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

Optimal. Leaf size=914 \[ \text{result too large to display} \]

[Out]

(-2*(e + f*x)^2)/(a*d) + (b^2*(e + f*x)^2)/(a*(a^2 + b^2)*d) + (4*b*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*d^2)
- (4*b^3*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (2*b*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2
*d) - (2*(e + f*x)^2*Coth[2*c + 2*d*x])/(a*d) + (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])
])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^
2)^(3/2)*d) - (2*b^2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^2) + (2*f*(e + f*x)*Log[1 - E^(4*(
c + d*x))])/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - ((2*I)*b*f^2*PolyLog[2, (-I)*E^(c
+ d*x)])/(a^2*d^3) + ((2*I)*b^3*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + ((2*I)*b*f^2*PolyLo
g[2, I*E^(c + d*x)])/(a^2*d^3) - ((2*I)*b^3*f^2*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (2*b*f*(e +
f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2
]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(
a^2*(a^2 + b^2)^(3/2)*d^2) - (b^2*f^2*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^3) + (f^2*PolyLog[2, E^(4
*(c + d*x))])/(2*a*d^3) - (2*b*f^2*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (2*b*f^2*PolyLog[3, E^(c + d*x)])/(a^
2*d^3) - (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (2*b^4
*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (b*(e + f*x)^2*Sech[c
+ d*x])/(a^2*d) + (b^3*(e + f*x)^2*Sech[c + d*x])/(a^2*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Tanh[c + d*x])/(a*(a
^2 + b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 2.03739, antiderivative size = 914, normalized size of antiderivative = 1., number of steps used = 51, number of rules used = 25, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.694, Rules used = {5589, 5461, 4184, 3716, 2190, 2279, 2391, 2622, 321, 207, 5462, 6741, 12, 6742, 6273, 4182, 2531, 2282, 6589, 4180, 5573, 3322, 2264, 3718, 5451} \[ \frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^4}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^2}+\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}-\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) b^3}{a^2 \left (a^2+b^2\right ) d^3}+\frac{(e+f x)^2 \text{sech}(c+d x) b^3}{a^2 \left (a^2+b^2\right ) d}+\frac{(e+f x)^2 b^2}{a \left (a^2+b^2\right ) d}-\frac{2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right ) b^2}{a \left (a^2+b^2\right ) d^2}-\frac{f^2 \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^2}{a \left (a^2+b^2\right ) d^3}+\frac{(e+f x)^2 \tanh (c+d x) b^2}{a \left (a^2+b^2\right ) d}+\frac{4 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right ) b}{a^2 d^2}+\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right ) b}{a^2 d}+\frac{2 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right ) b}{a^2 d^2}-\frac{2 i f^2 \text{PolyLog}\left (2,-i e^{c+d x}\right ) b}{a^2 d^3}+\frac{2 i f^2 \text{PolyLog}\left (2,i e^{c+d x}\right ) b}{a^2 d^3}-\frac{2 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right ) b}{a^2 d^2}-\frac{2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right ) b}{a^2 d^3}+\frac{2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right ) b}{a^2 d^3}-\frac{(e+f x)^2 \text{sech}(c+d x) b}{a^2 d}-\frac{2 (e+f x)^2}{a d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (2,e^{4 (c+d x)}\right )}{2 a d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e + f*x)^2)/(a*d) + (b^2*(e + f*x)^2)/(a*(a^2 + b^2)*d) + (4*b*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*d^2)
- (4*b^3*f*(e + f*x)*ArcTan[E^(c + d*x)])/(a^2*(a^2 + b^2)*d^2) + (2*b*(e + f*x)^2*ArcTanh[E^(c + d*x)])/(a^2
*d) - (2*(e + f*x)^2*Coth[2*c + 2*d*x])/(a*d) + (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])
])/(a^2*(a^2 + b^2)^(3/2)*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^
2)^(3/2)*d) - (2*b^2*f*(e + f*x)*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^2) + (2*f*(e + f*x)*Log[1 - E^(4*(
c + d*x))])/(a*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - ((2*I)*b*f^2*PolyLog[2, (-I)*E^(c
+ d*x)])/(a^2*d^3) + ((2*I)*b^3*f^2*PolyLog[2, (-I)*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) + ((2*I)*b*f^2*PolyLo
g[2, I*E^(c + d*x)])/(a^2*d^3) - ((2*I)*b^3*f^2*PolyLog[2, I*E^(c + d*x)])/(a^2*(a^2 + b^2)*d^3) - (2*b*f*(e +
f*x)*PolyLog[2, E^(c + d*x)])/(a^2*d^2) + (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2
]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(
