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

Optimal. Leaf size=714 \[ -\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac{2 b f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{2 b f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac{2 b^2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac{2 b f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{2 b f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^3}-\frac{b f^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{b \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac{b (e+f x)^2 \coth (c+d x)}{a^2 d}+\frac{b (e+f x)^2}{a^2 d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}-\frac{(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d} \]

[Out]

(b*(e + f*x)^2)/(a^2*d) - ((e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/
(a^3*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (b*(e + f*x)^2*Coth[c + d*x])/(a^2*d) - (f*(e + f*x)*Csch[c +
d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b
*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sq
rt[a^2 + b^2])])/(a^3*d) - (2*b*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (f*(e + f*x)*PolyLog[2, -E^(
c + d*x)])/(a*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (f*(e + f*x)*PolyLog[2, E^(c + d
*x)])/(a*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyL
og[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b*f^2*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) + (f^2*PolyLog
[3, -E^(c + d*x)])/(a*d^3) + (2*b^2*f^2*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) - (f^2*PolyLog[3, E^(c + d*x)])/(a
*d^3) - (2*b^2*f^2*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/
(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
]))])/(a^3*d^3)

________________________________________________________________________________________

Rubi [A]  time = 1.728, antiderivative size = 714, normalized size of antiderivative = 1., number of steps used = 52, number of rules used = 22, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.647, Rules used = {5587, 5457, 4182, 2531, 2282, 6589, 4186, 3770, 5569, 3720, 3716, 2190, 2279, 2391, 32, 5585, 5450, 3296, 2638, 5565, 3322, 2264} \[ -\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac{2 b f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{2 b f \sqrt{a^2+b^2} (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac{2 b^2 f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac{2 b f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^3}-\frac{2 b f^2 \sqrt{a^2+b^2} \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^3}-\frac{b f^2 \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac{f (e+f x) \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{f (e+f x) \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac{f^2 \text{PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac{b \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}-\frac{2 b^2 (e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac{b (e+f x)^2 \coth (c+d x)}{a^2 d}+\frac{b (e+f x)^2}{a^2 d}-\frac{f (e+f x) \text{csch}(c+d x)}{a d^2}-\frac{f^2 \tanh ^{-1}(\cosh (c+d x))}{a d^3}-\frac{(e+f x)^2 \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{(e+f x)^2 \coth (c+d x) \text{csch}(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e + f*x)^2)/(a^2*d) - ((e + f*x)^2*ArcTanh[E^(c + d*x)])/(a*d) - (2*b^2*(e + f*x)^2*ArcTanh[E^(c + d*x)])/
(a^3*d) - (f^2*ArcTanh[Cosh[c + d*x]])/(a*d^3) + (b*(e + f*x)^2*Coth[c + d*x])/(a^2*d) - (f*(e + f*x)*Csch[c +
d*x])/(a*d^2) - ((e + f*x)^2*Coth[c + d*x]*Csch[c + d*x])/(2*a*d) - (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b
*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^3*d) + (b*Sqrt[a^2 + b^2]*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sq
rt[a^2 + b^2])])/(a^3*d) - (2*b*f*(e + f*x)*Log[1 - E^(2*(c + d*x))])/(a^2*d^2) - (f*(e + f*x)*PolyLog[2, -E^(
c + d*x)])/(a*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -E^(c + d*x)])/(a^3*d^2) + (f*(e + f*x)*PolyLog[2, E^(c + d
*x)])/(a*d^2) + (2*b^2*f*(e + f*x)*PolyLog[2, E^(c + d*x)])/(a^3*d^2) - (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyL
og[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^3*d^2) + (2*b*Sqrt[a^2 + b^2]*f*(e + f*x)*PolyLog[2, -((b*
E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^2) - (b*f^2*PolyLog[2, E^(2*(c + d*x))])/(a^2*d^3) + (f^2*PolyLog
[3, -E^(c + d*x)])/(a*d^3) + (2*b^2*f^2*PolyLog[3, -E^(c + d*x)])/(a^3*d^3) - (f^2*PolyLog[3, E^(c + d*x)])/(a
*d^3) - (2*b^2*f^2*PolyLog[3, E^(c + d*x)])/(a^3*d^3) + (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/
(a - Sqrt[a^2 + b^2]))])/(a^3*d^3) - (2*b*Sqrt[a^2 + b^2]*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
]))])/(a^3*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 5457

Int[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d
*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; F
reeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 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 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 4186

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && 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 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 5585

