3.358 \(\int \frac{(e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\)
Optimal. Leaf size=1176 \[ \text{result too large to display} \]
[Out]
((e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (a^2*(
e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (f^2*ArcTan[Sinh[c + d*x]])/(b*d^3) + (a^2*f^2*ArcTan[Sinh
[c + d*x]])/(b*(a^2 + b^2)*d^3) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b
^2)^2*d) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a*b^2*(e +
f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (a*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^3) - (I*f*(e +
f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + ((2*I)*a^2*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^
2)^2*d^2) + (I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (I*f*(e + f*x)*PolyLog[2, I
*E^(c + d*x)])/(b*d^2) - ((2*I)*a^2*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^2*f*(e
+ f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (2*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (2*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))])/((a^2 + b^2)^2*d^2) + (a*b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + (I*f^2*P
olyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - ((2*I)*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I
*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) - (I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((2*I
)*a^2*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b
^2)*d^3) + (2*a*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (2*a*b^2*f
^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (a*b^2*f^2*PolyLog[3, -E^(2*(c
+ d*x))])/(2*(a^2 + b^2)^2*d^3) + (f*(e + f*x)*Sech[c + d*x])/(b*d^2) - (a^2*f*(e + f*x)*Sech[c + d*x])/(b*(a^
2 + b^2)*d^2) - (a*(e + f*x)^2*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) + (a*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)
*d^2) + ((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^2*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b
*(a^2 + b^2)*d)
________________________________________________________________________________________
Rubi [A] time = 1.69576, antiderivative size = 1176, normalized size of antiderivative = 1.,
number of steps used = 49, number of rules used = 15, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.441, Rules used
= {5583, 4186, 3770, 4180, 2531, 2282, 6589, 5573, 5561, 2190, 6742, 3718, 5451, 4184, 3475}
\[ -\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d}-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d}+\frac{f^2 \tan ^{-1}(\sinh (c+d x)) a^2}{b \left (a^2+b^2\right ) d^3}+\frac{i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}+\frac{2 i b f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac{i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}-\frac{2 i b f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^2}-\frac{i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}-\frac{2 i b f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}+\frac{i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{PolyLog}\left (3,i e^{c+d x}\right ) a^2}{\left (a^2+b^2\right )^2 d^3}-\frac{f (e+f x) \text{sech}(c+d x) a^2}{b \left (a^2+b^2\right ) d^2}-\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x) a^2}{2 b \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \text{sech}^2(c+d x) a}{2 \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a}{\left (a^2+b^2\right )^2 d}-\frac{b^2 (e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a}{\left (a^2+b^2\right )^2 d}+\frac{b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{\left (a^2+b^2\right )^2 d}-\frac{f^2 \log (\cosh (c+d x)) a}{\left (a^2+b^2\right ) d^3}-\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^2}-\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^2}+\frac{b^2 f (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a}{\left (a^2+b^2\right )^2 d^2}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^3}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a}{\left (a^2+b^2\right )^2 d^3}-\frac{b^2 f^2 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) a}{2 \left (a^2+b^2\right )^2 d^3}+\frac{f (e+f x) \tanh (c+d x) a}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}-\frac{i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}+\frac{i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}+\frac{i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}-\frac{i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d} \]
Antiderivative was successfully verified.
[In]
Int[((e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]
[Out]
((e + f*x)^2*ArcTan[E^(c + d*x)])/(b*d) - (2*a^2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (a^2*(
e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d) - (f^2*ArcTan[Sinh[c + d*x]])/(b*d^3) + (a^2*f^2*ArcTan[Sinh
[c + d*x]])/(b*(a^2 + b^2)*d^3) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/((a^2 + b
^2)^2*d) - (a*b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/((a^2 + b^2)^2*d) + (a*b^2*(e +
f*x)^2*Log[1 + E^(2*(c + d*x))])/((a^2 + b^2)^2*d) - (a*f^2*Log[Cosh[c + d*x]])/((a^2 + b^2)*d^3) - (I*f*(e +
f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + ((2*I)*a^2*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^
2)^2*d^2) + (I*a^2*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (I*f*(e + f*x)*PolyLog[2, I