a^2*(a^2 + b^2)^(3/2)*d^2) - (b^2*f^2*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^3) + (f^2*PolyLog[2, E^(4
*(c + d*x))])/(2*a*d^3) - (2*b*f^2*PolyLog[3, -E^(c + d*x)])/(a^2*d^3) + (2*b*f^2*PolyLog[3, E^(c + d*x)])/(a^
2*d^3) - (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) + (2*b^4
*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3) - (b*(e + f*x)^2*Sech[c
+ d*x])/(a^2*d) + (b^3*(e + f*x)^2*Sech[c + d*x])/(a^2*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Tanh[c + d*x])/(a*(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 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

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

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 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 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)^2 \text{csch}^2(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \text{csch}^2(c+d x) \text{sech}^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^2 \text{csch}(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac{4 \int (e+f x)^2 \text{csch}^2(2 c+2 d x) \, dx}{a}-\frac{b \int (e+f x)^2 \text{csch}(c+d x) \text{sech}^2(c+d x) \, dx}{a^2}+\frac{b^2 \int \frac{(e+f x)^2 \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}\\ &=\frac{b (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^2 \int (e+f x)^2 \text{sech}^2(c+d x) (a-b \sinh (c+d x)) \, dx}{a^2 \left (a^2+b^2\right )}+\frac{b^4 \int \frac{(e+f x)^2}{a+b \sinh (c+d x)} \, dx}{a^2 \left (a^2+b^2\right )}+\frac{(2 b f) \int (e+f x) \left (-\frac{\tanh ^{-1}(\cosh (c+d x))}{d}+\frac{\text{sech}(c+d x)}{d}\right ) \, dx}{a^2}+\frac{(4 f) \int (e+f x) \coth (2 c+2 d x) \, dx}{a d}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^2 \int \left (a (e+f x)^2 \text{sech}^2(c+d x)-b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a^2 \left (a^2+b^2\right )}+\frac{\left (2 b^4\right ) \int \frac{e^{c+d x} (e+f x)^2}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2 \left (a^2+b^2\right )}+\frac{(2 b f) \int \frac{(e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text{sech}(c+d x)\right )}{d} \, dx}{a^2}-\frac{(8 f) \int \frac{e^{2 (2 c+2 d x)} (e+f x)}{1-e^{2 (2 c+2 d x)}} \, dx}{a d}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{\left (2 b^5\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^{3/2}}-\frac{\left (2 b^5\right ) \int \frac{e^{c+d x} (e+f x)^2}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \left (a^2+b^2\right )^{3/2}}+\frac{b^2 \int (e+f x)^2 \text{sech}^2(c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac{(2 b f) \int (e+f x) \left (-\tanh ^{-1}(\cosh (c+d x))+\text{sech}(c+d x)\right ) \, dx}{a^2 d}-\frac{\left (2 f^2\right ) \int \log \left (1-e^{2 (2 c+2 d x)}\right ) \, dx}{a d^2}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}+\frac{(2 b f) \int \left (-(e+f x) \tanh ^{-1}(\cosh (c+d x))+(e+f x) \text{sech}(c+d x)\right ) \, dx}{a^2 d}-\frac{\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac{\left (2 b^4 f\right ) \int (e+f x) \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{\left (2 b^2 f\right ) \int (e+f x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right ) d}-\frac{\left (2 b^3 f\right ) \int (e+f x) \text{sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac{f^2 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 (2 c+2 d x)}\right )}{2 a d^3}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac{b (e+f x)^2 \tanh ^{-1}(\cosh (c+d x))}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac{(2 b f) \int (e+f x) \tanh ^{-1}(\cosh (c+d x)) \, dx}{a^2 d}+\frac{(2 b f) \int (e+f x) \text{sech}(c+d x) \, dx}{a^2 d}-\frac{\left (4 b^2 f\right ) \int \frac{e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right ) d}-\frac{\left (2 b^4 f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac{\left (2 b^4 f^2\right ) \int \text{Li}_2\left (-\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac{\left (2 i b^3 f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}-\frac{\left (2 i b^3 f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d^2}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac{4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac{b \int d (e+f x)^2 \text{csch}(c+d x) \, dx}{a^2 d}-\frac{\left (2 b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{b x}{-a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{\left (2 b^4 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{b