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

Optimal. Leaf size=1118 \[ \text{result too large to display} \]

[Out]

-((a*(e + f*x)^3)/(b^2*d)) + (a^3*(e + f*x)^3)/(b^2*(a^2 + b^2)*d) + (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*
d^2) - (6*a^2*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x)
)/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
])])/((a^2 + b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*d^2) - (3*a^3*f*(e + f*x)^2*Log
[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^2) - ((6*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^3) + ((
6*I)*a^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[2, I*E
^(c + d*x)])/(b*d^3) - ((6*I)*a^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (3*a^2*f*(e +
f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*a^2*f*(e + f*x)^2*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (3*a*f^2*(e + f*x)*PolyLog[2, -
E^(2*(c + d*x))])/(b^2*d^3) - (3*a^3*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^3) + ((6*I
)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^4) - ((6*I)*a^2*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^4)
- ((6*I)*f^3*PolyLog[3, I*E^(c + d*x)])/(b*d^4) + ((6*I)*a^2*f^3*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^
4) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (6*a
^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (3*a*f^3*Poly
Log[3, -E^(2*(c + d*x))])/(2*b^2*d^4) + (3*a^3*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^4) + (6*
a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - (6*a^2*f^3*PolyLog[4,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - ((e + f*x)^3*Sech[c + d*x])/(b*d) + (a^2*
(e + f*x)^3*Sech[c + d*x])/(b*(a^2 + b^2)*d) - (a*(e + f*x)^3*Tanh[c + d*x])/(b^2*d) + (a^3*(e + f*x)^3*Tanh[c
+ d*x])/(b^2*(a^2 + b^2)*d)

________________________________________________________________________________________

Rubi [A]  time = 1.98044, antiderivative size = 1118, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 15, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.536, Rules used = {5567, 5451, 4180, 2531, 2282, 6589, 5583, 4184, 3718, 2190, 5573, 3322, 2264, 6609, 6742} \[ \frac{(e+f x)^3 a^3}{b^2 \left (a^2+b^2\right ) d}-\frac{3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac{3 f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac{3 f^3 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^4}+\frac{(e+f x)^3 \tanh (c+d x) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac{6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^2}+\frac{(e+f x)^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^{3/2} d}-\frac{(e+f x)^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^2}{\left (a^2+b^2\right )^{3/2} d}+\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}-\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^3}+\frac{3 f (e+f x)^2 \text{PolyLog}\left (2,-\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{3 f (e+f x)^2 \text{PolyLog}\left (2,-\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{6 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^4}+\frac{6 