Int[(Cosh[(c_.) + (d_.)*(x_)]^(p_.)*Coth[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cosh[c + d*x]^p*Coth[c + d*x]^n, x], x] - Dis
t[b/a, Int[((e + f*x)^m*Cosh[c + d*x]^(p + 1)*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] && IGtQ[p, 0]

Rule 5450

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int
[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5565

Int[(Cosh[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symb
ol] :> -Dist[a/b^2, Int[(e + f*x)^m*Cosh[c + d*x]^(n - 2), x], x] + (Dist[1/b, Int[(e + f*x)^m*Cosh[c + d*x]^(
n - 2)*Sinh[c + d*x], x], x] + Dist[(a^2 + b^2)/b^2, Int[((e + f*x)^m*Cosh[c + d*x]^(n - 2))/(a + b*Sinh[c + d
*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 3322

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rubi steps

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

Mathematica [B]  time = 26.4172, size = 1529, normalized size = 2.14 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a*b*d^2*e*E^(2*c)*f*x + 4*a*b*d^2*E^(2*c)*f^2*x^2 + 2*a^2*d^2*e^2*ArcTanh[E^(c + d*x)] + 4*b^2*d^2*e^2*ArcT
anh[E^(c + d*x)] - 2*a^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] - 4*b^2*d^2*e^2*E^(2*c)*ArcTanh[E^(c + d*x)] + 4
*a^2*f^2*ArcTanh[E^(c + d*x)] - 4*a^2*E^(2*c)*f^2*ArcTanh[E^(c + d*x)] - 2*a^2*d^2*e*f*x*Log[1 - E^(c + d*x)]
- 4*b^2*d^2*e*f*x*Log[1 - E^(c + d*x)] + 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(c + d*x)] + 4*b^2*d^2*e*E^(2*c)*f*
x*Log[1 - E^(c + d*x)] - a^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - 2*b^2*d^2*f^2*x^2*Log[1 - E^(c + d*x)] + a^2*d
^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] + 2*a^2*d^2*e*f*x*Log
[1 + E^(c + d*x)] + 4*b^2*d^2*e*f*x*Log[1 + E^(c + d*x)] - 2*a^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] - 4*b^
2*d^2*e*E^(2*c)*f*x*Log[1 + E^(c + d*x)] + a^2*d^2*f^2*x^2*Log[1 + E^(c + d*x)] + 2*b^2*d^2*f^2*x^2*Log[1 + E^
(c + d*x)] - a^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] - 2*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] + 4
*a*b*d*e*f*Log[1 - E^(2*(c + d*x))] - 4*a*b*d*e*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + 4*a*b*d*f^2*x*Log[1 - E^(
2*(c + d*x))] - 4*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(2*(c + d*x))] - 2*(a^2 + 2*b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*
PolyLog[2, -E^(c + d*x)] + 2*(a^2 + 2*b^2)*d*(-1 + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + 2*a*b*f^2*Po
lyLog[2, E^(2*(c + d*x))] - 2*a*b*E^(2*c)*f^2*PolyLog[2, E^(2*(c + d*x))] - 2*a^2*f^2*PolyLog[3, -E^(c + d*x)]
- 4*b^2*f^2*PolyLog[3, -E^(c + d*x)] + 2*a^2*E^(2*c)*f^2*PolyLog[3, -E^(c + d*x)] + 4*b^2*E^(2*c)*f^2*PolyLog
[3, -E^(c + d*x)] + 2*a^2*f^2*PolyLog[3, E^(c + d*x)] + 4*b^2*f^2*PolyLog[3, E^(c + d*x)] - 2*a^2*E^(2*c)*f^2*
PolyLog[3, E^(c + d*x)] - 4*b^2*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)])/(2*a^3*d^3*(-1 + E^(2*c))) + (b*Sqrt[a^2
+ 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)*PolyL
og[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 + Sq
rt[a^2 + b^2]))]))/(a^3*d^3) + (Csch[c]*Csch[c + d*x]^2*(2*b*d*e^2*Cosh[c] + 4*b*d*e*f*x*Cosh[c] + 2*b*d*f^2*x
^2*Cosh[c] + 2*a*e*f*Cosh[d*x] + 2*a*f^2*x*Cosh[d*x] - 2*a*e*f*Cosh[2*c + d*x] - 2*a*f^2*x*Cosh[2*c + d*x] - 2
*b*d*e^2*Cosh[c + 2*d*x] - 4*b*d*e*f*x*Cosh[c + 2*d*x] - 2*b*d*f^2*x^2*Cosh[c + 2*d*x] + a*d*e^2*Sinh[d*x] + 2
*a*d*e*f*x*Sinh[d*x] + a*d*f^2*x^2*Sinh[d*x] - a*d*e^2*Sinh[2*c + d*x] - 2*a*d*e*f*x*Sinh[2*c + d*x] - a*d*f^2
*x^2*Sinh[2*c + d*x]))/(4*a^2*d^2)