x}{a+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{\left (2 i b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{\left (2 i b^3 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{\left (2 i b f^2\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac{\left (2 i b f^2\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac{\left (2 b^2 f^2\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac{4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{2 i b^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{2 i b^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac{b \int (e+f x)^2 \text{csch}(c+d x) \, dx}{a^2}-\frac{\left (2 i b f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}+\frac{\left (2 i b f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac{4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}-\frac{2 i b f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 d^3}+\frac{2 i b^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 d^3}-\frac{2 i b^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}+\frac{(2 b f) \int (e+f x) \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac{(2 b f) \int (e+f x) \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac{4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{2 b f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac{2 i b f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 d^3}+\frac{2 i b^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 d^3}-\frac{2 i b^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{2 b f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac{\left (2 b f^2\right ) \int \text{Li}_2\left (-e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac{\left (2 b f^2\right ) \int \text{Li}_2\left (e^{c+d x}\right ) \, dx}{a^2 d^2}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac{4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{2 b f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac{2 i b f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 d^3}+\frac{2 i b^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 d^3}-\frac{2 i b^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{2 b f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}-\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}+\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^3}\\ &=-\frac{2 (e+f x)^2}{a d}+\frac{b^2 (e+f x)^2}{a \left (a^2+b^2\right ) d}+\frac{4 b f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}-\frac{4 b^3 f (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}-\frac{2 (e+f x)^2 \coth (2 c+2 d x)}{a d}+\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{b^4 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d}-\frac{2 b^2 f (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{2 f (e+f x) \log \left (1-e^{4 (c+d x)}\right )}{a d^2}+\frac{2 b f (e+f x) \text{Li}_2\left (-e^{c+d x}\right )}{a^2 d^2}-\frac{2 i b f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 d^3}+\frac{2 i b^3 f^2 \text{Li}_2\left (-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 d^3}-\frac{2 i b^3 f^2 \text{Li}_2\left (i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^3}-\frac{2 b f (e+f x) \text{Li}_2\left (e^{c+d x}\right )}{a^2 d^2}+\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{2 b^4 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2}-\frac{b^2 f^2 \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_2\left (e^{4 (c+d x)}\right )}{2 a d^3}-\frac{2 b f^2 \text{Li}_3\left (-e^{c+d x}\right )}{a^2 d^3}+\frac{2 b f^2 \text{Li}_3\left (e^{c+d x}\right )}{a^2 d^3}-\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}+\frac{2 b^4 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3}-\frac{b (e+f x)^2 \text{sech}(c+d x)}{a^2 d}+\frac{b^3 (e+f x)^2 \text{sech}(c+d x)}{a^2 \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \tanh (c+d x)}{a \left (a^2+b^2\right ) d}\\ \end{align*}

Mathematica [B]  time = 28.6303, size = 2677, normalized size = 2.93 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

4*(((I/2)*a*(-(d*(e + f*x)*(d*(e + f*x) + (1 + I*E^(2*c))*f*Log[1 - E^(-c - d*x)] + (1 + I*E^(2*c))*f*Log[1 +
I*E^(-c - d*x)])) + (1 + I*E^(2*c))*f^2*PolyLog[2, (-I)*E^(-c - d*x)] + (1 + I*E^(2*c))*f^2*PolyLog[2, E^(-c -
d*x)]))/((a^2 + b^2)*d^3*(-I + E^(2*c))) - (b*(4*a*b*d^2*e*E^(2*c)*f*x + 2*a*b*d^2*E^(2*c)*f^2*x^2 + 2*a^2*d^
2*e^2*ArcTanh[E^(c + d*x)] + 2*b^2*d^2*e^2*ArcTanh[E^(c + d*x)] - 2*a^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] -