i f^3 \text{PolyLog}\left (3,i e^{c+d x}\right ) a^2}{b \left (a^2+b^2\right ) d^4}-\frac{6 f^2 (e+f x) \text{PolyLog}\left (3,-\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{6 f^2 (e+f x) \text{PolyLog}\left (3,-\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{6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}-\frac{6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}+\frac{(e+f x)^3 \text{sech}(c+d x) a^2}{b \left (a^2+b^2\right ) d}-\frac{(e+f x)^3 a}{b^2 d}+\frac{3 f (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{b^2 d^2}+\frac{3 f^2 (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a}{b^2 d^3}-\frac{3 f^3 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) a}{2 b^2 d^4}-\frac{(e+f x)^3 \tanh (c+d x) a}{b^2 d}+\frac{6 f (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d^2}-\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b d^3}+\frac{6 i f^2 (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{b d^3}+\frac{6 i f^3 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{b d^4}-\frac{6 i f^3 \text{PolyLog}\left (3,i e^{c+d x}\right )}{b d^4}-\frac{(e+f x)^3 \text{sech}(c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*(e + f*x)^3)/(b^2*d)) + (a^3*(e + f*x)^3)/(b^2*(a^2 + b^2)*d) + (6*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*
d^2) - (6*a^2*f*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b*(a^2 + b^2)*d^2) + (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x)
)/(a - Sqrt[a^2 + b^2])])/((a^2 + b^2)^(3/2)*d) - (a^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
])])/((a^2 + b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*d^2) - (3*a^3*f*(e + f*x)^2*Log
[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^2) - ((6*I)*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^3) + ((
6*I)*a^2*f^2*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[2, I*E
^(c + d*x)])/(b*d^3) - ((6*I)*a^2*f^2*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^3) + (3*a^2*f*(e +
f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) - (3*a^2*f*(e + f*x)^2*P
olyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^2) + (3*a*f^2*(e + f*x)*PolyLog[2, -
E^(2*(c + d*x))])/(b^2*d^3) - (3*a^3*f^2*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d^3) + ((6*I
)*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^4) - ((6*I)*a^2*f^3*PolyLog[3, (-I)*E^(c + d*x)])/(b*(a^2 + b^2)*d^4)
- ((6*I)*f^3*PolyLog[3, I*E^(c + d*x)])/(b*d^4) + ((6*I)*a^2*f^3*PolyLog[3, I*E^(c + d*x)])/(b*(a^2 + b^2)*d^
4) - (6*a^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) + (6*a
^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^3) - (3*a*f^3*Poly
Log[3, -E^(2*(c + d*x))])/(2*b^2*d^4) + (3*a^3*f^3*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^4) + (6*
a^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - (6*a^2*f^3*PolyLog[4,
-((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/((a^2 + b^2)^(3/2)*d^4) - ((e + f*x)^3*Sech[c + d*x])/(b*d) + (a^2*
(e + f*x)^3*Sech[c + d*x])/(b*(a^2 + b^2)*d) - (a*(e + f*x)^3*Tanh[c + d*x])/(b^2*d) + (a^3*(e + f*x)^3*Tanh[c
+ d*x])/(b^2*(a^2 + b^2)*d)

