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

Optimal. Leaf size=689 \[ -\frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}-\frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}+\frac{b^2 f (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^3 d^2}+\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^3}+\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^3}-\frac{b^2 f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}+\frac{2 b f^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac{2 b f^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}+\frac{f (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^2}-\frac{f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}-\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}+\frac{b^2 (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac{b^2 (e+f x)^3}{3 a^3 f}+\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 a^3 f}+\frac{4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac{b (e+f x)^2 \text{csch}(c+d x)}{a^2 d}-\frac{f (e+f x) \coth (c+d x)}{a d^2}+\frac{f^2 \log (\sinh (c+d x))}{a d^3}+\frac{(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac{(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{(e+f x)^3}{3 a f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.7085, antiderivative size = 689, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 20, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.714, Rules used = {5569, 3720, 3475, 3716, 2190, 2531, 2282, 6589, 5585, 5450, 3296, 2637, 5452, 4182, 2279, 2391, 5446, 3310, 5565, 5561} \[ -\frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}-\frac{2 f \left (a^2+b^2\right ) (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}+\frac{b^2 f (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^3 d^2}+\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^3}+\frac{2 f^2 \left (a^2+b^2\right ) \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^3}-\frac{b^2 f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}+\frac{2 b f^2 \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac{2 b f^2 \text{PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}+\frac{f (e+f x) \text{PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^2}-\frac{f^2 \text{PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^3}-\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}-\frac{\left (a^2+b^2\right ) (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}+\frac{b^2 (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}-\frac{b^2 (e+f x)^3}{3 a^3 f}+\frac{\left (a^2+b^2\right ) (e+f x)^3}{3 a^3 f}+\frac{4 b f (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^2 d^2}+\frac{b (e+f x)^2 \text{csch}(c+d x)}{a^2 d}-\frac{f (e+f x) \coth (c+d x)}{a d^2}+\frac{f^2 \log (\sinh (c+d x))}{a d^3}+\frac{(e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d}-\frac{(e+f x)^2 \coth ^2(c+d x)}{2 a d}+\frac{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{(e+f x)^3}{3 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3475

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

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 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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 5446

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

Rule 3310

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

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

Mathematica [B]  time = 42.2086, size = 2277, normalized size = 3.3 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b*(e + f*x)^2*Csch[c])/(a^2*d) - ((e + f*x)^2*Csch[(c + d*x)/2]^2)/(8*a*d) - (12*a^2*d^3*e^2*E^(2*c)*x + 12*b
^2*d^3*e^2*E^(2*c)*x + 12*a^2*d*E^(2*c)*f^2*x + 12*a^2*d^3*e*E^(2*c)*f*x^2 + 12*b^2*d^3*e*E^(2*c)*f*x^2 + 4*a^
2*d^3*E^(2*c)*f^2*x^3 + 4*b^2*d^3*E^(2*c)*f^2*x^3 + 24*a*b*d*e*f*ArcTanh[E^(c + d*x)] - 24*a*b*d*e*E^(2*c)*f*A
rcTanh[E^(c + d*x)] - 12*a*b*d*f^2*x*Log[1 - E^(c + d*x)] + 12*a*b*d*E^(2*c)*f^2*x*Log[1 - E^(c + d*x)] + 12*a
*b*d*f^2*x*Log[1 + E^(c + d*x)] - 12*a*b*d*E^(2*c)*f^2*x*Log[1 + E^(c + d*x)] + 6*a^2*d^2*e^2*Log[1 - E^(2*(c
+ d*x))] + 6*b^2*d^2*e^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*e^2*E^(2*c)*Log[1 - E^(2*(c + d*x))] - 6*b^2*d^2
*e^2*E^(2*c)*Log[1 - E^(2*(c + d*x))] + 6*a^2*f^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*E^(2*c)*f^2*Log[1 - E^(2*(c
+ d*x))] + 12*a^2*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] + 12*b^2*d^2*e*f*x*Log[1 - E^(2*(c + d*x))] - 12*a^2*d^2
*e*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] - 12*b^2*d^2*e*E^(2*c)*f*x*Log[1 - E^(2*(c + d*x))] + 6*a^2*d^2*f^2*x^
2*Log[1 - E^(2*(c + d*x))] + 6*b^2*d^2*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 6*a^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^
(2*(c + d*x))] - 6*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(2*(c + d*x))] - 12*a*b*(-1 + E^(2*c))*f^2*PolyLog[2, -E^
(c + d*x)] + 12*a*b*(-1 + E^(2*c))*f^2*PolyLog[2, E^(c + d*x)] + 6*a^2*d*e*f*PolyLog[2, E^(2*(c + d*x))] + 6*b
^2*d*e*f*PolyLog[2, E^(2*(c + d*x))] - 6*a^2*d*e*E^(2*c)*f*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*e*E^(2*c)*f*P
olyLog[2, E^(2*(c + d*x))] + 6*a^2*d*f^2*x*PolyLog[2, E^(2*(c + d*x))] + 6*b^2*d*f^2*x*PolyLog[2, E^(2*(c + d*
x))] - 6*a^2*d*E^(2*c)*f^2*x*PolyLog[2, E^(2*(c + d*x))] - 6*b^2*d*E^(2*c)*f^2*x*PolyLog[2, E^(2*(c + d*x))] -
3*a^2*f^2*PolyLog[3, E^(2*(c + d*x))] - 3*b^2*f^2*PolyLog[3, E^(2*(c + d*x))] + 3*a^2*E^(2*c)*f^2*PolyLog[3,
E^(2*(c + d*x))] + 3*b^2*E^(2*c)*f^2*PolyLog[3, E^(2*(c + d*x))])/(6*a^3*d^3*(-1 + E^(2*c))) + ((a^2 + b^2)*(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))/S
qrt[-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)*Lo
g[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2
)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*
x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c +
d*x))/(a*E^c - 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^3*(-1 + E^(2*c))) + ((e + f*x)^2*Sech[(c + d*x)/
2]^2)/(8*a*d) - ((e + f*x)*(-(a*f) + b*d*(e + f*x))*Csch[c/2]*Csch[(c + d*x)/2]*Sinh[(d*x)/2])/(2*a^2*d^2) - (
(e + f*x)*(a*f + b*d*(e + f*x))*Sech[c/2]*Sech[(c + d*x)/2]*Sinh[(d*x)/2])/(2*a^2*d^2)

