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

Optimal. Leaf size=419 \[ \frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}-\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^3}-\frac{b f (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}+\frac{b f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{2 f^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac{2 f^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}+\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}-\frac{b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^2 \text{csch}(c+d x)}{a d} \]

[Out]

(-4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + (b*(e + f*x)^2*Log[1 + (b*
E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/
(a^2*d) - (b*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*f^2
*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^
2*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*(e + f*x)*Pol
yLog[2, E^(2*(c + d*x))])/(a^2*d^2) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3)
- (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3) + (b*f^2*PolyLog[3, E^(2*(c + d*x)
)])/(2*a^2*d^3)

________________________________________________________________________________________

Rubi [A]  time = 0.817859, antiderivative size = 419, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5587, 5452, 4182, 2279, 2391, 5569, 3716, 2190, 2531, 2282, 6589, 5561} \[ \frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2}+\frac{2 b f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2}-\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3}-\frac{2 b f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^3}-\frac{b f (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^2}+\frac{b f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^2 d^3}-\frac{2 f^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^3}+\frac{2 f^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^3}+\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d}+\frac{b (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d}-\frac{b (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^2 d}-\frac{4 f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d^2}-\frac{(e+f x)^2 \text{csch}(c+d x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a*d^2) - ((e + f*x)^2*Csch[c + d*x])/(a*d) + (b*(e + f*x)^2*Log[1 + (b*
E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*d) + (b*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/
(a^2*d) - (b*(e + f*x)^2*Log[1 - E^(2*(c + d*x))])/(a^2*d) - (2*f^2*PolyLog[2, -E^(c + d*x)])/(a*d^3) + (2*f^2
*PolyLog[2, E^(c + d*x)])/(a*d^3) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^
2*d^2) + (2*b*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^2) - (b*f*(e + f*x)*Pol
yLog[2, E^(2*(c + d*x))])/(a^2*d^2) - (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*d^3)
- (2*b*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*d^3) + (b*f^2*PolyLog[3, E^(2*(c + d*x)
)])/(2*a^2*d^3)

Rule 5587

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*Csch[(c_.) + (d_.)*(x_)]^(p_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Csch[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(p - 1)*Coth[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 5452

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

Rule 4182

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

Rule 2279

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

Rule 2391

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

Rule 5569

Int[(Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/a, Int[(e + f*x)^m*Coth[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Cosh[c + d*x]*Coth[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 3716

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

Rule 2190

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

Rule 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 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]

Rubi steps

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

Mathematica [B]  time = 20.2357, size = 1595, normalized size = 3.81 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(((e + f*x)^2*Csch[c])/(a*d)) - (b*(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 - Sq
rt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d -
(3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b*E^(
2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt
[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3
*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(e +
f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e + f*x)
*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c + d
*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[
(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^
3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*a^2*(-1 + E^
(2*c))) + (b*d^3*(e + f*x)^3*(-1 + Coth[c]) + 3*d*e*f*(b*d*e - 2*a*f)*(d*x - Log[1 - Cosh[c + d*x] - Sinh[c +
d*x]]) - 6*d*f^2*(b*d*e + a*f)*x*Log[1 + Cosh[c + d*x] - Sinh[c + d*x]] - 3*b*d^2*f^3*x^2*Log[1 + Cosh[c + d*x
] - Sinh[c + d*x]] - 6*d*f^2*(b*d*e - a*f)*x*Log[1 - Cosh[c + d*x] + Sinh[c + d*x]] - 3*b*d^2*f^3*x^2*Log[1 -
Cosh[c + d*x] + Sinh[c + d*x]] + 3*d*e*f*(b*d*e + 2*a*f)*(d*x - Log[1 + Cosh[c + d*x] + Sinh[c + d*x]]) + 6*f^
2*(b*d*e - a*f)*PolyLog[2, Cosh[c + d*x] - Sinh[c + d*x]] + 6*f^2*(b*d*e + a*f)*PolyLog[2, -Cosh[c + d*x] + Si
nh[c + d*x]] + 6*b*f^3*(d*x*PolyLog[2, Cosh[c + d*x] - Sinh[c + d*x]] + PolyLog[3, Cosh[c + d*x] - Sinh[c + d*
x]]) + 6*b*f^3*(d*x*PolyLog[2, -Cosh[c + d*x] + Sinh[c + d*x]] + PolyLog[3, -Cosh[c + d*x] + Sinh[c + d*x]]))/
(3*a^2*d^3*f) + ((e + f*x)^2*Csch[c/2]*Csch[(c + d*x)/2]*Sinh[(d*x)/2])/(2*a*d) + ((e + f*x)^2*Sech[c/2]*Sech[
(c + d*x)/2]*Sinh[(d*x)/2])/(2*a*d)

