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

Optimal. Leaf size=1067 \[ \text{result too large to display} \]

[Out]

(a*(e + f*x)^3)/(3*b^2*f) + (2*a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)^2*ArcTan[E^(c + d*x
)])/(b*d) - (2*a^4*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2
) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)^2*Lo
g[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(
b^2*d) + (a^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - ((2*I)*a^2*f*(e + f*x)*PolyLog[2, (-
I)*E^(c + d*x)])/(b^3*d^2) + ((2*I)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + ((2*I)*a^4*f*(e + f*x)
*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + ((2*I)*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*
d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - ((2*I)*a^4*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)
])/(b^3*(a^2 + b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2
+ b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) -
(a*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*d^2) + (a^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*
(a^2 + b^2)*d^2) + ((2*I)*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*d^3) - ((2*I)*f^2*PolyLog[3, (-I)*E^(c +
d*x)])/(b*d^3) - ((2*I)*a^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - ((2*I)*a^2*f^2*PolyLog[3
, I*E^(c + d*x)])/(b^3*d^3) + ((2*I)*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((2*I)*a^4*f^2*PolyLog[3, I*E^(c
+ d*x)])/(b^3*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 +
b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (a*f^2*P
olyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^3) - (a^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^3) + (2
*f^2*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*d)

________________________________________________________________________________________

Rubi [A]  time = 1.69094, antiderivative size = 1067, normalized size of antiderivative = 1., number of steps used = 50, number of rules used = 14, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {5581, 5449, 3296, 2637, 4180, 2531, 2282, 6589, 3718, 2190, 5567, 5573, 5561, 6742} \[ -\frac{2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac{2 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}-\frac{2 i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}-\frac{2 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}+\frac{2 i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}-\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac{(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac{f (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac{f^2 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^3}+\frac{2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b^3 d}-\frac{2 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) a^2}{b^3 d^2}+\frac{2 i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) a^2}{b^3 d^2}+\frac{2 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right ) a^2}{b^3 d^3}-\frac{2 i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right ) a^2}{b^3 d^3}+\frac{(e+f x)^3 a}{3 b^2 f}-\frac{(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) a}{b^2 d}-\frac{f (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) a}{b^2 d^2}+\frac{f^2 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) a}{2 b^2 d^3}-\frac{2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac{2 f (e+f x) \cosh (c+d x)}{b d^2}+\frac{2 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{b d^2}-\frac{2 i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{b d^2}-\frac{2 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{b d^3}+\frac{2 i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right )}{b d^3}+\frac{2 f^2 \sinh (c+d x)}{b d^3}+\frac{(e+f x)^2 \sinh (c+d x)}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(e + f*x)^3)/(3*b^2*f) + (2*a^2*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)^2*ArcTan[E^(c + d*x
)])/(b*d) - (2*a^4*(e + f*x)^2*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (2*f*(e + f*x)*Cosh[c + d*x])/(b*d^2
) - (a^3*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a^3*(e + f*x)^2*Lo
g[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(
b^2*d) + (a^3*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - ((2*I)*a^2*f*(e + f*x)*PolyLog[2, (-
I)*E^(c + d*x)])/(b^3*d^2) + ((2*I)*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/(b*d^2) + ((2*I)*a^4*f*(e + f*x)
*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + ((2*I)*a^2*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b^3*
d^2) - ((2*I)*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - ((2*I)*a^4*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)
])/(b^3*(a^2 + b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2
+ b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) -
(a*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*d^2) + (a^3*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(b^2*
(a^2 + b^2)*d^2) + ((2*I)*a^2*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*d^3) - ((2*I)*f^2*PolyLog[3, (-I)*E^(c +
d*x)])/(b*d^3) - ((2*I)*a^4*f^2*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - ((2*I)*a^2*f^2*PolyLog[3
, I*E^(c + d*x)])/(b^3*d^3) + ((2*I)*f^2*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((2*I)*a^4*f^2*PolyLog[3, I*E^(c
+ d*x)])/(b^3*(a^2 + b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 +
b^2)*d^3) + (2*a^3*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^3) + (a*f^2*P
olyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^3) - (a^3*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^3) + (2
*f^2*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^2*Sinh[c + d*x])/(b*d)

Rule 5581

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(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*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[((e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n)/(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 5449

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 3296

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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 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 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 6742