*E^(c + d*x)])/(b*d^2) - ((2*I)*a^2*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)^2*d^2) - (I*a^2*f*(e
+ f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^2) - (2*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^2) - (2*a*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 +
b^2]))])/((a^2 + b^2)^2*d^2) + (a*b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/((a^2 + b^2)^2*d^2) + (I*f^2*P
olyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - ((2*I)*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I
*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) - (I*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((2*I
)*a^2*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*a^2*f^2*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b
^2)*d^3) + (2*a*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) + (2*a*b^2*f
^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^2*d^3) - (a*b^2*f^2*PolyLog[3, -E^(2*(c
+ d*x))])/(2*(a^2 + b^2)^2*d^3) + (f*(e + f*x)*Sech[c + d*x])/(b*d^2) - (a^2*f*(e + f*x)*Sech[c + d*x])/(b*(a^
2 + b^2)*d^2) - (a*(e + f*x)^2*Sech[c + d*x]^2)/(2*(a^2 + b^2)*d) + (a*f*(e + f*x)*Tanh[c + d*x])/((a^2 + b^2)
*d^2) + ((e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b*d) - (a^2*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*b
*(a^2 + b^2)*d)
Rule 5583
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[c + d*x]^(n - 1),
x], x] - Dist[a/b, Int[((e + f*x)^m*Sech[c + d*x]^(p + 1)*Tanh[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 4186
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Rule 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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 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 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 5561
Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
Rule 2190
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
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]
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 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int \frac{(e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \text{sech}^3(c+d x) \, dx}{b}-\frac{a \int \frac{(e+f x)^2 \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}+\frac{\int (e+f x)^2 \text{sech}(c+d x) \, dx}{2 b}-\frac{a \int (e+f x)^2 \text{sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{b \left (a^2+b^2\right )}-\frac{(a b) \int \frac{(e+f x)^2 \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2+b^2}-\frac{f^2 \int \text{sech}(c+d x) \, dx}{b d^2}\\ &=\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{(a b) \int (e+f x)^2 \text{sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^3\right ) \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{\left (a^2+b^2\right )^2}-\frac{a \int \left (a (e+f x)^2 \text{sech}^3(c+d x)-b (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{b \left (a^2+b^2\right )}-\frac{(i f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b d}+\frac{(i f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b d}\\ &=\frac{a b^2 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{(a b) \int \left (a (e+f x)^2 \text{sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^3\right ) \int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a b^3\right ) \int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{\left (a^2+b^2\right )^2}+\frac{a \int (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x) \, dx}{a^2+b^2}-\frac{a^2 \int (e+f x)^2 \text{sech}^3(c+d x) \, dx}{b \left (a^2+b^2\right )}+\frac{\left (i f^2\right ) \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{b d^2}-\frac{\left (i f^2\right ) \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{b d^2}\\ &=\frac{a b^2 (e+f x)^3}{3 \left (a^2+b^2\right )^2 f}+\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}-\frac{a^2 f (e+f x) \text{sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac{a (e+f x)^2 \text{sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{a^2 (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac{\left (a^2 b\right ) \int (e+f x)^2 \text{sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (a b^2\right ) \int (e+f x)^2 \tanh (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac{a^2 \int (e+f x)^2 \text{sech}(c+d x) \, dx}{2 b \left (a^2+b^2\right )}+\frac{\left (2 a b^2 f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac{\left (2 a b^2 f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac{(a f) \int (e+f x) \text{sech}^2(c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac{\left (i f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}-\frac{\left (i f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^3}+\frac{\left (a^2 f^2\right ) \int \text{sech}(c+d x) \, dx}{b \left (a^2+b^2\right ) d^2}\\ &=\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac{a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{i f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}-\frac{a^2 f (e+f x) \text{sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac{a (e+f x)^2 \text{sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{a^2 (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}+\frac{\left (2 a b^2\right ) \int \frac{e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{\left (a^2+b^2\right )^2}+\frac{\left (2 i a^2 b f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac{\left (2 i a^2 b f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac{\left (i a^2 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d}-\frac{\left (i a^2 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d}+\frac{\left (2 a b^2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac{\left (2 a b^2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac{\left (a f^2\right ) \int \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac{a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac{a b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i a^2 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{i a^2 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{i f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}-\frac{a^2 f (e+f x) \text{sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac{a (e+f x)^2 \text{sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{a^2 (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac{\left (2 a b^2 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac{\left (2 a b^2 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 )}{\left (a^2+b^2\right )^2 d^3}+\frac{\left (2 a b^2 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 )}{\left (a^2+b^2\right )^2 d^3}-\frac{\left (2 i a^2 b f^2\right ) \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac{\left (2 i a^2 b f^2\right ) \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac{\left (i a^2 f^2\right ) \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}+\frac{\left (i a^2 f^2\right ) \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{b \left (a^2+b^2\right ) d^2}\\ &=\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac{a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac{a b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i a^2 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{i a^2 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{a b^2 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{i f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{2 a b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{2 a b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}-\frac{a^2 f (e+f x) \text{sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac{a (e+f x)^2 \text{sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{a^2 (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac{\left (2 i a^2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{\left (2 i a^2 b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{\left (i a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac{\left (i a^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac{\left (a b^2 f^2\right ) \int \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}\\ &=\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac{a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac{a b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i a^2 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{i a^2 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{a b^2 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{2 i a^2 b f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{i a^2 f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac{i f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{2 i a^2 b f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{i a^2 f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac{2 a b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{2 a b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}-\frac{a^2 f (e+f x) \text{sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac{a (e+f x)^2 \text{sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{a^2 (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}-\frac{\left (a b^2 f^2\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 )^2 d^3}\\ &=\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 a^2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a^2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b \left (a^2+b^2\right ) d}-\frac{f^2 \tan ^{-1}(\sinh (c+d x))}{b d^3}+\frac{a^2 f^2 \tan ^{-1}(\sinh (c+d x))}{b \left (a^2+b^2\right ) d^3}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d}+\frac{a b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d}-\frac{a f^2 \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{i f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i a^2 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}+\frac{i f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac{2 i a^2 b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{i a^2 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{2 a b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{a b^2 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{i f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac{2 i a^2 b f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{i a^2 f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}-\frac{i f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac{2 i a^2 b f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{i a^2 f^2 \text{Li}_3\left (i e^{c+d x}\right )}{b \left (a^2+b^2\right ) d^3}+\frac{2 a b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{2 a b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{a b^2 f^2 \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^3}+\frac{f (e+f x) \text{sech}(c+d x)}{b d^2}-\frac{a^2 f (e+f x) \text{sech}(c+d x)}{b \left (a^2+b^2\right ) d^2}-\frac{a (e+f x)^2 \text{sech}^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac{a f (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b d}-\frac{a^2 (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 b \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [B] time = 31.4769, size = 3390, normalized size = 2.88 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[((e + f*x)^2*Sech[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]