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Maple [F]  time = 0.684, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ({\rm coth} \left (dx+c\right ) \right ) ^{2}{\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)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 3.62596, size = 17416, normalized size = 24.39 \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)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(4*a*b*d^2*e^2 - 8*a*b*c*d*e*f + 4*a*b*c^2*f^2 - 4*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a
*b*c^2*f^2)*cosh(d*x + c)^4 - 4*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*sinh(d*x + c
)^4 + 2*(a^2*d^2*f^2*x^2 + a^2*d^2*e^2 + 2*a^2*d*e*f + 2*(a^2*d^2*e*f + a^2*d*f^2)*x)*cosh(d*x + c)^3 + 2*(a^2
*d^2*f^2*x^2 + a^2*d^2*e^2 + 2*a^2*d*e*f + 2*(a^2*d^2*e*f + a^2*d*f^2)*x - 8*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*
x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x - a*b*d
^2*e^2 + 4*a*b*c*d*e*f - 2*a*b*c^2*f^2)*cosh(d*x + c)^2 + 2*(2*a*b*d^2*f^2*x^2 + 4*a*b*d^2*e*f*x - 2*a*b*d^2*e
^2 + 8*a*b*c*d*e*f - 4*a*b*c^2*f^2 - 12*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*cosh
(d*x + c)^2 + 3*(a^2*d^2*f^2*x^2 + a^2*d^2*e^2 + 2*a^2*d*e*f + 2*(a^2*d^2*e*f + a^2*d*f^2)*x)*cosh(d*x + c))*s
inh(d*x + c)^2 + 4*(b^2*d*f^2*x + b^2*d*e*f + (b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c)^4 + 4*(b^2*d*f^2*x + b^2
*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^2*x + b^2*d*e*f)*sinh(d*x + c)^4 - 2*(b^2*d*f^2*x + b^2*d*e*f
)*cosh(d*x + c)^2 - 2*(b^2*d*f^2*x + b^2*d*e*f - 3*(b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2
+ 4*((b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c)^3 - (b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*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) - 4*(b^2*d*f^2*x + b^2*d*e*f + (b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c)^4 + 4*(b^2*d*f^2*x
+ b^2*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f^2*x + b^2*d*e*f)*sinh(d*x + c)^4 - 2*(b^2*d*f^2*x + b^2*
d*e*f)*cosh(d*x + c)^2 - 2*(b^2*d*f^2*x + b^2*d*e*f - 3*(b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 4*((b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c)^3 - (b^2*d*f^2*x + b^2*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*
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^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2 + (b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c
^2*f^2)*cosh(d*x + c)^4 + 4*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d
^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x
+ c)^2 - 2*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2 - 3*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c)^3 - (b^2*d^2*e^2 - 2*
b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^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) + 2*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2 + (b^2*d^2*e^2 -
2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c)^4 + 4*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c)*sin
h(d*x + c)^3 + (b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^2*d^2*e^2 - 2*b^2*c*d*e*f +
b^2*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2 - 3*(b^2*d^2*e^2 - 2*b^2*c*d*e*f +
b^2*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c)^
3 - (b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*co
sh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 2*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*
b^2*c*d*e*f - b^2*c^2*f^2 + (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c)^4
+ 4*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d^2
*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*sinh(d*x + c)^4 - 2*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f
*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2
*c^2*f^2 - 3*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^
2 + 4*((b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c)^3 - (b^2*d^2*f^2*x^2 +
2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(-(a*cos
h(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^2*d^2*