Rule 5567

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sec
h[c + d*x]*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]

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 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 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 4184

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

Rule 3718

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

Rule 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 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 6609

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

Rule 6742

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

Rubi steps

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

Mathematica [A]  time = 13.669, size = 1147, normalized size = 1.03 \[ \frac{\left (-2 e^3 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^3+f^3 x^3 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3+3 e^2 f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^3-f^3 x^3 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3-3 e f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3-3 e^2 f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^3+3 f (e+f x)^2 \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d^2-3 f (e+f x)^2 \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d^2-6 e f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d-6 f^3 x \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d+6 e f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d+6 f^3 x \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d+6 f^3 \text{PolyLog}\left (4,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )-6 f^3 \text{PolyLog}\left (4,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right ) a^2}{\left (a^2+b^2\right )^{3/2} d^4}+\frac{f \left (4 a f^2 x^3 d^3+12 a e f x^2 d^3-12 a e^2 e^{2 c} x d^3+12 a e^2 \left (1+e^{2 c}\right ) x d^3+12 b e^2 \left (1+e^{2 c}\right ) \tan ^{-1}\left (e^{c+d x}\right ) d^2-6 a e^2 \left (1+e^{2 c}\right ) \left (2 d x-\log \left (1+e^{2 (c+d x)}\right )\right ) d^2+12 i b e \left (1+e^{2 c}\right ) f \left (d x \left (\log \left (1-i e^{c+d x}\right )-\log \left (1+i e^{c+d x}\right )\right )-\text{PolyLog}\left (2,-i e^{c+d x}\right )+\text{PolyLog}\left (2,i e^{c+d x}\right )\right ) d-6 a e \left (1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1+e^{2 (c+d x)}\right )\right )-\text{PolyLog}\left (2,-e^{2 (c+d x)}\right )\right ) d+6 i b \left (1+e^{2 c}\right ) f^2 \left (d^2 \log \left (1-i e^{c+d x}\right ) x^2-d^2 \log \left (1+i e^{c+d x}\right ) x^2-2 d \text{PolyLog}\left (2,-i e^{c+d x}\right ) x+2 d \text{PolyLog}\left (2,i e^{c+d x}\right ) x+2 \text{PolyLog}\left (3,-i e^{c+d x}\right )-2 \text{PolyLog}\left (3,i e^{c+d x}\right )\right )-a \left (1+e^{2 c}\right ) f^2 \left (2 d^2 \left (2 d x-3 \log \left (1+e^{2 (c+d x)}\right )\right ) x^2-6 d \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) x+3 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right )\right )\right )}{2 \left (a^2+b^2\right ) d^4 \left (1+e^{2 c}\right )}+\frac{\text{sech}(c) \text{sech}(c+d x) \left (-b \cosh (c) e^3-a \sinh (d x) e^3-3 b f x \cosh (c) e^2-3 a f x \sinh (d x) e^2-3 b f^2 x^2 \cosh (c) e-3 a f^2 x^2 \sinh (d x) e-b f^3 x^3 \cosh (c)-a f^3 x^3 \sinh (d x)\right )}{\left (a^2+b^2\right ) d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(f*(-12*a*d^3*e^2*E^(2*c)*x + 12*a*d^3*e^2*(1 + E^(2*c))*x + 12*a*d^3*e*f*x^2 + 4*a*d^3*f^2*x^3 + 12*b*d^2*e^2
*(1 + E^(2*c))*ArcTan[E^(c + d*x)] - 6*a*d^2*e^2*(1 + E^(2*c))*(2*d*x - Log[1 + E^(2*(c + d*x))]) + (12*I)*b*d
*e*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1 + I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)] + Pol
yLog[2, I*E^(c + d*x)]) - 6*a*d*e*(1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(
c + d*x))]) + (6*I)*b*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*Log[1 + I*E^(c + d*x)] - 2*d
*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-I)*E^(c + d*x)] - 2*PolyLog
[3, I*E^(c + d*x)]) - a*(1 + E^(2*c))*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, -
E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))])))/(2*(a^2 + b^2)*d^4*(1 + E^(2*c))) + (a^2*(-2*d^3*e^3*ArcT
anh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^
3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2])] - 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d
*x))/(a + Sqrt[a^2 + b^2])] - d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*f*(e + f*x)^2
*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sq
rt[a^2 + b^2]))] - 6*d*e*f^2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*f^3*x*PolyLog[3, (b*E^(c
+ d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*e*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*f^3*x*
PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])
] - 6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/((a^2 + b^2)^(3/2)*d^4) + (Sech[c]*Sech[c + d
*x]*(-(b*e^3*Cosh[c]) - 3*b*e^2*f*x*Cosh[c] - 3*b*e*f^2*x^2*Cosh[c] - b*f^3*x^3*Cosh[c] - a*e^3*Sinh[d*x] - 3*
a*e^2*f*x*Sinh[d*x] - 3*a*e*f^2*x^2*Sinh[d*x] - a*f^3*x^3*Sinh[d*x]))/((a^2 + b^2)*d)