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Maple [F]  time = 0.668, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}{\rm coth} \left (dx+c\right ){\rm csch} \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*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*coth(d*x+c)*csch(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*} e^{2}{\left (\frac{2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} + \frac{b \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{2} d} - \frac{b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac{b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d}\right )} - \frac{2 \,{\left (f^{2} x^{2} e^{c} + 2 \, e f x e^{c}\right )} e^{\left (d x\right )}}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} - \frac{2 \, e f \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac{2 \, e f \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}} - \frac{{\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x{\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} b f^{2}}{a^{2} d^{3}} - \frac{{\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x{\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (d x + c\right )})\right )} b f^{2}}{a^{2} d^{3}} - \frac{2 \,{\left (b d e f + a f^{2}\right )}{\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )}}{a^{2} d^{3}} - \frac{2 \,{\left (b d e f - a f^{2}\right )}{\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )}}{a^{2} d^{3}} + \frac{b d^{3} f^{2} x^{3} + 3 \,{\left (b d e f + a f^{2}\right )} d^{2} x^{2}}{3 \, a^{2} d^{3}} + \frac{b d^{3} f^{2} x^{3} + 3 \,{\left (b d e f - a f^{2}\right )} d^{2} x^{2}}{3 \, a^{2} d^{3}} - \int -\frac{2 \,{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x -{\left (a b f^{2} x^{2} e^{c} + 2 \, a b e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a^{3} e^{\left (d x + c\right )} - a^{2} b}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^2*(2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) + b*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^2*d) -
b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d)) - 2*(f^2*x^2*e^c + 2*e*f*x*e^c)*e^(d*x)/(a
*d*e^(2*d*x + 2*c) - a*d) - 2*e*f*log(e^(d*x + c) + 1)/(a*d^2) + 2*e*f*log(e^(d*x + c) - 1)/(a*d^2) - (d^2*x^2
*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*b*f^2/(a^2*d^3) - (d^2*x^2*log
(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*b*f^2/(a^2*d^3) - 2*(b*d*e*f + a*f^
2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^2*d^3) - 2*(b*d*e*f - a*f^2)*(d*x*log(-e^(d*x + c) + 1)
+ dilog(e^(d*x + c)))/(a^2*d^3) + 1/3*(b*d^3*f^2*x^3 + 3*(b*d*e*f + a*f^2)*d^2*x^2)/(a^2*d^3) + 1/3*(b*d^3*f^
2*x^3 + 3*(b*d*e*f - a*f^2)*d^2*x^2)/(a^2*d^3) - integrate(-2*(b^2*f^2*x^2 + 2*b^2*e*f*x - (a*b*f^2*x^2*e^c +
2*a*b*e*f*x*e^c)*e^(d*x))/(a^2*b*e^(2*d*x + 2*c) + 2*a^3*e^(d*x + c) - a^2*b), x)