2*b^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] - 2*a^2*d^2*e*f*x*Log[1 - E^(c + d*x)] - 2*b^2*d^2*e*f*x*Log[1 - E
^(c + d*x)] + 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 2*b^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] - a^
2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - b^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] + a^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^
(c + d*x)] + b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*a^2*d^2*e*f*x*Log[1 + E^(c + d*x)] + 2*b^2*d^2*e
*f*x*Log[1 + E^(c + d*x)] - 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 2*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(
c + d*x)] + a^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + b^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] - a^2*d^2*E^(2*c)*f^2*
x^2*Log[1 + E^(c + d*x)] - b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] + 2*a*b*d*e*f*Log[1 - E^(2*(c + d*x))]
- 2*a*b*d*e*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + 2*a*b*d*f^2*x*Log[1 - E^(2*(c + d*x))] - 2*a*b*d*E^(2*c)*f^2
*x*Log[1 - E^(2*(c + d*x))] - 2*(a^2 + b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -E^(c + d*x)] + 2*(a^2 + b
^2)*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + a*b*f^2*PolyLog[2, E^(2*(c + d*x))] - a*b*E^(2*c)*f
^2*PolyLog[2, E^(2*(c + d*x))] - 2*a^2*f^2*PolyLog[3, -E^(c + d*x)] - 2*b^2*f^2*PolyLog[3, -E^(c + d*x)] + 2*a
^2*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] + 2*b^2*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] + 2*a^2*f^2*PolyLog[3, E^
(c + d*x)] + 2*b^2*f^2*PolyLog[3, E^(c + d*x)] - 2*a^2*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)] - 2*b^2*E^(2*c)*f^2
*PolyLog[3, E^(c + d*x)]))/(4*a^2*(a^2 + b^2)*d^3*(-1 + E^(2*c))) + (b^4*(-2*d^2*e^2*ArcTanh[(a + b*E^(c + d*x
))/Sqrt[a^2 + b^2]] + 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^2*f^2*x^2*Log[1 + (b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2])] - 2*d^2*e*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - d^2*f^2*x^2*Log
[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*f*(e + f*x)*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2]
)] - 2*d*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*f^2*PolyLog[3, (b*E^(c + d*x))/(
-a + Sqrt[a^2 + b^2])] + 2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(4*a^2*(a^2 + b^2)^(3/2)
*d^3) + (a*e*f*Sech[c/2]*(Cosh[c/2]*Log[Cosh[c/2]*Cosh[(d*x)/2] + Sinh[c/2]*Sinh[(d*x)/2]] - (d*x*Sinh[c/2])/2
))/(2*(a^2 + b^2)*d^2*(Cosh[c/2]^2 - Sinh[c/2]^2)) - (a*f^2*Csch[c/2]*(-(d^2*x^2)/(4*E^ArcTanh[Coth[c/2]]) + (
I*Coth[c/2]*(-(d*x*(-Pi + (2*I)*ArcTanh[Coth[c/2]]))/2 - Pi*Log[1 + E^(d*x)] - 2*((I/2)*d*x + I*ArcTanh[Coth[c
/2]])*Log[1 - E^((2*I)*((I/2)*d*x + I*ArcTanh[Coth[c/2]]))] + Pi*Log[Cosh[(d*x)/2]] + (2*I)*ArcTanh[Coth[c/2]]
*Log[I*Sinh[(d*x)/2 + ArcTanh[Coth[c/2]]]] + I*PolyLog[2, E^((2*I)*((I/2)*d*x + I*ArcTanh[Coth[c/2]]))]))/Sqrt
[1 - Coth[c/2]^2])*Sech[c/2])/(2*(a^2 + b^2)*d^3*Sqrt[Csch[c/2]^2*(-Cosh[c/2]^2 + Sinh[c/2]^2)]) - (e*f*x*Csch
[c/2]*Sech[c/2]*(a^2*Cosh[c] - b^2*Cosh[c] + a^2*Cosh[2*c] - I*a^2*Sinh[c] - I*b^2*Sinh[c]))/(8*a*(a^2 + b^2)*
d*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c] + I*Sinh[c])) - (f^2*x^2*Csch[c/2]*Sech[c/2]*(a
^2*Cosh[c] - b^2*Cosh[c] + a^2*Cosh[2*c] - I*a^2*Sinh[c] - I*b^2*Sinh[c]))/(16*a*(a^2 + b^2)*d*(Cosh[c/2] - I*
Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c] + I*Sinh[c])) + (b*e*f*ArcTan[(Sinh[c] + Cosh[c]*Tanh[(d*x)/2])/
Sqrt[Cosh[c]^2 - Sinh[c]^2]])/((a^2 + b^2)*d^2*Sqrt[Cosh[c]^2 - Sinh[c]^2]) - ((I/2)*a*f*(Cosh[c] + Sinh[c])*(
((e + f*x)^2*(Cosh[c] - Sinh[c]))/(2*f) + ((e + f*x)*Log[1 - I*Cosh[c + d*x] + I*Sinh[c + d*x]]*(I + Cosh[c] -
Sinh[c]))/d - (I*f*PolyLog[2, I*(Cosh[c + d*x] - Sinh[c + d*x])]*(Cosh[c] - Sinh[c])*(-I + Cosh[c] + Sinh[c])
)/d^2))/((a^2 + b^2)*d*(-I + Cosh[c] + Sinh[c])) + (b*f^2*(((-I)*Csch[c]*(I*(d*x + ArcTanh[Coth[c]])*(Log[1 -
E^(-(d*x) - ArcTanh[Coth[c]])] - Log[1 + E^(-(d*x) - ArcTanh[Coth[c]])]) + I*(PolyLog[2, -E^(-(d*x) - ArcTanh[
Coth[c]])] - PolyLog[2, E^(-(d*x) - ArcTanh[Coth[c]])])))/Sqrt[1 - Coth[c]^2] - (2*ArcTan[(Sinh[c] + Cosh[c]*T
anh[(d*x)/2])/Sqrt[Cosh[c]^2 - Sinh[c]^2]]*ArcTanh[Coth[c]])/Sqrt[Cosh[c]^2 - Sinh[c]^2]))/(2*(a^2 + b^2)*d^3)
+ (Csch[2*c]*Csch[2*c + 2*d*x]*(a*b*e^2*Cosh[c - d*x] + 2*a*b*e*f*x*Cosh[c - d*x] + a*b*f^2*x^2*Cosh[c - d*x]
- a*b*e^2*Cosh[3*c + d*x] - 2*a*b*e*f*x*Cosh[3*c + d*x] - a*b*f^2*x^2*Cosh[3*c + d*x] - b^2*e^2*Sinh[2*c] - 2