________________________________________________________________________________________

Maple [F]  time = 0.732, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{2}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [C]  time = 4.22311, size = 14864, normalized size = 13.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*(a^3 + a*b^2)*d^3*e^3 - 12*(a^3 + a*b^2)*c*d^2*e^2*f + 12*(a^3 + a*b^2)*c^2*d*e*f^2 - 4*(a^3 + a*b^2)*c
^3*f^3 - 4*((a^3 + a*b^2)*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + 3*(a^3 +
a*b^2)*c*d^2*e^2*f - 3*(a^3 + a*b^2)*c^2*d*e*f^2 + (a^3 + a*b^2)*c^3*f^3)*cosh(d*x + c)^2 - 4*((a^3 + a*b^2)*
d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + 3*(a^3 + a*b^2)*c*d^2*e^2*f - 3*(a
^3 + a*b^2)*c^2*d*e*f^2 + (a^3 + a*b^2)*c^3*f^3)*sinh(d*x + c)^2 + 6*(a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x
+ a^2*b*d^2*e^2*f + (a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(a^2*b*d^2
*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^2*f^3*x^2 + 2*a^2*b*d
^2*e*f^2*x + a^2*b*d^2*e^2*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c)
+ (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 6*(a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e
*f^2*x + a^2*b*d^2*e^2*f + (a^2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c)^2 + 2*(a^
2*b*d^2*f^3*x^2 + 2*a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^2*f^3*x^2 + 2*
a^2*b*d^2*e*f^2*x + a^2*b*d^2*e^2*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*dilog((a*cosh(d*x + c) + a*sinh(d*
x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(a^2*b*d^3*e^3 - 3*a^2*b*c*
d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3 + (a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 -
a^2*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3
)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3)*si
nh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2
*a) + 2*(a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3 + (a^2*b*d^3*e^3 - 3*a^2*b*
c*d^2*e^2*f + 3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f +
3*a^2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^3*e^3 - 3*a^2*b*c*d^2*e^2*f + 3*a^
2*b*c^2*d*e*f^2 - a^2*b*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x +
c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x +
3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3 + (a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2
*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a^2*b*d^3*f^3
*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3
)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d
^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) +
a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - 2*(a^2*b*d^3*f^3*x^3 +
3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3 + (a^2
*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2
*b*c^3*f^3)*cosh(d*x + c)^2 + 2*(a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d
^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*cosh(d*x + c)*sinh(d*x + c) + (a^2*b*d^3*f^3*x^3 + 3*a^2*b*d^3
*e*f^2*x^2 + 3*a^2*b*d^3*e^2*f*x + 3*a^2*b*c*d^2*e^2*f - 3*a^2*b*c^2*d*e*f^2 + a^2*b*c^3*f^3)*sinh(d*x + c)^2)
*sqrt((a^2 + b^2)/b^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2
+ b^2)/b^2) - b)/b) + 12*(a^2*b*f^3*cosh(d*x + c)^2 + 2*a^2*b*f^3*cosh(d*x + c)*sinh(d*x + c) + a^2*b*f^3*sin
h(d*x + c)^2 + a^2*b*f^3)*sqrt((a^2 + b^2)/b^2)*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x +
c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 12*(a^2*b*f^3*cosh(d*x + c)^2 + 2*a^2*b*f^3*cosh(d*x + c)*si