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

Rubi steps

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

Mathematica [B]  time = 19.6768, size = 3418, normalized size = 3.2 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(-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)] + PolyL
og[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))]))/(6*(a^2 + b^2)*d^3*(1 + E^(2*c))) + (a^3*(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 - S
qrt[(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 + Sqr
t[(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*b^2*(a^2 + b^2)*(-1 + E^(2*c))) + (Csch[c]*Sech[c]*(Cosh[c + d*x]/(
24*b^2*d^3) - Sinh[c + d*x]/(24*b^2*d^3))*(-3*a^2*b*d^2*e^2*Cosh[2*d*x] - 3*b^3*d^2*e^2*Cosh[2*d*x] + 6*a^2*b*
d*e*f*Cosh[2*d*x] + 6*b^3*d*e*f*Cosh[2*d*x] - 6*a^2*b*f^2*Cosh[2*d*x] - 6*b^3*f^2*Cosh[2*d*x] - 6*a^2*b*d^2*e*
f*x*Cosh[2*d*x] - 6*b^3*d^2*e*f*x*Cosh[2*d*x] + 6*a^2*b*d*f^2*x*Cosh[2*d*x] + 6*b^3*d*f^2*x*Cosh[2*d*x] - 3*a^
2*b*d^2*f^2*x^2*Cosh[2*d*x] - 3*b^3*d^2*f^2*x^2*Cosh[2*d*x] - 6*a^3*d^3*e^2*x*Cosh[c - d*x] - 6*a*b^2*d^3*e^2*
x*Cosh[c - d*x] - 6*a^3*d^3*e*f*x^2*Cosh[c - d*x] - 6*a*b^2*d^3*e*f*x^2*Cosh[c - d*x] - 2*a^3*d^3*f^2*x^3*Cosh
[c - d*x] - 2*a*b^2*d^3*f^2*x^3*Cosh[c - d*x] - 12*a^3*d^3*e^2*x*Cosh[c + d*x] + 12*a*b^2*d^3*e^2*x*Cosh[c + d
*x] - 12*a^3*d^3*e*f*x^2*Cosh[c + d*x] + 12*a*b^2*d^3*e*f*x^2*Cosh[c + d*x] - 4*a^3*d^3*f^2*x^3*Cosh[c + d*x]
+ 4*a*b^2*d^3*f^2*x^3*Cosh[c + d*x] - 6*a^3*d^3*e^2*x*Cosh[3*c + d*x] - 6*a*b^2*d^3*e^2*x*Cosh[3*c + d*x] - 6*
a^3*d^3*e*f*x^2*Cosh[3*c + d*x] - 6*a*b^2*d^3*e*f*x^2*Cosh[3*c + d*x] - 2*a^3*d^3*f^2*x^3*Cosh[3*c + d*x] - 2*
a*b^2*d^3*f^2*x^3*Cosh[3*c + d*x] + 3*a^2*b*d^2*e^2*Cosh[4*c + 2*d*x] + 3*b^3*d^2*e^2*Cosh[4*c + 2*d*x] - 6*a^
2*b*d*e*f*Cosh[4*c + 2*d*x] - 6*b^3*d*e*f*Cosh[4*c + 2*d*x] + 6*a^2*b*f^2*Cosh[4*c + 2*d*x] + 6*b^3*f^2*Cosh[4
*c + 2*d*x] + 6*a^2*b*d^2*e*f*x*Cosh[4*c + 2*d*x] + 6*b^3*d^2*e*f*x*Cosh[4*c + 2*d*x] - 6*a^2*b*d*f^2*x*Cosh[4
*c + 2*d*x] - 6*b^3*d*f^2*x*Cosh[4*c + 2*d*x] + 3*a^2*b*d^2*f^2*x^2*Cosh[4*c + 2*d*x] + 3*b^3*d^2*f^2*x^2*Cosh
[4*c + 2*d*x] - 6*a^2*b*d^2*e^2*Sinh[2*c] - 6*b^3*d^2*e^2*Sinh[2*c] - 12*a^2*b*d*e*f*Sinh[2*c] - 12*b^3*d*e*f*
Sinh[2*c] - 12*a^2*b*f^2*Sinh[2*c] - 12*b^3*f^2*Sinh[2*c] - 12*a^2*b*d^2*e*f*x*Sinh[2*c] - 12*b^3*d^2*e*f*x*Si
nh[2*c] - 12*a^2*b*d*f^2*x*Sinh[2*c] - 12*b^3*d*f^2*x*Sinh[2*c] - 6*a^2*b*d^2*f^2*x^2*Sinh[2*c] - 6*b^3*d^2*f^
2*x^2*Sinh[2*c] - 3*a^2*b*d^2*e^2*Sinh[2*d*x] - 3*b^3*d^2*e^2*Sinh[2*d*x] + 6*a^2*b*d*e*f*Sinh[2*d*x] + 6*b^3*
d*e*f*Sinh[2*d*x] - 6*a^2*b*f^2*Sinh[2*d*x] - 6*b^3*f^2*Sinh[2*d*x] - 6*a^2*b*d^2*e*f*x*Sinh[2*d*x] - 6*b^3*d^
2*e*f*x*Sinh[2*d*x] + 6*a^2*b*d*f^2*x*Sinh[2*d*x] + 6*b^3*d*f^2*x*Sinh[2*d*x] - 3*a^2*b*d^2*f^2*x^2*Sinh[2*d*x
] - 3*b^3*d^2*f^2*x^2*Sinh[2*d*x] + 6*a^3*d^3*e^2*x*Sinh[c - d*x] + 6*a*b^2*d^3*e^2*x*Sinh[c - d*x] + 6*a^3*d^
3*e*f*x^2*Sinh[c - d*x] + 6*a*b^2*d^3*e*f*x^2*Sinh[c - d*x] + 2*a^3*d^3*f^2*x^3*Sinh[c - d*x] + 2*a*b^2*d^3*f^
2*x^3*Sinh[c - d*x] - 12*a^3*d^3*e^2*x*Sinh[c + d*x] + 12*a*b^2*d^3*e^2*x*Sinh[c + d*x] - 12*a^3*d^3*e*f*x^2*S
inh[c + d*x] + 12*a*b^2*d^3*e*f*x^2*Sinh[c + d*x] - 4*a^3*d^3*f^2*x^3*Sinh[c + d*x] + 4*a*b^2*d^3*f^2*x^3*Sinh
[c + d*x] - 6*a^3*d^3*e^2*x*Sinh[3*c + d*x] - 6*a*b^2*d^3*e^2*x*Sinh[3*c + d*x] - 6*a^3*d^3*e*f*x^2*Sinh[3*c +
d*x] - 6*a*b^2*d^3*e*f*x^2*Sinh[3*c + d*x] - 2*a^3*d^3*f^2*x^3*Sinh[3*c + d*x] - 2*a*b^2*d^3*f^2*x^3*Sinh[3*c
+ d*x] + 3*a^2*b*d^2*e^2*Sinh[4*c + 2*d*x] + 3*b^3*d^2*e^2*Sinh[4*c + 2*d*x] - 6*a^2*b*d*e*f*Sinh[4*c + 2*d*x
] - 6*b^3*d*e*f*Sinh[4*c + 2*d*x] + 6*a^2*b*f^2*Sinh[4*c + 2*d*x] + 6*b^3*f^2*Sinh[4*c + 2*d*x] + 6*a^2*b*d^2*
e*f*x*Sinh[4*c + 2*d*x] + 6*b^3*d^2*e*f*x*Sinh[4*c + 2*d*x] - 6*a^2*b*d*f^2*x*Sinh[4*c + 2*d*x] - 6*b^3*d*f^2*
x*Sinh[4*c + 2*d*x] + 3*a^2*b*d^2*f^2*x^2*Sinh[4*c + 2*d*x] + 3*b^3*d^2*f^2*x^2*Sinh[4*c + 2*d*x]))/(a^2 + b^2
)