*b^2*e*f*x*Sinh[2*c] - b^2*f^2*x^2*Sinh[2*c] + 2*a^2*e^2*Sinh[2*d*x] + b^2*e^2*Sinh[2*d*x] + 4*a^2*e*f*x*Sinh[
2*d*x] + 2*b^2*e*f*x*Sinh[2*d*x] + 2*a^2*f^2*x^2*Sinh[2*d*x] + b^2*f^2*x^2*Sinh[2*d*x]))/(4*a*(a^2 + b^2)*d))

________________________________________________________________________________________

Maple [F]  time = 1.658, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ({\rm csch} \left (dx+c\right ) \right ) ^{2} \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)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 5.02976, size = 23058, normalized size = 25.23 \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)^2*csch(d*x+c)^2*sech(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(4*(2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*e^2 - 8*(2*a^5 + 3*a^3*b^2 + a*b^4)*c*d*e*f + 4*(2*a^5 + 3*a^3*b^2 + a
*b^4)*c^2*f^2 + 4*((2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*e*f*x + 2*(2*a^
5 + 3*a^3*b^2 + a*b^4)*c*d*e*f - (2*a^5 + 3*a^3*b^2 + a*b^4)*c^2*f^2)*cosh(d*x + c)^4 + 4*((2*a^5 + 3*a^3*b^2
+ a*b^4)*d^2*f^2*x^2 + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*e*f*x + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*c*d*e*f - (2*a^
5 + 3*a^3*b^2 + a*b^4)*c^2*f^2)*sinh(d*x + c)^4 + 4*((a^4*b + a^2*b^3)*d^2*f^2*x^2 + 2*(a^4*b + a^2*b^3)*d^2*e
*f*x + (a^4*b + a^2*b^3)*d^2*e^2)*cosh(d*x + c)^3 + 4*((a^4*b + a^2*b^3)*d^2*f^2*x^2 + 2*(a^4*b + a^2*b^3)*d^2
*e*f*x + (a^4*b + a^2*b^3)*d^2*e^2 + 4*((2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^5 + 3*a^3*b^2 + a*b^4
)*d^2*e*f*x + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*c*d*e*f - (2*a^5 + 3*a^3*b^2 + a*b^4)*c^2*f^2)*cosh(d*x + c))*sinh
(d*x + c)^3 + 4*((a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(a^3*b^2 + a*b^4)*d^2*e*f*x + (a^3*b^2 + a*b^4)*d^2*e^2)*co
sh(d*x + c)^2 + 4*((a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(a^3*b^2 + a*b^4)*d^2*e*f*x + (a^3*b^2 + a*b^4)*d^2*e^2 +
6*((2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*e*f*x + 2*(2*a^5 + 3*a^3*b^2 +
a*b^4)*c*d*e*f - (2*a^5 + 3*a^3*b^2 + a*b^4)*c^2*f^2)*cosh(d*x + c)^2 + 3*((a^4*b + a^2*b^3)*d^2*f^2*x^2 + 2*
(a^4*b + a^2*b^3)*d^2*e*f*x + (a^4*b + a^2*b^3)*d^2*e^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 4*(b^5*d*f^2*x + b^5
*d*e*f - (b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)^4 - 4*(b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)^3*sinh(d*x + c)
- 6*(b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)*sin
h(d*x + c)^3 - (b^5*d*f^2*x + b^5*d*e*f)*sinh(d*x + c)^4)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sin
h(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 4*(b^5*d*f^2*x + b^5*d*e*
f - (b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)^4 - 4*(b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)^3*sinh(d*x + c) - 6*
(b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d*f^2*x + b^5*d*e*f)*cosh(d*x + c)*sinh(d*x
+ c)^3 - (b^5*d*f^2*x + b^5*d*e*f)*sinh(d*x + c)^4)*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^5*d^2*e^2 - 2*b^5*c*d*e*f
+ b^5*c^2*f^2 - (b^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2)*cosh(d*x + c)^4 - 4*(b^5*d^2*e^2 - 2*b^5*c*d*e*f
+ b^5*c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c) - 6*(b^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2)*cosh(d*x + c)^2*s
inh(d*x + c)^2 - 4*(b^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 - (b^5*d^2*e^2 -
2*b^5*c*d*e*f + b^5*c^2*f^2)*sinh(d*x + c)^4)*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^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2 - (b^5*d^2*e^2 - 2*b^5*c*d*e
*f + b^5*c^2*f^2)*cosh(d*x + c)^4 - 4*(b^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2)*cosh(d*x + c)^3*sinh(d*x + c
) - 6*(b^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d^2*e^2 - 2*b^5*c*d
*e*f + b^5*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 - (b^5*d^2*e^2 - 2*b^5*c*d*e*f + b^5*c^2*f^2)*sinh(d*x + c)^
4)*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^5
*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2 - (b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*
e*f - b^5*c^2*f^2)*cosh(d*x + c)^4 - 4*(b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*cosh(
d*x + c)^3*sinh(d*x + c) - 6*(b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*cosh(d*x + c)^2
*sinh(d*x + c)^2 - 4*(b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*cosh(d*x + c)*sinh(d*x
