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

Optimal. Leaf size=535 \[ \frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2 \sqrt{a^2+b^2}}-\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2 \sqrt{a^2+b^2}}-\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3 \sqrt{a^2+b^2}}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^3 \sqrt{a^2+b^2}}+\frac{2 b f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac{2 b f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac{2 b f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac{2 b f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}+\frac{f^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}+\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d \sqrt{a^2+b^2}}-\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d \sqrt{a^2+b^2}}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac{2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{(e+f x)^2 \coth (c+d x)}{a d}-\frac{(e+f x)^2}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.07963, antiderivative size = 535, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5575, 4184, 3716, 2190, 2279, 2391, 4182, 2531, 2282, 6589, 3322, 2264} \[ \frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2 \sqrt{a^2+b^2}}-\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2 \sqrt{a^2+b^2}}-\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^3 \sqrt{a^2+b^2}}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^3 \sqrt{a^2+b^2}}+\frac{2 b f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac{2 b f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac{2 b f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac{2 b f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}+\frac{f^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}+\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d \sqrt{a^2+b^2}}-\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d \sqrt{a^2+b^2}}+\frac{2 b (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d}+\frac{2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac{(e+f x)^2 \coth (c+d x)}{a d}-\frac{(e+f x)^2}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5575

Int[(Csch[(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*Csch[c + d*x]^n, x], x] - Dist[b/a, Int[((e + f*x)^m*Csch[c + d*x]^(n - 1))/
(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4184

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

Rule 3716

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

Rule 2190

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

Rule 2279

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

Rule 2391

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

Rule 4182

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 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]