[Out]
(-12*a*b^2*d^3*e^2*E^(2*c)*x + 12*a^3*d*E^(2*c)*f^2*x + 12*a*b^2*d*E^(2*c)*f^2*x - 12*a*b^2*d^3*e*E^(2*c)*f*x^
2 - 4*a*b^2*d^3*E^(2*c)*f^2*x^3 - 6*a^2*b*d^2*e^2*ArcTan[E^(c + d*x)] + 6*b^3*d^2*e^2*ArcTan[E^(c + d*x)] - 6*
a^2*b*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 6*b^3*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] - 12*a^2*b*f^2*ArcTan[E^
(c + d*x)] - 12*b^3*f^2*ArcTan[E^(c + d*x)] - 12*a^2*b*E^(2*c)*f^2*ArcTan[E^(c + d*x)] - 12*b^3*E^(2*c)*f^2*Ar
cTan[E^(c + d*x)] - (6*I)*a^2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*b^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)]
- (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] -
(3*I)*a^2*b*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] - (3*I)*a^2*b*d^
2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (6*I)*a^2*b*
d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (6*I)*b^3*d^2*e*f*x*Log[1 + I*E^(c + d*x)] + (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*
Log[1 + I*E^(c + d*x)] - (6*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] + (3*I)*a^2*b*d^2*f^2*x^2*Log[1 +
I*E^(c + d*x)] - (3*I)*b^3*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] + (3*I)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c
+ d*x)] - (3*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*a*b^2*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 6
*a*b^2*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 6*a^3*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*f^2*Log[1 + E^(
2*(c + d*x))] - 6*a^3*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] + 12
*a*b^2*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 6*a*b^2*d^2*
f^2*x^2*Log[1 + E^(2*(c + d*x))] + 6*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] + (6*I)*b*(a^2 - b^2)*
d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (6*I)*b*(-a^2 + b^2)*d*(1 + E^(2*c))*f*(e + f*x)*Po
lyLog[2, I*E^(c + d*x)] + 6*a*b^2*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 6*a*b^2*d*e*E^(2*c)*f*PolyLog[2, -E^(2*
(c + d*x))] + 6*a*b^2*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*a*b^2*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x
))] - (6*I)*a^2*b*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^2*b*
E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*b^3*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*a^2*b*f^
2*PolyLog[3, I*E^(c + d*x)] - (6*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)] + (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[3, I*E
^(c + d*x)] - (6*I)*b^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*a*b^2*f^2*PolyLog[3, -E^(2*(c + d*x))] - 3*a
*b^2*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))])/(6*(a^2 + b^2)^2*d^3*(1 + E^(2*c))) + (a*b^2*(6*e^2*E^(2*c)*x +
6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]]
)/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]
])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2
- b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 -
b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x
) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d
- (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E
^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c -
Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d -
(6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(
2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sq
rt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2
+ b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b
^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E
^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*
E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E
^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*(a^2 + b^2)^2*(-1 + E^(2*c))) + (Csch[c]*Sech[c]*Sech[c + d*x]^2*(
6*a^3*e*f + 6*a*b^2*e*f - 12*a*b^2*d^2*e^2*x + 6*a^3*f^2*x + 6*a*b^2*f^2*x - 12*a*b^2*d^2*e*f*x^2 - 4*a*b^2*d^
2*f^2*x^3 - 6*a^3*e*f*Cosh[2*c] - 6*a*b^2*e*f*Cosh[2*c] - 6*a^3*f^2*x*Cosh[2*c] - 6*a*b^2*f^2*x*Cosh[2*c] - 6*
a^3*e*f*Cosh[2*d*x] - 6*a*b^2*e*f*Cosh[2*d*x] - 6*a^3*f^2*x*Cosh[2*d*x] - 6*a*b^2*f^2*x*Cosh[2*d*x] - 3*a^2*b*
d*e^2*Cosh[c - d*x] - 3*b^3*d*e^2*Cosh[c - d*x] - 6*a^2*b*d*e*f*x*Cosh[c - d*x] - 6*b^3*d*e*f*x*Cosh[c - d*x]
- 3*a^2*b*d*f^2*x^2*Cosh[c - d*x] - 3*b^3*d*f^2*x^2*Cosh[c - d*x] + 3*a^2*b*d*e^2*Cosh[3*c + d*x] + 3*b^3*d*e^
2*Cosh[3*c + d*x] + 6*a^2*b*d*e*f*x*Cosh[3*c + d*x] + 6*b^3*d*e*f*x*Cosh[3*c + d*x] + 3*a^2*b*d*f^2*x^2*Cosh[3
*c + d*x] + 3*b^3*d*f^2*x^2*Cosh[3*c + d*x] + 6*a^3*e*f*Cosh[2*c + 2*d*x] + 6*a*b^2*e*f*Cosh[2*c + 2*d*x] - 12