f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2 + (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f -
b^2*c^2*f^2)*cosh(d*x + c)^4 + 4*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x +
c)*sinh(d*x + c)^3 + (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*sinh(d*x + c)^4 - 2*(b
^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c)^2 - 2*(b^2*d^2*f^2*x^2 + 2*b^2*d
^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2 - 3*(b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*c
osh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*
x + c)^3 - (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*sqr
t((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) - 4*(b^2*f^2*cosh(d*x + c)^4 + 4*b^2*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f^2*sinh(d*x + c
)^4 - 2*b^2*f^2*cosh(d*x + c)^2 + b^2*f^2 + 2*(3*b^2*f^2*cosh(d*x + c)^2 - b^2*f^2)*sinh(d*x + c)^2 + 4*(b^2*f
^2*cosh(d*x + c)^3 - b^2*f^2*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*polylog(3, (a*cosh(d*x + c) +
a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 4*(b^2*f^2*cosh(d*x + c)^4
+ 4*b^2*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f^2*sinh(d*x + c)^4 - 2*b^2*f^2*cosh(d*x + c)^2 + b^2*f^2 + 2*
(3*b^2*f^2*cosh(d*x + c)^2 - b^2*f^2)*sinh(d*x + c)^2 + 4*(b^2*f^2*cosh(d*x + c)^3 - b^2*f^2*cosh(d*x + c))*si
nh(d*x + c))*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*(a^2*d^2*f^2*x^2 + a^2*d^2*e^2 - 2*a^2*d*e*f + 2*(a^2*d^2*e*f - a^2*d*f
^2)*x)*cosh(d*x + c) - 2*((a^2 + 2*b^2)*d*f^2*x + ((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*co
sh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((
a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*sinh(d*x + c)^4 + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2 - 2*
((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d*f^2*x + (a^2 +
2*b^2)*d*e*f - 2*a*b*f^2 - 3*((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*cosh(d*x + c)^2)*sinh(d
*x + c)^2 + 4*(((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d*f^
2*x + (a^2 + 2*b^2)*d*e*f - 2*a*b*f^2)*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) + 2*
((a^2 + 2*b^2)*d*f^2*x + ((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f + 2*a*b*f^2)*cosh(d*x + c)^4 + 4*((a^2 +
2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f + 2*a*b*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d*f^2*x + (a
^2 + 2*b^2)*d*e*f + 2*a*b*f^2)*sinh(d*x + c)^4 + (a^2 + 2*b^2)*d*e*f + 2*a*b*f^2 - 2*((a^2 + 2*b^2)*d*f^2*x +
(a^2 + 2*b^2)*d*e*f + 2*a*b*f^2)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f + 2*a*b*f^2
- 3*((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f + 2*a*b*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + 2*
b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f + 2*a*b*f^2)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d*f^2*x + (a^2 + 2*b^2)*d*e*f
+ 2*a*b*f^2)*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) + ((a^2 + 2*b^2)*d^2*f^2*x^2
+ (a^2 + 2*b^2)*d^2*e^2 + 4*a*b*d*e*f + ((a^2 + 2*b^2)*d^2*f^2*x^2 + (a^2 + 2*b^2)*d^2*e^2 + 4*a*b*d*e*f + 2*
a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*a*b*d*f^2)*x)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^2*f^2*x^2 + (a^2 + 2
*b^2)*d^2*e^2 + 4*a*b*d*e*f + 2*a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*a*b*d*f^2)*x)*cosh(d*x + c)*sinh(d*x +
c)^3 + ((a^2 + 2*b^2)*d^2*f^2*x^2 + (a^2 + 2*b^2)*d^2*e^2 + 4*a*b*d*e*f + 2*a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f
+ 2*a*b*d*f^2)*x)*sinh(d*x + c)^4 + 2*a^2*f^2 - 2*((a^2 + 2*b^2)*d^2*f^2*x^2 + (a^2 + 2*b^2)*d^2*e^2 + 4*a*b*
d*e*f + 2*a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*a*b*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^2*f^2*x^2
+ (a^2 + 2*b^2)*d^2*e^2 + 4*a*b*d*e*f + 2*a^2*f^2 - 3*((a^2 + 2*b^2)*d^2*f^2*x^2 + (a^2 + 2*b^2)*d^2*e^2 + 4*a
*b*d*e*f + 2*a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*a*b*d*f^2)*x)*cosh(d*x + c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f +
2*a*b*d*f^2)*x)*sinh(d*x + c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*a*b*d*f^2)*x + 4*(((a^2 + 2*b^2)*d^2*f^2*x^2 +