nh(d*x + c) + a^2*b*f^3*sinh(d*x + c)^2 + a^2*b*f^3)*sqrt((a^2 + b^2)/b^2)*polylog(4, (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) - 12*(a^2*b*d*f^3*x + a^2*b*d*e*f^2
+ (a^2*b*d*f^3*x + a^2*b*d*e*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d*f^3*x + a^2*b*d*e*f^2)*cosh(d*x + c)*sinh(d*x
+ c) + (a^2*b*d*f^3*x + a^2*b*d*e*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) + a*
sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 12*(a^2*b*d*f^3*x + a^2*b*d*e*
f^2 + (a^2*b*d*f^3*x + a^2*b*d*e*f^2)*cosh(d*x + c)^2 + 2*(a^2*b*d*f^3*x + a^2*b*d*e*f^2)*cosh(d*x + c)*sinh(d
*x + c) + (a^2*b*d*f^3*x + a^2*b*d*e*f^2)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) +
a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 4*((a^2*b + b^3)*d^3*f^3*x^
3 + 3*(a^2*b + b^3)*d^3*e*f^2*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + (a^2*b + b^3)*d^3*e^3)*cosh(d*x + c) + (12*(
a^3 + a*b^2)*d*f^3*x + 12*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 + 12*I*(a^2*b + b^3)*d*e*f^2 + (1
2*(a^3 + a*b^2)*d*f^3*x + 12*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 + 12*I*(a^2*b + b^3)*d*e*f^2)*
cosh(d*x + c)^2 + (24*(a^3 + a*b^2)*d*f^3*x + 24*I*(a^2*b + b^3)*d*f^3*x + 24*(a^3 + a*b^2)*d*e*f^2 + 24*I*(a^
2*b + b^3)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (12*(a^3 + a*b^2)*d*f^3*x + 12*I*(a^2*b + b^3)*d*f^3*x + 12*
(a^3 + a*b^2)*d*e*f^2 + 12*I*(a^2*b + b^3)*d*e*f^2)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c))
+ (12*(a^3 + a*b^2)*d*f^3*x - 12*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 - 12*I*(a^2*b + b^3)*d*e*f
^2 + (12*(a^3 + a*b^2)*d*f^3*x - 12*I*(a^2*b + b^3)*d*f^3*x + 12*(a^3 + a*b^2)*d*e*f^2 - 12*I*(a^2*b + b^3)*d*
e*f^2)*cosh(d*x + c)^2 + (24*(a^3 + a*b^2)*d*f^3*x - 24*I*(a^2*b + b^3)*d*f^3*x + 24*(a^3 + a*b^2)*d*e*f^2 - 2
4*I*(a^2*b + b^3)*d*e*f^2)*cosh(d*x + c)*sinh(d*x + c) + (12*(a^3 + a*b^2)*d*f^3*x - 12*I*(a^2*b + b^3)*d*f^3*
x + 12*(a^3 + a*b^2)*d*e*f^2 - 12*I*(a^2*b + b^3)*d*e*f^2)*sinh(d*x + c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*
x + c)) + (6*(a^3 + a*b^2)*d^2*e^2*f + 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b
+ b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*e^2*f + 6*I*(a^2
*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 + 6*
I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*(a^3 + a*b^2)*d^2*e^2*f + 12*I*(a^2*b + b^3)*d^2*e^2*f - 24*(a^
3 + a*b^2)*c*d*e*f^2 - 24*I*(a^2*b + b^3)*c*d*e*f^2 + 12*(a^3 + a*b^2)*c^2*f^3 + 12*I*(a^2*b + b^3)*c^2*f^3)*c
osh(d*x + c)*sinh(d*x + c) + (6*(a^3 + a*b^2)*d^2*e^2*f + 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e
*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*lo
g(cosh(d*x + c) + sinh(d*x + c) + I) + (6*(a^3 + a*b^2)*d^2*e^2*f - 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*
b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3
+ a*b^2)*d^2*e^2*f - 6*I*(a^2*b + b^3)*d^2*e^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 +
6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*(a^3 + a*b^2)*d^2*e^2*f - 12*I*(a^
2*b + b^3)*d^2*e^2*f - 24*(a^3 + a*b^2)*c*d*e*f^2 + 24*I*(a^2*b + b^3)*c*d*e*f^2 + 12*(a^3 + a*b^2)*c^2*f^3 -
12*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*(a^3 + a*b^2)*d^2*e^2*f - 6*I*(a^2*b + b^3)*d^2*e
^2*f - 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*d*e*f^2 + 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)
*c^2*f^3)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - I) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 - 6*I*(a^2*b