*a*b^2*d^2*e^2*x*Cosh[2*c + 2*d*x] + 6*a^3*f^2*x*Cosh[2*c + 2*d*x] + 6*a*b^2*f^2*x*Cosh[2*c + 2*d*x] - 12*a*b^
2*d^2*e*f*x^2*Cosh[2*c + 2*d*x] - 4*a*b^2*d^2*f^2*x^3*Cosh[2*c + 2*d*x] - 6*a^3*d*e^2*Sinh[2*c] - 6*a*b^2*d*e^
2*Sinh[2*c] - 12*a^3*d*e*f*x*Sinh[2*c] - 12*a*b^2*d*e*f*x*Sinh[2*c] - 6*a^3*d*f^2*x^2*Sinh[2*c] - 6*a*b^2*d*f^
2*x^2*Sinh[2*c] + 6*a^2*b*e*f*Sinh[c - d*x] + 6*b^3*e*f*Sinh[c - d*x] + 6*a^2*b*f^2*x*Sinh[c - d*x] + 6*b^3*f^
2*x*Sinh[c - d*x] + 6*a^2*b*e*f*Sinh[3*c + d*x] + 6*b^3*e*f*Sinh[3*c + d*x] + 6*a^2*b*f^2*x*Sinh[3*c + d*x] +
6*b^3*f^2*x*Sinh[3*c + d*x]))/(24*(a^2 + b^2)^2*d^2)
________________________________________________________________________________________
Maple [F] time = 0.454, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}\tanh \left ( dx+c \right ) }{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((f*x+e)^2*sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
[Out]
int((f*x+e)^2*sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((f*x+e)^2*sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-a^2*b*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^
(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + b^3*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d
*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) -
2*a*b^2*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2
*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 2*a^2*b*d^2*e*f*integrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c
) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 2*b^3*d
^2*e*f*integrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2
*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 4*a*b^2*d^2*e*f*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^
2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + a^3*f^2*(2*(d*x + c
)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) + a*b^2*f^2*(2*(d*x
+ c)/((a^4 + 2*a^2*b^2 + b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - (a*b^2*log(-2*a
*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b^4)*d) - a*b^2*log(e^(-2*d*x - 2*c) + 1)/((a^4 +
2*a^2*b^2 + b^4)*d) - (a^2*b - b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) - (b*e^(-d*x - c) - 2*a*e
^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4
*c))*d))*e^2 - 2*a^2*b*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) - 2*b^3*f^2*arctan(e^(d*x + c))/(
(a^4 + 2*a^2*b^2 + b^4)*d^3) - (2*a*f^2*x + 2*a*e*f - (b*d*f^2*x^2*e^(3*c) + 2*b*e*f*e^(3*c) + 2*(d*e*f + f^2)
*b*x*e^(3*c))*e^(3*d*x) + 2*(a*d*f^2*x^2*e^(2*c) + a*e*f*e^(2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) + (b
*d*f^2*x^2*e^c - 2*b*e*f*e^c + 2*(d*e*f - f^2)*b*x*e^c)*e^(d*x))/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2*d
^2*e^(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c))*e^(2*d*x)) + integrate(2*(a*b^3*f^2*x^2 + 2*a*b^
3*e*f*x - (a^2*b^2*f^2*x^2*e^c + 2*a^2*b^2*e*f*x*e^c)*e^(d*x))/(a^4*b + 2*a^2*b^3 + b^5 - (a^4*b*e^(2*c) + 2*a
^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^5*e^c + 2*a^3*b^2*e^c + a*b^4*e^c)*e^(d*x)), x)
________________________________________________________________________________________
Fricas [C] time = 5.5478, size = 24520, normalized size = 20.85 \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*sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")
[Out]
1/2*(4*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*c*f^2)*cosh(d*x + c)^4 + 4*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2
)*c*f^2)*sinh(d*x + c)^4 - 4*(a^3 + a*b^2)*d*e*f + 4*(a^3 + a*b^2)*c*f^2 + 2*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2
*b + b^3)*d^2*e^2 + 2*(a^2*b + b^3)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f + (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^3
+ 2*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 + 2*(a^2*b + b^3)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f + (
a^2*b + b^3)*d*f^2)*x + 8*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*((a
^3 + a*b^2)*d^2*f^2*x^2 + (a^3 + a*b^2)*d^2*e^2 + (a^3 + a*b^2)*d*e*f - 2*(a^3 + a*b^2)*c*f^2 + (2*(a^3 + a*b^
2)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c)^2 - 2*(2*(a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e^
2 + 2*(a^3 + a*b^2)*d*e*f - 4*(a^3 + a*b^2)*c*f^2 - 12*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*c*f^2)*cosh(d*x
+ c)^2 + 2*(2*(a^3 + a*b^2)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x - 3*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^
2*e^2 + 2*(a^2*b + b^3)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f + (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c
)^2 - 2*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f
- (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c) - 4*(a*b^2*d*f^2*x + a*b^2*d*e*f + (a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(
d*x + c)^4 + 4*(a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*f^2*x + a*b^2*d*e*f)*sin
h(d*x + c)^4 + 2*(a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f^2*x + a*b^2*d*e*f + 3*(a*b^2*d*f
^2*x + a*b^2*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)^3 + (a*b
^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x
+ c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 4*(a*b^2*d*f^2*x + a*b^2*d*e*f + (a*b^2*d*f^2*x +
a*b^2*d*e*f)*cosh(d*x + c)^4 + 4*(a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d*f^2*x