(a^2 + 2*b^2)*d^2*e^2 + 4*a*b*d*e*f + 2*a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*a*b*d*f^2)*x)*cosh(d*x + c)^3
- ((a^2 + 2*b^2)*d^2*f^2*x^2 + (a^2 + 2*b^2)*d^2*e^2 + 4*a*b*d*e*f + 2*a^2*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f + 2*
a*b*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - ((a^2 + 2*b^2)*d^2*e^2 +
((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*f^2)*cosh(d
*x + c)^4 + 4*((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^
2)*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c
+ (a^2 + 2*b^2)*c^2 + 2*a^2)*f^2)*sinh(d*x + c)^4 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c + (a^2 + 2*b^
2)*c^2 + 2*a^2)*f^2 - 2*((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c + (a^2 + 2*b^2)*
c^2 + 2*a^2)*f^2)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c +
(a^2 + 2*b^2)*c^2 + 2*a^2)*f^2 - 3*((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c + (a^
2 + 2*b^2)*c^2 + 2*a^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a*b + (a^2 +
2*b^2)*c)*d*e*f + (4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*f^2)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d^2*e^2 - 2*(2*a
*b + (a^2 + 2*b^2)*c)*d*e*f + (4*a*b*c + (a^2 + 2*b^2)*c^2 + 2*a^2)*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(cos
h(d*x + c) + sinh(d*x + c) - 1) - ((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f + ((a^2 + 2*b^2)*d^2*f^
2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*
x)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^
2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*
(a^2 + 2*b^2)*c*d*e*f - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x)*sinh(d*
x + c)^4 - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^2 - 2*((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f - (4*a*b
*c + (a^2 + 2*b^2)*c^2)*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d^
2*f^2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^2 - 3*((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*(a^
2 + 2*b^2)*c*d*e*f - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x)*cosh(d*x +
c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x)*sinh(d*x + c)^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x
+ 4*(((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f - (4*a*b*c + (a^2 + 2*b^2)*c^2)*f^2 + 2*((a^2 + 2*b
^2)*d^2*e*f - 2*a*b*d*f^2)*x)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d^2*f^2*x^2 + 2*(a^2 + 2*b^2)*c*d*e*f - (4*a*b*
c + (a^2 + 2*b^2)*c^2)*f^2 + 2*((a^2 + 2*b^2)*d^2*e*f - 2*a*b*d*f^2)*x)*cosh(d*x + c))*sinh(d*x + c))*log(-cos
h(d*x + c) - sinh(d*x + c) + 1) + 2*((a^2 + 2*b^2)*f^2*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*f^2*cosh(d*x + c)*sin
h(d*x + c)^3 + (a^2 + 2*b^2)*f^2*sinh(d*x + c)^4 - 2*(a^2 + 2*b^2)*f^2*cosh(d*x + c)^2 + (a^2 + 2*b^2)*f^2 + 2
*(3*(a^2 + 2*b^2)*f^2*cosh(d*x + c)^2 - (a^2 + 2*b^2)*f^2)*sinh(d*x + c)^2 + 4*((a^2 + 2*b^2)*f^2*cosh(d*x + c
)^3 - (a^2 + 2*b^2)*f^2*cosh(d*x + c))*sinh(d*x + c))*polylog(3, cosh(d*x + c) + sinh(d*x + c)) - 2*((a^2 + 2*
b^2)*f^2*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*b^2)*f^2*sinh(d*x + c)
^4 - 2*(a^2 + 2*b^2)*f^2*cosh(d*x + c)^2 + (a^2 + 2*b^2)*f^2 + 2*(3*(a^2 + 2*b^2)*f^2*cosh(d*x + c)^2 - (a^2 +
2*b^2)*f^2)*sinh(d*x + c)^2 + 4*((a^2 + 2*b^2)*f^2*cosh(d*x + c)^3 - (a^2 + 2*b^2)*f^2*cosh(d*x + c))*sinh(d*
x + c))*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) + 2*(a^2*d^2*f^2*x^2 + a^2*d^2*e^2 - 2*a^2*d*e*f - 8*(a*b*d
^2*f^2*x^2 + 2*a*b*d^2*e*f*x + 2*a*b*c*d*e*f - a*b*c^2*f^2)*cosh(d*x + c)^3 + 3*(a^2*d^2*f^2*x^2 + a^2*d^2*e^2
+ 2*a^2*d*e*f + 2*(a^2*d^2*e*f + a^2*d*f^2)*x)*cosh(d*x + c)^2 + 2*(a^2*d^2*e*f - a^2*d*f^2)*x + 4*(a*b*d^2*f
^2*x^2 + 2*a*b*d^2*e*f*x - a*b*d^2*e^2 + 4*a*b*c*d*e*f - 2*a*b*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^3
*cosh(d*x + c)^4 + 4*a^3*d^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d^3*sinh(d*x + c)^4 - 2*a^3*d^3*cosh(d*x + c)
^2 + a^3*d^3 + 2*(3*a^3*d^3*cosh(d*x + c)^2 - a^3*d^3)*sinh(d*x + c)^2 + 4*(a^3*d^3*cosh(d*x + c)^3 - a^3*d^3*
cosh(d*x + c))*sinh(d*x + c))

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

[Out]

Timed out