a*b^3)*c^2*f^3)*sinh(d*x + c)^2)*
log(cosh(d*x + c) + sinh(d*x + c) + I) + (6*I*(a^4 + a^2*b^2)*d^2*e^2*f + 6*(a^3*b + a*b^3)*d^2*e^2*f - 12*I*(
a^4 + a^2*b^2)*c*d*e*f^2 - 12*(a^3*b + a*b^3)*c*d*e*f^2 + 6*I*(a^4 + a^2*b^2)*c^2*f^3 + 6*(a^3*b + a*b^3)*c^2*
f^3 + (6*I*(a^4 + a^2*b^2)*d^2*e^2*f + 6*(a^3*b + a*b^3)*d^2*e^2*f - 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 - 12*(a^3*
b + a*b^3)*c*d*e*f^2 + 6*I*(a^4 + a^2*b^2)*c^2*f^3 + 6*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*I*(a^4 +
a^2*b^2)*d^2*e^2*f + 12*(a^3*b + a*b^3)*d^2*e^2*f - 24*I*(a^4 + a^2*b^2)*c*d*e*f^2 - 24*(a^3*b + a*b^3)*c*d*e
*f^2 + 12*I*(a^4 + a^2*b^2)*c^2*f^3 + 12*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*I*(a^4 + a^
2*b^2)*d^2*e^2*f + 6*(a^3*b + a*b^3)*d^2*e^2*f - 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 - 12*(a^3*b + a*b^3)*c*d*e*f^2
+ 6*I*(a^4 + a^2*b^2)*c^2*f^3 + 6*(a^3*b + a*b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c)
- I) + 2*((a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(a^4 + 2*a^2*b^2 + b^4)*c*d^2*e^2*f + 3*(a^4 + 2*a^2*b^2 + b^4)
*c^2*d*e*f^2 - (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3 + ((a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(a^4 + 2*a^2*b^2 + b^4)*
c*d^2*e^2*f + 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d*e*f^2 - (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3)*cosh(d*x + c)^2 + 2*((a
^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(a^4 + 2*a^2*b^2 + b^4)*c*d^2*e^2*f + 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d*e*f^2
- (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d^3*e^3 - 3*(a^4 + 2
*a^2*b^2 + b^4)*c*d^2*e^2*f + 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d*e*f^2 - (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3)*sinh(d*
x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (6*I*(a^4 + a^2*b^2)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*f^
3*x^2 + 12*I*(a^4 + a^2*b^2)*d^2*e*f^2*x + 12*(a^3*b + a*b^3)*d^2*e*f^2*x + 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 1
2*(a^3*b + a*b^3)*c*d*e*f^2 - 6*I*(a^4 + a^2*b^2)*c^2*f^3 - 6*(a^3*b + a*b^3)*c^2*f^3 + (6*I*(a^4 + a^2*b^2)*d
^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*f^3*x^2 + 12*I*(a^4 + a^2*b^2)*d^2*e*f^2*x + 12*(a^3*b + a*b^3)*d^2*e*f^2*x
+ 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 12*(a^3*b + a*b^3)*c*d*e*f^2 - 6*I*(a^4 + a^2*b^2)*c^2*f^3 - 6*(a^3*b + a*
b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*I*(a^4 + a^2*b^2)*d^2*f^3*x^2 + 12*(a^3*b + a*b^3)*d^2*f^3*x^2 + 24*I*(a^4
+ a^2*b^2)*d^2*e*f^2*x + 24*(a^3*b + a*b^3)*d^2*e*f^2*x + 24*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 24*(a^3*b + a*b^3)