Rubi steps

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

Mathematica [A]  time = 16.1467, size = 795, normalized size = 1.49 \[ \frac{\left (-2 e^2 \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right ) d^2+f^2 x^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^2+2 e f x \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) d^2-f^2 x^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^2-2 e f x \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) d^2+2 f (e+f x) \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right ) d-2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) d-2 f^2 \text{PolyLog}\left (3,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )\right ) b^2}{a^2 \sqrt{a^2+b^2} d^3}-\frac{b d^2 x^2 \log \left (1-e^{-c-d x}\right ) f^2-b d^2 x^2 \log \left (1+e^{-c-d x}\right ) f^2+2 b \left (d x \text{PolyLog}\left (2,-e^{-c-d x}\right )+\text{PolyLog}\left (3,-e^{-c-d x}\right )\right ) f^2-2 b \left (d x \text{PolyLog}\left (2,e^{-c-d x}\right )+\text{PolyLog}\left (3,e^{-c-d x}\right )\right ) f^2+2 d (b d e-a f) x \log \left (1-e^{-c-d x}\right ) f-2 d (b d e+a f) x \log \left (1+e^{-c-d x}\right ) f+2 (b d e+a f) \text{PolyLog}\left (2,-e^{-c-d x}\right ) f+2 (a f-b d e) \text{PolyLog}\left (2,e^{-c-d x}\right ) f+\frac{2 a d^2 (e+f x)^2}{-1+e^{2 c}}-d e (b d e-2 a f) \left (d x-\log \left (1-e^{c+d x}\right )\right )+d e (b d e+2 a f) \left (d x-\log \left (1+e^{c+d x}\right )\right )}{a^2 d^3}+\frac{\text{sech}\left (\frac{c}{2}\right ) \text{sech}\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (-\sinh \left (\frac{d x}{2}\right ) e^2-2 f x \sinh \left (\frac{d x}{2}\right ) e-f^2 x^2 \sinh \left (\frac{d x}{2}\right )\right )}{2 a d}+\frac{\text{csch}\left (\frac{c}{2}\right ) \text{csch}\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\sinh \left (\frac{d x}{2}\right ) e^2+2 f x \sinh \left (\frac{d x}{2}\right ) e+f^2 x^2 \sinh \left (\frac{d x}{2}\right )\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.743, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.49746, size = 8581, normalized size = 16.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a^3 + a*b^2)*d^2*e^2 - 4*(a^3 + a*b^2)*c*d*e*f + 2*(a^3 + a*b^2)*c^2*f^2 + 2*((a^3 + a*b^2)*d^2*f^2*x^2 +
2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)^2 + 4*((a^3 + a*b^
2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^2)*c^2*f^2)*cosh(d*x + c)*si
nh(d*x + c) + 2*((a^3 + a*b^2)*d^2*f^2*x^2 + 2*(a^3 + a*b^2)*d^2*e*f*x + 2*(a^3 + a*b^2)*c*d*e*f - (a^3 + a*b^
2)*c^2*f^2)*sinh(d*x + c)^2 + 2*(b^3*d*f^2*x + b^3*d*e*f - (b^3*d*f^2*x + b^3*d*e*f)*cosh(d*x + c)^2 - 2*(b^3*
d*f^2*x + b^3*d*e*f)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f^2*x + b^3*d*e*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*(b^3*d*f^2*x + b^3*d*e*f - (b^3*d*f^2*x + b^3*d*e*f)*cosh(d*x + c)^2 - 2*(b^3*d*f^2*x + b^3*d*e*f
)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f^2*x + b^3*d*e*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) - (b^3*d^2
*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2 - (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^3*d^2*
e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*s
inh(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) + (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2 - (b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)^
2 - 2*(b^3*d^2*e^2 - 2*b^3*c*d*e*f + b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^2*e^2 - 2*b^3*c*d*e*f +
b^3*c^2*f^2)*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) + (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2 - (b^3*d^2*f^2*x^2 + 2*
b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*
d*e*f - b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^
2*f^2)*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*s
inh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2
- (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^3*d^2*f^2*x^2 + 2*
b^3*d^2*e*f*x + 2*b^3*c*d*e*f - b^3*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d^2*f^2*x^2 + 2*b^3*d^2*e*f*x
+ 2*b^3*c*d*e*f - b^3*c^2*f^2)*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*(b^3*f^2*cosh(d*x + c)^2 + 2*b^3*f^2*c
osh(d*x + c)*sinh(d*x + c) + b^3*f^2*sinh(d*x + c)^2 - b^3*f^2)*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x +
c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 2*(b^3*f^2*cosh(d*x +
c)^2 + 2*b^3*f^2*cosh(d*x + c)*sinh(d*x + c) + b^3*f^2*sinh(d*x + c)^2 - b^3*f^2)*sqrt((a^2 + b^2)/b^2)*polylo
g(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*((
a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*f^2 - ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f
- (a^3 + a*b^2)*f^2)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*f^2)*cos
h(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f - (a^3 + a*b^2)*f^2)*sinh(d*x + c)^2)*
dilog(cosh(d*x + c) + sinh(d*x + c)) + 2*((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f + (a^3 + a*b^2)*f^2 - ((
a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d*e*f + (a^3 + a*b^2)*f^2)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d*f^2*x + (
a^2*b + b^3)*d*e*f + (a^3 + a*b^2)*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d*f^2*x + (a^2*b + b^3)*d
*e*f + (a^3 + a*b^2)*f^2)*sinh(d*x + c)^2)*dilog(-cosh(d*x + c) - sinh(d*x + c)) + ((a^2*b + b^3)*d^2*f^2*x^2
+ (a^2*b + b^3)*d^2*e^2 + 2*(a^3 + a*b^2)*d*e*f - ((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 + 2*(a^3
+ a*b^2)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d^2*f^2
*x^2 + (a^2*b + b^3)*d^2*e^2 + 2*(a^3 + a*b^2)*d*e*f + 2*((a^2*b + b^3)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*x)*cosh
(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d^2*f^2*x^2 + (a^2*b + b^3)*d^2*e^2 + 2*(a^3 + a*b^2)*d*e*f + 2*((a^2
*b + b^3)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*x)*sinh(d*x + c)^2 + 2*((a^2*b + b^3)*d^2*e*f + (a^3 + a*b^2)*d*f^2)*
x)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - ((a^2*b + b^3)*d^2*e^2 - 2*(a^3 + a*b^2 + (a^2*b + b^3)*c)*d*e*f +
((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2 - ((a^2*b + b^3)*d^2*e^2 - 2*(a^3 + a*b^2 + (a^2*b + b^3)*c)*d*e*
f + ((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d^2*e^2 - 2*(a^3 + a*b^2 +
(a^2*b + b^3)*c)*d*e*f + ((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b +
b^3)*d^2*e^2 - 2*(a^3 + a*b^2 + (a^2*b + b^3)*c)*d*e*f + ((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2)*sinh(d*
x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - 1) - ((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*e*f - ((a^
2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2 - ((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*e*f - ((a^2*b + b^3
)*c^2 + 2*(a^3 + a*b^2)*c)*f^2 + 2*((a^2*b + b^3)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^2*
b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)*c*d*e*f - ((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2 + 2*((a^2*b + b^3
)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d^2*f^2*x^2 + 2*(a^2*b + b^3)
*c*d*e*f - ((a^2*b + b^3)*c^2 + 2*(a^3 + a*b^2)*c)*f^2 + 2*((a^2*b + b^3)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x)*si
nh(d*x + c)^2 + 2*((a^2*b + b^3)*d^2*e*f - (a^3 + a*b^2)*d*f^2)*x)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2
*((a^2*b + b^3)*f^2*cosh(d*x + c)^2 + 2*(a^2*b + b^3)*f^2*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f^2*sinh
(d*x + c)^2 - (a^2*b + b^3)*f^2)*polylog(3, cosh(d*x + c) + sinh(d*x + c)) + 2*((a^2*b + b^3)*f^2*cosh(d*x + c
)^2 + 2*(a^2*b + b^3)*f^2*cosh(d*x + c)*sinh(d*x + c) + (a^2*b + b^3)*f^2*sinh(d*x + c)^2 - (a^2*b + b^3)*f^2)
*polylog(3, -cosh(d*x + c) - sinh(d*x + c)))/((a^4 + a^2*b^2)*d^3*cosh(d*x + c)^2 + 2*(a^4 + a^2*b^2)*d^3*cosh
(d*x + c)*sinh(d*x + c) + (a^4 + a^2*b^2)*d^3*sinh(d*x + c)^2 - (a^4 + a^2*b^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*csch(d*x+c)**2/(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*csch(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out