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Fricas [C]  time = 2.93885, size = 6053, normalized size = 14.45 \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*coth(d*x+c)*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*cosh(d*x + c) + 2*(b*d*f^2*x + b*d*e*f - (b*d*f^2*x + b*d*e*f)
*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*e*f)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x + b*d*e*f)*sinh(d*x + c)^2
)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b
+ 1) + 2*(b*d*f^2*x + b*d*e*f - (b*d*f^2*x + b*d*e*f)*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*e*f)*cosh(d*x + c)*
sinh(d*x + c) - (b*d*f^2*x + b*d*e*f)*sinh(d*x + c)^2)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x
+ c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*(b*d*f^2*x + b*d*e*f - a*f^2 - (b*d*f^2*x + b*d*
e*f - a*f^2)*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*e*f - a*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2*x + b*d*
e*f - a*f^2)*sinh(d*x + c)^2)*dilog(cosh(d*x + c) + sinh(d*x + c)) - 2*(b*d*f^2*x + b*d*e*f + a*f^2 - (b*d*f^2
*x + b*d*e*f + a*f^2)*cosh(d*x + c)^2 - 2*(b*d*f^2*x + b*d*e*f + a*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d*f^2
*x + b*d*e*f + a*f^2)*sinh(d*x + c)^2)*dilog(-cosh(d*x + c) - sinh(d*x + c)) + (b*d^2*e^2 - 2*b*c*d*e*f + b*c^
2*f^2 - (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh(d*x + c)^2 - 2*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh(d
*x + c)*sinh(d*x + c) - (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sin
h(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + (b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2 - (b*d^2*e^2 - 2*b*c*d*e*
f + b*c^2*f^2)*cosh(d*x + c)^2 - 2*(b*d^2*e^2 - 2*b*c*d*e*f + b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*
e^2 - 2*b*c*d*e*f + b*c^2*f^2)*sinh(d*x + c)^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) + (b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2 - (b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2
*b*c*d*e*f - b*c^2*f^2)*cosh(d*x + c)^2 - 2*(b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*cosh(d*x
+ c)*sinh(d*x + c) - (b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sinh(d*x + c)^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) + (b*d^2*f^2*x^
2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2 - (b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*cosh(d
*x + c)^2 - 2*(b*d^2*f^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f
^2*x^2 + 2*b*d^2*e*f*x + 2*b*c*d*e*f - b*c^2*f^2)*sinh(d*x + c)^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) - (b*d^2*f^2*x^2 + b*d^2*e^2 + 2*a*d*e*f - (b
*d^2*f^2*x^2 + b*d^2*e^2 + 2*a*d*e*f + 2*(b*d^2*e*f + a*d*f^2)*x)*cosh(d*x + c)^2 - 2*(b*d^2*f^2*x^2 + b*d^2*e
^2 + 2*a*d*e*f + 2*(b*d^2*e*f + a*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^2*x^2 + b*d^2*e^2 + 2*a*d*e
*f + 2*(b*d^2*e*f + a*d*f^2)*x)*sinh(d*x + c)^2 + 2*(b*d^2*e*f + a*d*f^2)*x)*log(cosh(d*x + c) + sinh(d*x + c)
+ 1) - (b*d^2*e^2 - 2*(b*c + a)*d*e*f + (b*c^2 + 2*a*c)*f^2 - (b*d^2*e^2 - 2*(b*c + a)*d*e*f + (b*c^2 + 2*a*c
)*f^2)*cosh(d*x + c)^2 - 2*(b*d^2*e^2 - 2*(b*c + a)*d*e*f + (b*c^2 + 2*a*c)*f^2)*cosh(d*x + c)*sinh(d*x + c) -
(b*d^2*e^2 - 2*(b*c + a)*d*e*f + (b*c^2 + 2*a*c)*f^2)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1)
- (b*d^2*f^2*x^2 + 2*b*c*d*e*f - (b*c^2 + 2*a*c)*f^2 - (b*d^2*f^2*x^2 + 2*b*c*d*e*f - (b*c^2 + 2*a*c)*f^2 + 2
*(b*d^2*e*f - a*d*f^2)*x)*cosh(d*x + c)^2 - 2*(b*d^2*f^2*x^2 + 2*b*c*d*e*f - (b*c^2 + 2*a*c)*f^2 + 2*(b*d^2*e*
f - a*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) - (b*d^2*f^2*x^2 + 2*b*c*d*e*f - (b*c^2 + 2*a*c)*f^2 + 2*(b*d^2*e*
f - a*d*f^2)*x)*sinh(d*x + c)^2 + 2*(b*d^2*e*f - a*d*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 2*(b*f^
2*cosh(d*x + c)^2 + 2*b*f^2*cosh(d*x + c)*sinh(d*x + c) + b*f^2*sinh(d*x + c)^2 - b*f^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) + 2*(b*f^2*cosh(d*x +
c)^2 + 2*b*f^2*cosh(d*x + c)*sinh(d*x + c) + b*f^2*sinh(d*x + c)^2 - b*f^2)*polylog(3, (a*cosh(d*x + c) + a*s
inh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 2*(b*f^2*cosh(d*x + c)^2 + 2*b*
f^2*cosh(d*x + c)*sinh(d*x + c) + b*f^2*sinh(d*x + c)^2 - b*f^2)*polylog(3, cosh(d*x + c) + sinh(d*x + c)) - 2
*(b*f^2*cosh(d*x + c)^2 + 2*b*f^2*cosh(d*x + c)*sinh(d*x + c) + b*f^2*sinh(d*x + c)^2 - b*f^2)*polylog(3, -cos
h(d*x + c) - sinh(d*x + c)) + 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*sinh(d*x + c))/(a^2*d^3*cosh(d*x +
c)^2 + 2*a^2*d^3*cosh(d*x + c)*sinh(d*x + c) + a^2*d^3*sinh(d*x + c)^2 - a^2*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out