+ a*b^2*d*e*f)*sinh(d*x + c)^4 + 2*(a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)^2 + 2*(a*b^2*d*f^2*x + a*b^2*d
*e*f + 3*(a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d*f^2*x + a*b^2*d*e*f)*cos
h(d*x + c)^3 + (a*b^2*d*f^2*x + a*b^2*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x
+ c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + (4*a*b^2*d*f^2*x + 4*a*b^2*d*e
*f - 2*I*(a^2*b - b^3)*d*f^2*x + (4*a*b^2*d*f^2*x + 4*a*b^2*d*e*f - 2*I*(a^2*b - b^3)*d*f^2*x - 2*I*(a^2*b - b
^3)*d*e*f)*cosh(d*x + c)^4 + (16*a*b^2*d*f^2*x + 16*a*b^2*d*e*f - 8*I*(a^2*b - b^3)*d*f^2*x - 8*I*(a^2*b - b^3
)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a*b^2*d*f^2*x + 4*a*b^2*d*e*f - 2*I*(a^2*b - b^3)*d*f^2*x - 2*I*(a
^2*b - b^3)*d*e*f)*sinh(d*x + c)^4 - 2*I*(a^2*b - b^3)*d*e*f + (8*a*b^2*d*f^2*x + 8*a*b^2*d*e*f - 4*I*(a^2*b -
b^3)*d*f^2*x - 4*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)^2 + (8*a*b^2*d*f^2*x + 8*a*b^2*d*e*f - 4*I*(a^2*b - b^3
)*d*f^2*x - 4*I*(a^2*b - b^3)*d*e*f + (24*a*b^2*d*f^2*x + 24*a*b^2*d*e*f - 12*I*(a^2*b - b^3)*d*f^2*x - 12*I*(
a^2*b - b^3)*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a*b^2*d*f^2*x + 16*a*b^2*d*e*f - 8*I*(a^2*b - b^3)
*d*f^2*x - 8*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)^3 + (16*a*b^2*d*f^2*x + 16*a*b^2*d*e*f - 8*I*(a^2*b - b^3)*d
*f^2*x - 8*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + (4*
a*b^2*d*f^2*x + 4*a*b^2*d*e*f + 2*I*(a^2*b - b^3)*d*f^2*x + (4*a*b^2*d*f^2*x + 4*a*b^2*d*e*f + 2*I*(a^2*b - b^
3)*d*f^2*x + 2*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)^4 + (16*a*b^2*d*f^2*x + 16*a*b^2*d*e*f + 8*I*(a^2*b - b^3)
*d*f^2*x + 8*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a*b^2*d*f^2*x + 4*a*b^2*d*e*f + 2*I*(a^
2*b - b^3)*d*f^2*x + 2*I*(a^2*b - b^3)*d*e*f)*sinh(d*x + c)^4 + 2*I*(a^2*b - b^3)*d*e*f + (8*a*b^2*d*f^2*x + 8
*a*b^2*d*e*f + 4*I*(a^2*b - b^3)*d*f^2*x + 4*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)^2 + (8*a*b^2*d*f^2*x + 8*a*b
^2*d*e*f + 4*I*(a^2*b - b^3)*d*f^2*x + 4*I*(a^2*b - b^3)*d*e*f + (24*a*b^2*d*f^2*x + 24*a*b^2*d*e*f + 12*I*(a^
2*b - b^3)*d*f^2*x + 12*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a*b^2*d*f^2*x + 16*a*b^
2*d*e*f + 8*I*(a^2*b - b^3)*d*f^2*x + 8*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c)^3 + (16*a*b^2*d*f^2*x + 16*a*b^2*
d*e*f + 8*I*(a^2*b - b^3)*d*f^2*x + 8*I*(a^2*b - b^3)*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog(-I*cosh(d*x +
c) - I*sinh(d*x + c)) - 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f
+ a*b^2*c^2*f^2)*cosh(d*x + c)^4 + 4*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x
+ c)^3 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sinh(d*x + c)^4 + 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e
*f + a*b^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2 + 3*(a*b^2*d^2*e^2 -
2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^
2*c^2*f^2)*cosh(d*x + c)^3 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*l
og(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) - 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e
*f + a*b^2*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)^4 + 4*(a*b^2*d^2*e^2 - 2*
a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^
2)*sinh(d*x + c)^4 + 2*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b^2*d^2*e^2 -
2*a*b^2*c*d*e*f + a*b^2*c^2*f^2 + 3*(a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + 4*((a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*cosh(d*x + c)^3 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d
*e*f + a*b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*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*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2 + (a*b^2*d^2*f
^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^4 + 4*(a*b^2*d^2*f^2*x^2 + 2*a*b^2
*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2
*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*sinh(d*x + c)^4 + 2*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2
*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^2 + 2*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2
*c^2*f^2 + 3*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^2)*sinh(d
*x + c)^2 + 4*((a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^3 + (a*
b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*c
osh(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*(a*b^2*
d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2 + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2
*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^4 + 4*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f -
a*b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b
^2*c^2*f^2)*sinh(d*x + c)^4 + 2*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh
(d*x + c)^2 + 2*(a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2 + 3*(a*b^2*d^2*f^2*x^
2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a*b^2*d^2*f^2*
x^2 + 2*a*b^2*d^2*e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c)^3 + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*d^2*
e*f*x + 2*a*b^2*c*d*e*f - a*b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*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*a*b^2*d^2*e^2 - 4*a*b^2*c*d*e*f - I*