+ c)^3 - (b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*sinh(d*x + c)^4)*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^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2 - (b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x +
2*b^5*c*d*e*f - b^5*c^2*f^2)*cosh(d*x + c)^4 - 4*(b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*
f^2)*cosh(d*x + c)^3*sinh(d*x + c) - 6*(b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*cosh(
d*x + c)^2*sinh(d*x + c)^2 - 4*(b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*cosh(d*x + c)
*sinh(d*x + c)^3 - (b^5*d^2*f^2*x^2 + 2*b^5*d^2*e*f*x + 2*b^5*c*d*e*f - b^5*c^2*f^2)*sinh(d*x + c)^4)*sqrt((a^
2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b
^2) - b)/b) + 4*(b^5*f^2*cosh(d*x + c)^4 + 4*b^5*f^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*f^2*cosh(d*x + c)^2
*sinh(d*x + c)^2 + 4*b^5*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*f^2*sinh(d*x + c)^4 - b^5*f^2)*sqrt((a^2 + b^
2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b
^2))/b) - 4*(b^5*f^2*cosh(d*x + c)^4 + 4*b^5*f^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*b^5*f^2*cosh(d*x + c)^2*sin
h(d*x + c)^2 + 4*b^5*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^5*f^2*sinh(d*x + c)^4 - b^5*f^2)*sqrt((a^2 + b^2)/b
^2)*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))
/b) - 4*((a^4*b + a^2*b^3)*d^2*f^2*x^2 + 2*(a^4*b + a^2*b^3)*d^2*e*f*x + (a^4*b + a^2*b^3)*d^2*e^2)*cosh(d*x +
c) - 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x - ((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*
e*f - (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x + c)^4 - 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b
^3 + b^5)*d*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x + c)^3*sinh(d*x + c) - 6*((a^4*b + 2*a^2*b^3 + b^5)*
d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4
*((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*
x + c)*sinh(d*x + c)^3 - ((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f - (a^5 + 2*a^3*b
^2 + a*b^4)*f^2)*sinh(d*x + c)^4 + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*dilog(cosh
(d*x + c) + sinh(d*x + c)) - (4*((a^5 + a^3*b^2)*f^2 + I*(a^4*b + a^2*b^3)*f^2)*cosh(d*x + c)^4 + 16*((a^5 + a
^3*b^2)*f^2 + I*(a^4*b + a^2*b^3)*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + 24*((a^5 + a^3*b^2)*f^2 + I*(a^4*b + a^
2*b^3)*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 16*((a^5 + a^3*b^2)*f^2 + I*(a^4*b + a^2*b^3)*f^2)*cosh(d*x + c)
*sinh(d*x + c)^3 + 4*((a^5 + a^3*b^2)*f^2 + I*(a^4*b + a^2*b^3)*f^2)*sinh(d*x + c)^4 - 4*(a^5 + a^3*b^2)*f^2 -
4*I*(a^4*b + a^2*b^3)*f^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - (4*((a^5 + a^3*b^2)*f^2 - I*(a^4*b + a^
2*b^3)*f^2)*cosh(d*x + c)^4 + 16*((a^5 + a^3*b^2)*f^2 - I*(a^4*b + a^2*b^3)*f^2)*cosh(d*x + c)^3*sinh(d*x + c)
+ 24*((a^5 + a^3*b^2)*f^2 - I*(a^4*b + a^2*b^3)*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + 16*((a^5 + a^3*b^2)*f^
2 - I*(a^4*b + a^2*b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + 4*((a^5 + a^3*b^2)*f^2 - I*(a^4*b + a^2*b^3)*f^2)
*sinh(d*x + c)^4 - 4*(a^5 + a^3*b^2)*f^2 + 4*I*(a^4*b + a^2*b^3)*f^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)
) + 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x - ((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*
f + (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x + c)^4 - 4*((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3
+ b^5)*d*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x + c)^3*sinh(d*x + c) - 6*((a^4*b + 2*a^2*b^3 + b^5)*d*
f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*(
(a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*cosh(d*x
+ c)*sinh(d*x + c)^3 - ((a^4*b + 2*a^2*b^3 + b^5)*d*f^2*x + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f + (a^5 + 2*a^3*b^2
+ a*b^4)*f^2)*sinh(d*x + c)^4 + (a^4*b + 2*a^2*b^3 + b^5)*d*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*f^2)*dilog(-cosh(
d*x + c) - sinh(d*x + c)) + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*f^2*x^2 + (a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 - ((a
^4*b + 2*a^2*b^3 + b^5)*d^2*f^2*x^2 + (a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*e*f +
2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^4 - 4*((a^4*b + 2*a^2
*b^3 + b^5)*d^2*f^2*x^2 + (a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*e*f + 2*((a^4*b +
2*a^2*b^3 + b^5)*d^2*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^3*sinh(d*x + c) - 6*((a^4*b + 2*a
^2*b^3 + b^5)*d^2*f^2*x^2 + (a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*e*f + 2*((a^4*b