*c*d*e*f^2 - 12*I*(a^4 + a^2*b^2)*c^2*f^3 - 12*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*I*(a^
4 + a^2*b^2)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*f^3*x^2 + 12*I*(a^4 + a^2*b^2)*d^2*e*f^2*x + 12*(a^3*b + a*b^
3)*d^2*e*f^2*x + 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 12*(a^3*b + a*b^3)*c*d*e*f^2 - 6*I*(a^4 + a^2*b^2)*c^2*f^3 -
6*(a^3*b + a*b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (-6*I*(a^4 + a^2*b^2
)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*f^3*x^2 - 12*I*(a^4 + a^2*b^2)*d^2*e*f^2*x + 12*(a^3*b + a*b^3)*d^2*e*f^
2*x - 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 12*(a^3*b + a*b^3)*c*d*e*f^2 + 6*I*(a^4 + a^2*b^2)*c^2*f^3 - 6*(a^3*b +
a*b^3)*c^2*f^3 + (-6*I*(a^4 + a^2*b^2)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*f^3*x^2 - 12*I*(a^4 + a^2*b^2)*d^2
*e*f^2*x + 12*(a^3*b + a*b^3)*d^2*e*f^2*x - 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 12*(a^3*b + a*b^3)*c*d*e*f^2 + 6*
I*(a^4 + a^2*b^2)*c^2*f^3 - 6*(a^3*b + a*b^3)*c^2*f^3)*cosh(d*x + c)^2 + (-12*I*(a^4 + a^2*b^2)*d^2*f^3*x^2 +
12*(a^3*b + a*b^3)*d^2*f^3*x^2 - 24*I*(a^4 + a^2*b^2)*d^2*e*f^2*x + 24*(a^3*b + a*b^3)*d^2*e*f^2*x - 24*I*(a^4
+ a^2*b^2)*c*d*e*f^2 + 24*(a^3*b + a*b^3)*c*d*e*f^2 + 12*I*(a^4 + a^2*b^2)*c^2*f^3 - 12*(a^3*b + a*b^3)*c^2*f
^3)*cosh(d*x + c)*sinh(d*x + c) + (-6*I*(a^4 + a^2*b^2)*d^2*f^3*x^2 + 6*(a^3*b + a*b^3)*d^2*f^3*x^2 - 12*I*(a^
4 + a^2*b^2)*d^2*e*f^2*x + 12*(a^3*b + a*b^3)*d^2*e*f^2*x - 12*I*(a^4 + a^2*b^2)*c*d*e*f^2 + 12*(a^3*b + a*b^3
)*c*d*e*f^2 + 6*I*(a^4 + a^2*b^2)*c^2*f^3 - 6*(a^3*b + a*b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) -
I*sinh(d*x + c) + 1) + 2*((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 3*(
a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f*x + 3*(a^4 + 2*a^2*b^2 + b^4)*c*d^2*e^2*f - 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d*e
*f^2 + (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3 + ((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*
e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f*x + 3*(a^4 + 2*a^2*b^2 + b^4)*c*d^2*e^2*f - 3*(a^4 + 2*a^2*b^2
+ b^4)*c^2*d*e*f^2 + (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^
3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e^2*f*x + 3*(a^4 + 2*a^2*b^2 + b^4
)*c*d^2*e^2*f - 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d*e*f^2 + (a^4 + 2*a^2*b^2 + b^4)*c^3*f^3)*cosh(d*x + c)*sinh(d*
x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d^3*f^3*x^3 + 3*(a^4 + 2*a^2*b^2 + b^4)*d^3*e*f^2*x^2 + 3*(a^4 + 2*a^2*b^2 +
b^4)*d^3*e^2*f*x + 3*(a^4 + 2*a^2*b^2 + b^4)*c*d^2*e^2*f - 3*(a^4 + 2*a^2*b^2 + b^4)*c^2*d*e*f^2 + (a^4 + 2*a