(a^2*b - b^3)*d^2*e^2 + 2*I*(a^2*b - b^3)*c*d*e*f + (2*a*b^2*d^2*e^2 - 4*a*b^2*c*d*e*f - I*(a^2*b - b^3)*d^2*e
^2 + 2*I*(a^2*b - b^3)*c*d*e*f + 2*(a*b^2*c^2 - a^3 - a*b^2)*f^2 - I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2
)*cosh(d*x + c)^4 + (8*a*b^2*d^2*e^2 - 16*a*b^2*c*d*e*f - 4*I*(a^2*b - b^3)*d^2*e^2 + 8*I*(a^2*b - b^3)*c*d*e*
f + 8*(a*b^2*c^2 - a^3 - a*b^2)*f^2 - 4*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c)*sinh(d*x +
c)^3 + (2*a*b^2*d^2*e^2 - 4*a*b^2*c*d*e*f - I*(a^2*b - b^3)*d^2*e^2 + 2*I*(a^2*b - b^3)*c*d*e*f + 2*(a*b^2*c^2
- a^3 - a*b^2)*f^2 - I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*sinh(d*x + c)^4 + 2*(a*b^2*c^2 - a^3 - a*b^
2)*f^2 - I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2 + (4*a*b^2*d^2*e^2 - 8*a*b^2*c*d*e*f - 2*I*(a^2*b - b^3)*
d^2*e^2 + 4*I*(a^2*b - b^3)*c*d*e*f + 4*(a*b^2*c^2 - a^3 - a*b^2)*f^2 - 2*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c
^2)*f^2)*cosh(d*x + c)^2 + (4*a*b^2*d^2*e^2 - 8*a*b^2*c*d*e*f - 2*I*(a^2*b - b^3)*d^2*e^2 + 4*I*(a^2*b - b^3)*
c*d*e*f + 4*(a*b^2*c^2 - a^3 - a*b^2)*f^2 - 2*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2 + (12*a*b^2*d^2*e^2
- 24*a*b^2*c*d*e*f - 6*I*(a^2*b - b^3)*d^2*e^2 + 12*I*(a^2*b - b^3)*c*d*e*f + 12*(a*b^2*c^2 - a^3 - a*b^2)*f^2
- 6*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a*b^2*d^2*e^2 - 16*a*
b^2*c*d*e*f - 4*I*(a^2*b - b^3)*d^2*e^2 + 8*I*(a^2*b - b^3)*c*d*e*f + 8*(a*b^2*c^2 - a^3 - a*b^2)*f^2 - 4*I*(2
*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c)^3 + (8*a*b^2*d^2*e^2 - 16*a*b^2*c*d*e*f - 4*I*(a^2*b -
b^3)*d^2*e^2 + 8*I*(a^2*b - b^3)*c*d*e*f + 8*(a*b^2*c^2 - a^3 - a*b^2)*f^2 - 4*I*(2*a^2*b + 2*b^3 + (a^2*b - b
^3)*c^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) + (2*a*b^2*d^2*e^2 - 4*a*b^
2*c*d*e*f + I*(a^2*b - b^3)*d^2*e^2 - 2*I*(a^2*b - b^3)*c*d*e*f + (2*a*b^2*d^2*e^2 - 4*a*b^2*c*d*e*f + I*(a^2*
b - b^3)*d^2*e^2 - 2*I*(a^2*b - b^3)*c*d*e*f + 2*(a*b^2*c^2 - a^3 - a*b^2)*f^2 + I*(2*a^2*b + 2*b^3 + (a^2*b -
b^3)*c^2)*f^2)*cosh(d*x + c)^4 + (8*a*b^2*d^2*e^2 - 16*a*b^2*c*d*e*f + 4*I*(a^2*b - b^3)*d^2*e^2 - 8*I*(a^2*b
- b^3)*c*d*e*f + 8*(a*b^2*c^2 - a^3 - a*b^2)*f^2 + 4*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x +
c)*sinh(d*x + c)^3 + (2*a*b^2*d^2*e^2 - 4*a*b^2*c*d*e*f + I*(a^2*b - b^3)*d^2*e^2 - 2*I*(a^2*b - b^3)*c*d*e*f
+ 2*(a*b^2*c^2 - a^3 - a*b^2)*f^2 + I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*sinh(d*x + c)^4 + 2*(a*b^2*c^
2 - a^3 - a*b^2)*f^2 + I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2 + (4*a*b^2*d^2*e^2 - 8*a*b^2*c*d*e*f + 2*I*
(a^2*b - b^3)*d^2*e^2 - 4*I*(a^2*b - b^3)*c*d*e*f + 4*(a*b^2*c^2 - a^3 - a*b^2)*f^2 + 2*I*(2*a^2*b + 2*b^3 + (
a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c)^2 + (4*a*b^2*d^2*e^2 - 8*a*b^2*c*d*e*f + 2*I*(a^2*b - b^3)*d^2*e^2 - 4*I*
(a^2*b - b^3)*c*d*e*f + 4*(a*b^2*c^2 - a^3 - a*b^2)*f^2 + 2*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2 + (12*
a*b^2*d^2*e^2 - 24*a*b^2*c*d*e*f + 6*I*(a^2*b - b^3)*d^2*e^2 - 12*I*(a^2*b - b^3)*c*d*e*f + 12*(a*b^2*c^2 - a^
3 - a*b^2)*f^2 + 6*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a*b^2*d
^2*e^2 - 16*a*b^2*c*d*e*f + 4*I*(a^2*b - b^3)*d^2*e^2 - 8*I*(a^2*b - b^3)*c*d*e*f + 8*(a*b^2*c^2 - a^3 - a*b^2
)*f^2 + 4*I*(2*a^2*b + 2*b^3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c)^3 + (8*a*b^2*d^2*e^2 - 16*a*b^2*c*d*e*f +
4*I*(a^2*b - b^3)*d^2*e^2 - 8*I*(a^2*b - b^3)*c*d*e*f + 8*(a*b^2*c^2 - a^3 - a*b^2)*f^2 + 4*I*(2*a^2*b + 2*b^
3 + (a^2*b - b^3)*c^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) + (2*a*b^2*d^
2*f^2*x^2 + 4*a*b^2*d^2*e*f*x + 4*a*b^2*c*d*e*f - 2*a*b^2*c^2*f^2 + I*(a^2*b - b^3)*d^2*f^2*x^2 + 2*I*(a^2*b -
b^3)*d^2*e*f*x + 2*I*(a^2*b - b^3)*c*d*e*f - I*(a^2*b - b^3)*c^2*f^2 + (2*a*b^2*d^2*f^2*x^2 + 4*a*b^2*d^2*e*f
*x + 4*a*b^2*c*d*e*f - 2*a*b^2*c^2*f^2 + I*(a^2*b - b^3)*d^2*f^2*x^2 + 2*I*(a^2*b - b^3)*d^2*e*f*x + 2*I*(a^2*
b - b^3)*c*d*e*f - I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)^4 + (8*a*b^2*d^2*f^2*x^2 + 16*a*b^2*d^2*e*f*x + 16*a
*b^2*c*d*e*f - 8*a*b^2*c^2*f^2 + 4*I*(a^2*b - b^3)*d^2*f^2*x^2 + 8*I*(a^2*b - b^3)*d^2*e*f*x + 8*I*(a^2*b - b^
3)*c*d*e*f - 4*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*d^2*f^2*x^2 + 4*a*b^2*d^2*e*f
*x + 4*a*b^2*c*d*e*f - 2*a*b^2*c^2*f^2 + I*(a^2*b - b^3)*d^2*f^2*x^2 + 2*I*(a^2*b - b^3)*d^2*e*f*x + 2*I*(a^2*
b - b^3)*c*d*e*f - I*(a^2*b - b^3)*c^2*f^2)*sinh(d*x + c)^4 + (4*a*b^2*d^2*f^2*x^2 + 8*a*b^2*d^2*e*f*x + 8*a*b
^2*c*d*e*f - 4*a*b^2*c^2*f^2 + 2*I*(a^2*b - b^3)*d^2*f^2*x^2 + 4*I*(a^2*b - b^3)*d^2*e*f*x + 4*I*(a^2*b - b^3)
*c*d*e*f - 2*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)^2 + (4*a*b^2*d^2*f^2*x^2 + 8*a*b^2*d^2*e*f*x + 8*a*b^2*c*d
*e*f - 4*a*b^2*c^2*f^2 + 2*I*(a^2*b - b^3)*d^2*f^2*x^2 + 4*I*(a^2*b - b^3)*d^2*e*f*x + 4*I*(a^2*b - b^3)*c*d*e
*f - 2*I*(a^2*b - b^3)*c^2*f^2 + (12*a*b^2*d^2*f^2*x^2 + 24*a*b^2*d^2*e*f*x + 24*a*b^2*c*d*e*f - 12*a*b^2*c^2*
f^2 + 6*I*(a^2*b - b^3)*d^2*f^2*x^2 + 12*I*(a^2*b - b^3)*d^2*e*f*x + 12*I*(a^2*b - b^3)*c*d*e*f - 6*I*(a^2*b -
b^3)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a*b^2*d^2*f^2*x^2 + 16*a*b^2*d^2*e*f*x + 16*a*b^2*c*d*e*
f - 8*a*b^2*c^2*f^2 + 4*I*(a^2*b - b^3)*d^2*f^2*x^2 + 8*I*(a^2*b - b^3)*d^2*e*f*x + 8*I*(a^2*b - b^3)*c*d*e*f
- 4*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)^3 + (8*a*b^2*d^2*f^2*x^2 + 16*a*b^2*d^2*e*f*x + 16*a*b^2*c*d*e*f -