+ 2*a^2*b^3 + b^5)*d^2*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*((a^4*b +
2*a^2*b^3 + b^5)*d^2*f^2*x^2 + (a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*e*f + 2*((a^
4*b + 2*a^2*b^3 + b^5)*d^2*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c)^3 - ((a^4*b +
2*a^2*b^3 + b^5)*d^2*f^2*x^2 + (a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*e*f + 2*((a^
4*b + 2*a^2*b^3 + b^5)*d^2*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*sinh(d*x + c)^4 + 2*(a^5 + 2*a^3*b^2 + a*
b^4)*d*e*f + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f + (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*log(cosh(d*x + c) + si
nh(d*x + c) + 1) - ((4*(a^5 + a^3*b^2)*d*e*f + 4*I*(a^4*b + a^2*b^3)*d*e*f - 4*(a^5 + a^3*b^2)*c*f^2 - 4*I*(a^
4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^4 + (16*(a^5 + a^3*b^2)*d*e*f + 16*I*(a^4*b + a^2*b^3)*d*e*f - 16*(a^5 + a
^3*b^2)*c*f^2 - 16*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + (24*(a^5 + a^3*b^2)*d*e*f + 24*I
*(a^4*b + a^2*b^3)*d*e*f - 24*(a^5 + a^3*b^2)*c*f^2 - 24*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^2*sinh(d*x +
c)^2 + (16*(a^5 + a^3*b^2)*d*e*f + 16*I*(a^4*b + a^2*b^3)*d*e*f - 16*(a^5 + a^3*b^2)*c*f^2 - 16*I*(a^4*b + a^
2*b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*(a^5 + a^3*b^2)*d*e*f + 4*I*(a^4*b + a^2*b^3)*d*e*f - 4*(a^5
+ a^3*b^2)*c*f^2 - 4*I*(a^4*b + a^2*b^3)*c*f^2)*sinh(d*x + c)^4 - 4*(a^5 + a^3*b^2)*d*e*f - 4*I*(a^4*b + a^2*b
^3)*d*e*f + 4*(a^5 + a^3*b^2)*c*f^2 + 4*I*(a^4*b + a^2*b^3)*c*f^2)*log(cosh(d*x + c) + sinh(d*x + c) + I) - ((
4*(a^5 + a^3*b^2)*d*e*f - 4*I*(a^4*b + a^2*b^3)*d*e*f - 4*(a^5 + a^3*b^2)*c*f^2 + 4*I*(a^4*b + a^2*b^3)*c*f^2)
*cosh(d*x + c)^4 + (16*(a^5 + a^3*b^2)*d*e*f - 16*I*(a^4*b + a^2*b^3)*d*e*f - 16*(a^5 + a^3*b^2)*c*f^2 + 16*I*
(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^3*sinh(d*x + c) + (24*(a^5 + a^3*b^2)*d*e*f - 24*I*(a^4*b + a^2*b^3)*d*
e*f - 24*(a^5 + a^3*b^2)*c*f^2 + 24*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 + (16*(a^5 + a^
3*b^2)*d*e*f - 16*I*(a^4*b + a^2*b^3)*d*e*f - 16*(a^5 + a^3*b^2)*c*f^2 + 16*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*
x + c)*sinh(d*x + c)^3 + (4*(a^5 + a^3*b^2)*d*e*f - 4*I*(a^4*b + a^2*b^3)*d*e*f - 4*(a^5 + a^3*b^2)*c*f^2 + 4*
I*(a^4*b + a^2*b^3)*c*f^2)*sinh(d*x + c)^4 - 4*(a^5 + a^3*b^2)*d*e*f + 4*I*(a^4*b + a^2*b^3)*d*e*f + 4*(a^5 +
a^3*b^2)*c*f^2 - 4*I*(a^4*b + a^2*b^3)*c*f^2)*log(cosh(d*x + c) + sinh(d*x + c) - I) - 2*((a^4*b + 2*a^2*b^3 +
b^5)*d^2*e^2 - ((a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)
*d*e*f + ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2)*cosh(d*x + c)^4 - 4*((a^4*b + 2*
a^2*b^3 + b^5)*d^2*e^2 - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*d*e*f + ((a^4*b + 2*a^2*b^3
+ b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2)*cosh(d*x + c)^3*sinh(d*x + c) - 6*((a^4*b + 2*a^2*b^3 + b^5)
*d^2*e^2 - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*d*e*f + ((a^4*b + 2*a^2*b^3 + b^5)*c^2 +
2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 - 2
*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*d*e*f + ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a
^3*b^2 + a*b^4)*c)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 - ((a^4*b + 2*a^2*b^3 + b^5)*d^2*e^2 - 2*(a^5 + 2*a^3*b^
2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*d*e*f + ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*
c)*f^2)*sinh(d*x + c)^4 - 2*(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*c)*d*e*f + ((a^4*b + 2*a^2*b^
3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (4*(a^5 + a^3*b^2)
*d*f^2*x - 4*I*(a^4*b + a^2*b^3)*d*f^2*x - (4*(a^5 + a^3*b^2)*d*f^2*x - 4*I*(a^4*b + a^2*b^3)*d*f^2*x + 4*(a^5
+ a^3*b^2)*c*f^2 - 4*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^4 - (16*(a^5 + a^3*b^2)*d*f^2*x - 16*I*(a^4*b +
a^2*b^3)*d*f^2*x + 16*(a^5 + a^3*b^2)*c*f^2 - 16*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^3*sinh(d*x + c) - (
24*(a^5 + a^3*b^2)*d*f^2*x - 24*I*(a^4*b + a^2*b^3)*d*f^2*x + 24*(a^5 + a^3*b^2)*c*f^2 - 24*I*(a^4*b + a^2*b^3
)*c*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 - (16*(a^5 + a^3*b^2)*d*f^2*x - 16*I*(a^4*b + a^2*b^3)*d*f^2*x + 16*(
a^5 + a^3*b^2)*c*f^2 - 16*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 - (4*(a^5 + a^3*b^2)*d*f^2*
x - 4*I*(a^4*b + a^2*b^3)*d*f^2*x + 4*(a^5 + a^3*b^2)*c*f^2 - 4*I*(a^4*b + a^2*b^3)*c*f^2)*sinh(d*x + c)^4 + 4
*(a^5 + a^3*b^2)*c*f^2 - 4*I*(a^4*b + a^2*b^3)*c*f^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (4*(a^5 + a