^2*b^2 + b^4)*c^3*f^3)*sinh(d*x + c)^2)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 12*((a^4 + 2*a^2*b^2 + b^4)*
f^3*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*f^3*
sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f^3)*polylog(4, cosh(d*x + c) + sinh(d*x + c)) - 12*((a^4 + 2*a^2*b^
2 + b^4)*f^3*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*f^3*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 +
b^4)*f^3*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f^3)*polylog(4, -cosh(d*x + c) - sinh(d*x + c)) - 12*((a^4
+ 2*a^2*b^2 + b^4)*d*f^3*x + (a^4 + 2*a^2*b^2 + b^4)*d*e*f^2 + ((a^4 + 2*a^2*b^2 + b^4)*d*f^3*x + (a^4 + 2*a^2
*b^2 + b^4)*d*e*f^2)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f^3*x + (a^4 + 2*a^2*b^2 + b^4)*d*e*f^2)*c
osh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d*f^3*x + (a^4 + 2*a^2*b^2 + b^4)*d*e*f^2)*sinh(d*x + c)
^2)*polylog(3, cosh(d*x + c) + sinh(d*x + c)) + (12*I*(a^4 + a^2*b^2)*f^3 - 12*(a^3*b + a*b^3)*f^3 + (12*I*(a^
4 + a^2*b^2)*f^3 - 12*(a^3*b + a*b^3)*f^3)*cosh(d*x + c)^2 + (24*I*(a^4 + a^2*b^2)*f^3 - 24*(a^3*b + a*b^3)*f^
3)*cosh(d*x + c)*sinh(d*x + c) + (12*I*(a^4 + a^2*b^2)*f^3 - 12*(a^3*b + a*b^3)*f^3)*sinh(d*x + c)^2)*polylog(
3, I*cosh(d*x + c) + I*sinh(d*x + c)) + (-12*I*(a^4 + a^2*b^2)*f^3 - 12*(a^3*b + a*b^3)*f^3 + (-12*I*(a^4 + a^
2*b^2)*f^3 - 12*(a^3*b + a*b^3)*f^3)*cosh(d*x + c)^2 + (-24*I*(a^4 + a^2*b^2)*f^3 - 24*(a^3*b + a*b^3)*f^3)*co
sh(d*x + c)*sinh(d*x + c) + (-12*I*(a^4 + a^2*b^2)*f^3 - 12*(a^3*b + a*b^3)*f^3)*sinh(d*x + c)^2)*polylog(3, -
I*cosh(d*x + c) - I*sinh(d*x + c)) + 12*((a^4 + 2*a^2*b^2 + b^4)*d*f^3*x + (a^4 + 2*a^2*b^2 + b^4)*d*e*f^2 + (
(a^4 + 2*a^2*b^2 + b^4)*d*f^3*x + (a^4 + 2*a^2*b^2 + b^4)*d*e*f^2)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4
)*d*f^3*x + (a^4 + 2*a^2*b^2 + b^4)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d*f^3*x +
(a^4 + 2*a^2*b^2 + b^4)*d*e*f^2)*sinh(d*x + c)^2)*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) + 4*((a^4 + a^2*b
^2)*d^3*f^3*x^3 + 3*(a^4 + a^2*b^2)*d^3*e*f^2*x^2 + 3*(a^4 + a^2*b^2)*d^3*e^2*f*x + (a^4 + a^2*b^2)*d^3*e^3 -
2*((a^3*b + a*b^3)*d^3*f^3*x^3 + 3*(a^3*b + a*b^3)*d^3*e*f^2*x^2 + 3*(a^3*b + a*b^3)*d^3*e^2*f*x + 3*(a^3*b +
a*b^3)*c*d^2*e^2*f - 3*(a^3*b + a*b^3)*c^2*d*e*f^2 + (a^3*b + a*b^3)*c^3*f^3)*cosh(d*x + c))*sinh(d*x + c))/((
a^5 + 2*a^3*b^2 + a*b^4)*d^4*cosh(d*x + c)^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d^4*cosh(d*x + c)*sinh(d*x + c) + (
a^5 + 2*a^3*b^2 + a*b^4)*d^4*sinh(d*x + c)^2 + (a^5 + 2*a^3*b^2 + a*b^4)*d^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*csch(d*x+c)*sech(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*csch(d*x+c)*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out