8*a*b^2*c^2*f^2 + 4*I*(a^2*b - b^3)*d^2*f^2*x^2 + 8*I*(a^2*b - b^3)*d^2*e*f*x + 8*I*(a^2*b - b^3)*c*d*e*f - 4*
I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (2*a*b^2*d
^2*f^2*x^2 + 4*a*b^2*d^2*e*f*x + 4*a*b^2*c*d*e*f - 2*a*b^2*c^2*f^2 - I*(a^2*b - b^3)*d^2*f^2*x^2 - 2*I*(a^2*b
- b^3)*d^2*e*f*x - 2*I*(a^2*b - b^3)*c*d*e*f + I*(a^2*b - b^3)*c^2*f^2 + (2*a*b^2*d^2*f^2*x^2 + 4*a*b^2*d^2*e*
f*x + 4*a*b^2*c*d*e*f - 2*a*b^2*c^2*f^2 - I*(a^2*b - b^3)*d^2*f^2*x^2 - 2*I*(a^2*b - b^3)*d^2*e*f*x - 2*I*(a^2
*b - b^3)*c*d*e*f + I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)^4 + (8*a*b^2*d^2*f^2*x^2 + 16*a*b^2*d^2*e*f*x + 16*
a*b^2*c*d*e*f - 8*a*b^2*c^2*f^2 - 4*I*(a^2*b - b^3)*d^2*f^2*x^2 - 8*I*(a^2*b - b^3)*d^2*e*f*x - 8*I*(a^2*b - b
^3)*c*d*e*f + 4*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (2*a*b^2*d^2*f^2*x^2 + 4*a*b^2*d^2*e*
f*x + 4*a*b^2*c*d*e*f - 2*a*b^2*c^2*f^2 - I*(a^2*b - b^3)*d^2*f^2*x^2 - 2*I*(a^2*b - b^3)*d^2*e*f*x - 2*I*(a^2
*b - b^3)*c*d*e*f + I*(a^2*b - b^3)*c^2*f^2)*sinh(d*x + c)^4 + (4*a*b^2*d^2*f^2*x^2 + 8*a*b^2*d^2*e*f*x + 8*a*
b^2*c*d*e*f - 4*a*b^2*c^2*f^2 - 2*I*(a^2*b - b^3)*d^2*f^2*x^2 - 4*I*(a^2*b - b^3)*d^2*e*f*x - 4*I*(a^2*b - b^3
)*c*d*e*f + 2*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)^2 + (4*a*b^2*d^2*f^2*x^2 + 8*a*b^2*d^2*e*f*x + 8*a*b^2*c*
d*e*f - 4*a*b^2*c^2*f^2 - 2*I*(a^2*b - b^3)*d^2*f^2*x^2 - 4*I*(a^2*b - b^3)*d^2*e*f*x - 4*I*(a^2*b - b^3)*c*d*
e*f + 2*I*(a^2*b - b^3)*c^2*f^2 + (12*a*b^2*d^2*f^2*x^2 + 24*a*b^2*d^2*e*f*x + 24*a*b^2*c*d*e*f - 12*a*b^2*c^2
*f^2 - 6*I*(a^2*b - b^3)*d^2*f^2*x^2 - 12*I*(a^2*b - b^3)*d^2*e*f*x - 12*I*(a^2*b - b^3)*c*d*e*f + 6*I*(a^2*b
- b^3)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((8*a*b^2*d^2*f^2*x^2 + 16*a*b^2*d^2*e*f*x + 16*a*b^2*c*d*e
*f - 8*a*b^2*c^2*f^2 - 4*I*(a^2*b - b^3)*d^2*f^2*x^2 - 8*I*(a^2*b - b^3)*d^2*e*f*x - 8*I*(a^2*b - b^3)*c*d*e*f
+ 4*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c)^3 + (8*a*b^2*d^2*f^2*x^2 + 16*a*b^2*d^2*e*f*x + 16*a*b^2*c*d*e*f -
8*a*b^2*c^2*f^2 - 4*I*(a^2*b - b^3)*d^2*f^2*x^2 - 8*I*(a^2*b - b^3)*d^2*e*f*x - 8*I*(a^2*b - b^3)*c*d*e*f + 4
*I*(a^2*b - b^3)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) + 4*(a*b^2
*f^2*cosh(d*x + c)^4 + 4*a*b^2*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + a*b^2*f^2*sinh(d*x + c)^4 + 2*a*b^2*f^2*cos
h(d*x + c)^2 + a*b^2*f^2 + 2*(3*a*b^2*f^2*cosh(d*x + c)^2 + a*b^2*f^2)*sinh(d*x + c)^2 + 4*(a*b^2*f^2*cosh(d*x
+ c)^3 + a*b^2*f^2*cosh(d*x + c))*sinh(d*x + c))*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*b^2*f^2*cosh(d*x + c)^4 + 4*a*b^2*f^2*cosh(d*x + c)*s
inh(d*x + c)^3 + a*b^2*f^2*sinh(d*x + c)^4 + 2*a*b^2*f^2*cosh(d*x + c)^2 + a*b^2*f^2 + 2*(3*a*b^2*f^2*cosh(d*x
+ c)^2 + a*b^2*f^2)*sinh(d*x + c)^2 + 4*(a*b^2*f^2*cosh(d*x + c)^3 + a*b^2*f^2*cosh(d*x + c))*sinh(d*x + c))*
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*b^2*f^2 + (4*a*b^2*f^2 - 2*I*(a^2*b - b^3)*f^2)*cosh(d*x + c)^4 + (16*a*b^2*f^2 - 8*I*(a^2*b - b^3)*f^2
)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a*b^2*f^2 - 2*I*(a^2*b - b^3)*f^2)*sinh(d*x + c)^4 - 2*I*(a^2*b - b^3)*f^
2 + (8*a*b^2*f^2 - 4*I*(a^2*b - b^3)*f^2)*cosh(d*x + c)^2 + (8*a*b^2*f^2 - 4*I*(a^2*b - b^3)*f^2 + (24*a*b^2*f
^2 - 12*I*(a^2*b - b^3)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a*b^2*f^2 - 8*I*(a^2*b - b^3)*f^2)*cosh(d
*x + c)^3 + (16*a*b^2*f^2 - 8*I*(a^2*b - b^3)*f^2)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, I*cosh(d*x + c) +
I*sinh(d*x + c)) - (4*a*b^2*f^2 + (4*a*b^2*f^2 + 2*I*(a^2*b - b^3)*f^2)*cosh(d*x + c)^4 + (16*a*b^2*f^2 + 8*I*
(a^2*b - b^3)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (4*a*b^2*f^2 + 2*I*(a^2*b - b^3)*f^2)*sinh(d*x + c)^4 + 2*I
*(a^2*b - b^3)*f^2 + (8*a*b^2*f^2 + 4*I*(a^2*b - b^3)*f^2)*cosh(d*x + c)^2 + (8*a*b^2*f^2 + 4*I*(a^2*b - b^3)*
f^2 + (24*a*b^2*f^2 + 12*I*(a^2*b - b^3)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + ((16*a*b^2*f^2 + 8*I*(a^2*b -
b^3)*f^2)*cosh(d*x + c)^3 + (16*a*b^2*f^2 + 8*I*(a^2*b - b^3)*f^2)*cosh(d*x + c))*sinh(d*x + c))*polylog(3, -
I*cosh(d*x + c) - I*sinh(d*x + c)) - 2*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 - 2*(a^2*b + b^3)*d*
e*f - 8*((a^3 + a*b^2)*d*f^2*x + (a^3 + a*b^2)*c*f^2)*cosh(d*x + c)^3 - 3*((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b
+ b^3)*d^2*e^2 + 2*(a^2*b + b^3)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f + (a^2*b + b^3)*d*f^2)*x)*cosh(d*x + c)^2 +
2*((a^2*b + b^3)*d^2*e*f - (a^2*b + b^3)*d*f^2)*x + 4*((a^3 + a*b^2)*d^2*f^2*x^2 + (a^3 + a*b^2)*d^2*e^2 + (a^
3 + a*b^2)*d*e*f - 2*(a^3 + a*b^2)*c*f^2 + (2*(a^3 + a*b^2)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c))*s
inh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)*sinh(
d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^3*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^2 + (a^
4 + 2*a^2*b^2 + b^4)*d^3 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*d^3)*sin
h(d*x + c)^2 + 4*((a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d^3*cosh(d*x + c))*sin
h(d*x + c))
________________________________________________________________________________________
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*sech(d*x+c)**2*tanh(d*x+c)/(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*sech(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out