^3*b^2)*d*f^2*x + 4*I*(a^4*b + a^2*b^3)*d*f^2*x - (4*(a^5 + a^3*b^2)*d*f^2*x + 4*I*(a^4*b + a^2*b^3)*d*f^2*x +
4*(a^5 + a^3*b^2)*c*f^2 + 4*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^4 - (16*(a^5 + a^3*b^2)*d*f^2*x + 16*I*(
a^4*b + a^2*b^3)*d*f^2*x + 16*(a^5 + a^3*b^2)*c*f^2 + 16*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)^3*sinh(d*x +
c) - (24*(a^5 + a^3*b^2)*d*f^2*x + 24*I*(a^4*b + a^2*b^3)*d*f^2*x + 24*(a^5 + a^3*b^2)*c*f^2 + 24*I*(a^4*b +
a^2*b^3)*c*f^2)*cosh(d*x + c)^2*sinh(d*x + c)^2 - (16*(a^5 + a^3*b^2)*d*f^2*x + 16*I*(a^4*b + a^2*b^3)*d*f^2*x
+ 16*(a^5 + a^3*b^2)*c*f^2 + 16*I*(a^4*b + a^2*b^3)*c*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 - (4*(a^5 + a^3*b^2)
*d*f^2*x + 4*I*(a^4*b + a^2*b^3)*d*f^2*x + 4*(a^5 + a^3*b^2)*c*f^2 + 4*I*(a^4*b + a^2*b^3)*c*f^2)*sinh(d*x + c
)^4 + 4*(a^5 + a^3*b^2)*c*f^2 + 4*I*(a^4*b + a^2*b^3)*c*f^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - 2*(
(a^4*b + 2*a^2*b^3 + b^5)*d^2*f^2*x^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*c*d*e*f - ((a^4*b + 2*a^2*b^3 + b^5)*d^2*f
^2*x^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*c*d*e*f - ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)
*f^2 + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^4 - 4*((a^4*b
+ 2*a^2*b^3 + b^5)*d^2*f^2*x^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*c*d*e*f - ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5
+ 2*a^3*b^2 + a*b^4)*c)*f^2 + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh
(d*x + c)^3*sinh(d*x + c) - 6*((a^4*b + 2*a^2*b^3 + b^5)*d^2*f^2*x^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*c*d*e*f - (
(a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2 + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f - (a
^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)^2*sinh(d*x + c)^2 - 4*((a^4*b + 2*a^2*b^3 + b^5)*d^2*f^2*x^2 +
2*(a^4*b + 2*a^2*b^3 + b^5)*c*d*e*f - ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2 + 2
*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c)^3 - ((a^
4*b + 2*a^2*b^3 + b^5)*d^2*f^2*x^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*c*d*e*f - ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*
(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2 + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*
sinh(d*x + c)^4 - ((a^4*b + 2*a^2*b^3 + b^5)*c^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*c)*f^2 + 2*((a^4*b + 2*a^2*b^3
+ b^5)*d^2*e*f - (a^5 + 2*a^3*b^2 + a*b^4)*d*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 4*((a^4*b + 2*a
^2*b^3 + b^5)*f^2*cosh(d*x + c)^4 + 4*(a^4*b + 2*a^2*b^3 + b^5)*f^2*cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^4*b +
2*a^2*b^3 + b^5)*f^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4*b + 2*a^2*b^3 + b^5)*f^2*cosh(d*x + c)*sinh(d*x
+ c)^3 + (a^4*b + 2*a^2*b^3 + b^5)*f^2*sinh(d*x + c)^4 - (a^4*b + 2*a^2*b^3 + b^5)*f^2)*polylog(3, cosh(d*x +
c) + sinh(d*x + c)) + 4*((a^4*b + 2*a^2*b^3 + b^5)*f^2*cosh(d*x + c)^4 + 4*(a^4*b + 2*a^2*b^3 + b^5)*f^2*cosh
(d*x + c)^3*sinh(d*x + c) + 6*(a^4*b + 2*a^2*b^3 + b^5)*f^2*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^4*b + 2*a^2
*b^3 + b^5)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4*b + 2*a^2*b^3 + b^5)*f^2*sinh(d*x + c)^4 - (a^4*b + 2*a^2
*b^3 + b^5)*f^2)*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 4*((a^4*b + a^2*b^3)*d^2*f^2*x^2 + 2*(a^4*b + a^
2*b^3)*d^2*e*f*x + (a^4*b + a^2*b^3)*d^2*e^2 - 4*((2*a^5 + 3*a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(2*a^5 + 3*a^3*b
^2 + a*b^4)*d^2*e*f*x + 2*(2*a^5 + 3*a^3*b^2 + a*b^4)*c*d*e*f - (2*a^5 + 3*a^3*b^2 + a*b^4)*c^2*f^2)*cosh(d*x
+ c)^3 - 3*((a^4*b + a^2*b^3)*d^2*f^2*x^2 + 2*(a^4*b + a^2*b^3)*d^2*e*f*x + (a^4*b + a^2*b^3)*d^2*e^2)*cosh(d*
x + c)^2 - 2*((a^3*b^2 + a*b^4)*d^2*f^2*x^2 + 2*(a^3*b^2 + a*b^4)*d^2*e*f*x + (a^3*b^2 + a*b^4)*d^2*e^2)*cosh(
d*x + c))*sinh(d*x + c))/((a^6 + 2*a^4*b^2 + a^2*b^4)*d^3*cosh(d*x + c)^4 + 4*(a^6 + 2*a^4*b^2 + a^2*b^4)*d^3*
cosh(d*x + c)^3*sinh(d*x + c) + 6*(a^6 + 2*a^4*b^2 + a^2*b^4)*d^3*cosh(d*x + c)^2*sinh(d*x + c)^2 + 4*(a^6 + 2
*a^4*b^2 + a^2*b^4)*d^3*cosh(d*x + c)*sinh(d*x + c)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*d^3*sinh(d*x + c)^4 - (a^6
+ 2*a^4*b^2 + a^2*b^4)*d^3)

________________________________________________________________________________________

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)**2*csch(d*x+c)**2*sech(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out