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Maple [F]  time = 0.853, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ( \sinh \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*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \, a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{2} b^{2} + b^{4}\right )} d} - \frac{4 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{2 \, a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{2 \,{\left (d x + c\right )} a}{b^{2} d} - \frac{e^{\left (d x + c\right )}}{b d} + \frac{e^{\left (-d x - c\right )}}{b d}\right )} e^{2} - \frac{{\left (2 \, a d^{3} f^{2} x^{3} e^{c} + 6 \, a d^{3} e f x^{2} e^{c} - 3 \,{\left (b d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \,{\left (d^{2} e f - d f^{2}\right )} b x e^{\left (2 \, c\right )} - 2 \,{\left (d e f - f^{2}\right )} b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 3 \,{\left (b d^{2} f^{2} x^{2} + 2 \,{\left (d^{2} e f + d f^{2}\right )} b x + 2 \,{\left (d e f + f^{2}\right )} b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{6 \, b^{2} d^{3}} + \int \frac{2 \,{\left (a^{3} b f^{2} x^{2} + 2 \, a^{3} b e f x -{\left (a^{4} f^{2} x^{2} e^{c} + 2 \, a^{4} e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} b^{3} + b^{5} -{\left (a^{2} b^{3} e^{\left (2 \, c\right )} + b^{5} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{3} b^{2} e^{c} + a b^{4} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int -\frac{2 \,{\left (a f^{2} x^{2} + 2 \, a e f x -{\left (b f^{2} x^{2} e^{c} + 2 \, b e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} + b^{2} +{\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b^2 + b^4)*d) - 4*b*arctan(e^(-d*x - c))/((a
^2 + b^2)*d) + 2*a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(
-d*x - c)/(b*d))*e^2 - 1/6*(2*a*d^3*f^2*x^3*e^c + 6*a*d^3*e*f*x^2*e^c - 3*(b*d^2*f^2*x^2*e^(2*c) + 2*(d^2*e*f
- d*f^2)*b*x*e^(2*c) - 2*(d*e*f - f^2)*b*e^(2*c))*e^(d*x) + 3*(b*d^2*f^2*x^2 + 2*(d^2*e*f + d*f^2)*b*x + 2*(d*
e*f + f^2)*b)*e^(-d*x))*e^(-c)/(b^2*d^3) + integrate(2*(a^3*b*f^2*x^2 + 2*a^3*b*e*f*x - (a^4*f^2*x^2*e^c + 2*a
^4*e*f*x*e^c)*e^(d*x))/(a^2*b^3 + b^5 - (a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^3*b^2*e^c + a*b^4*e^c
)*e^(d*x)), x) - integrate(-2*(a*f^2*x^2 + 2*a*e*f*x - (b*f^2*x^2*e^c + 2*b*e*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (
a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [C]  time = 3.65942, size = 6611, normalized size = 6.2 \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*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out