+ b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x - 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^
2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*
f^3*x^2 - 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x - 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(
a^3 + a*b^2)*c*d*e*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*c
osh(d*x + c)^2 + (12*(a^3 + a*b^2)*d^2*f^3*x^2 - 12*I*(a^2*b + b^3)*d^2*f^3*x^2 + 24*(a^3 + a*b^2)*d^2*e*f^2*x
- 24*I*(a^2*b + b^3)*d^2*e*f^2*x + 24*(a^3 + a*b^2)*c*d*e*f^2 - 24*I*(a^2*b + b^3)*c*d*e*f^2 - 12*(a^3 + a*b^
2)*c^2*f^3 + 12*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 - 6*I*(a^2
*b + b^3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x - 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e
*f^2 - 12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 + 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*lo
g(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 + 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 12*(
a^3 + a*b^2)*d^2*e*f^2*x + 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)*c*
d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3 + (6*(a^3 + a*b^2)*d^2*f^3*x^2 + 6*I*(a^2*b + b^
3)*d^2*f^3*x^2 + 12*(a^3 + a*b^2)*d^2*e*f^2*x + 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 +
12*I*(a^2*b + b^3)*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3)*cosh(d*x + c)^2 + (12*(a^3
+ a*b^2)*d^2*f^3*x^2 + 12*I*(a^2*b + b^3)*d^2*f^3*x^2 + 24*(a^3 + a*b^2)*d^2*e*f^2*x + 24*I*(a^2*b + b^3)*d^2
*e*f^2*x + 24*(a^3 + a*b^2)*c*d*e*f^2 + 24*I*(a^2*b + b^3)*c*d*e*f^2 - 12*(a^3 + a*b^2)*c^2*f^3 - 12*I*(a^2*b
+ b^3)*c^2*f^3)*cosh(d*x + c)*sinh(d*x + c) + (6*(a^3 + a*b^2)*d^2*f^3*x^2 + 6*I*(a^2*b + b^3)*d^2*f^3*x^2 + 1
2*(a^3 + a*b^2)*d^2*e*f^2*x + 12*I*(a^2*b + b^3)*d^2*e*f^2*x + 12*(a^3 + a*b^2)*c*d*e*f^2 + 12*I*(a^2*b + b^3)
*c*d*e*f^2 - 6*(a^3 + a*b^2)*c^2*f^3 - 6*I*(a^2*b + b^3)*c^2*f^3)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*si
nh(d*x + c) + 1) - 12*((a^3 + a*b^2)*f^3 + I*(a^2*b + b^3)*f^3 + ((a^3 + a*b^2)*f^3 + I*(a^2*b + b^3)*f^3)*cos
h(d*x + c)^2 + 2*((a^3 + a*b^2)*f^3 + I*(a^2*b + b^3)*f^3)*cosh(d*x + c)*sinh(d*x + c) + ((a^3 + a*b^2)*f^3 +
I*(a^2*b + b^3)*f^3)*sinh(d*x + c)^2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) - (12*(a^3 + a*b^2)*f^3 -
12*I*(a^2*b + b^3)*f^3 + 12*((a^3 + a*b^2)*f^3 - I*(a^2*b + b^3)*f^3)*cosh(d*x + c)^2 + 24*((a^3 + a*b^2)*f^3
- I*(a^2*b + b^3)*f^3)*cosh(d*x + c)*sinh(d*x + c) + 12*((a^3 + a*b^2)*f^3 - I*(a^2*b + b^3)*f^3)*sinh(d*x + c
)^2)*polylog(3, -I*cosh(d*x + c) - I*sinh(d*x + c)) - 4*((a^2*b + b^3)*d^3*f^3*x^3 + 3*(a^2*b + b^3)*d^3*e*f^2
*x^2 + 3*(a^2*b + b^3)*d^3*e^2*f*x + (a^2*b + b^3)*d^3*e^3 + 2*((a^3 + a*b^2)*d^3*f^3*x^3 + 3*(a^3 + a*b^2)*d^
3*e*f^2*x^2 + 3*(a^3 + a*b^2)*d^3*e^2*f*x + 3*(a^3 + a*b^2)*c*d^2*e^2*f - 3*(a^3 + a*b^2)*c^2*d*e*f^2 + (a^3 +
a*b^2)*c^3*f^3)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^4*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b
^2 + b^4)*d^4*cosh(d*x + c)*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*d^4*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b
^4)*d^4)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e + f x\right )^{3} \tanh ^{2}{\left (c + d x \right )}}{a + b \sinh{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*tanh(d*x+c)**2/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**3*tanh(c + d*x)**2/(a + b*sinh(c + d*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out