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Maple [F]  time = 0.671, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2} \left ({\rm coth} \left (dx+c\right ) \right ) ^{3}}{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)^3/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-e^2*(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x -
4*c) - a^2)*d) + (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x
- c) + 1)/(a^3*d) - (a^2 + b^2)*log(e^(-d*x - c) - 1)/(a^3*d)) + 2*(a*f^2*x + a*e*f + (b*d*f^2*x^2*e^(3*c) +
2*b*d*e*f*x*e^(3*c))*e^(3*d*x) - (a*d*f^2*x^2*e^(2*c) + a*e*f*e^(2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x)
- (b*d*f^2*x^2*e^c + 2*b*d*e*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2
) - (2*b*d*e*f + a*f^2)*x/(a^2*d^2) + (2*b*d*e*f - a*f^2)*x/(a^2*d^2) + (2*b*d*e*f + a*f^2)*log(e^(d*x + c) +
1)/(a^2*d^3) - (2*b*d*e*f - a*f^2)*log(e^(d*x + c) - 1)/(a^2*d^3) + (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilo
g(-e^(d*x + c)) - 2*polylog(3, -e^(d*x + c)))*(a^2*f^2 + b^2*f^2)/(a^3*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)))*(a^2*f^2 + b^2*f^2)/(a^3*d^3) + 2*(a^2*d*e*f + b^2*d*e*
f + a*b*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3) + 2*(a^2*d*e*f + b^2*d*e*f - a*b*f^2)*
(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) - 1/3*((a^2*f^2 + b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f +
b^2*d*e*f + a*b*f^2)*d^2*x^2)/(a^3*d^3) - 1/3*((a^2*f^2 + b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f + b^2*d*e*f - a*b*f
^2)*d^2*x^2)/(a^3*d^3) + integrate(-2*((a^2*b*f^2 + b^3*f^2)*x^2 + 2*(a^2*b*e*f + b^3*e*f)*x - ((a^3*f^2*e^c +
a*b^2*f^2*e^c)*x^2 + 2*(a^3*e*f*e^c + a*b^2*e*f*e^c)*x)*e^(d*x))/(a^3*b*e^(2*d*x + 2*c) + 2*a^4*e^(d*x + c) -
a^3*b), x)

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Fricas [C]  time = 2.99641, size = 18048, normalized size = 26.19 \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)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(2*a^2*d*e*f - 2*a^2*c*f^2 - 2*(a^2*d*f^2*x + a^2*c*f^2)*cosh(d*x + c)^4 - 2*(a^2*d*f^2*x + a^2*c*f^2)*sinh(d*
x + c)^4 + 2*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + a*b*d^2*e^2)*cosh(d*x + c)^3 + 2*(a*b*d^2*f^2*x^2 + 2*a*b*d^
2*e*f*x + a*b*d^2*e^2 - 4*(a^2*d*f^2*x + a^2*c*f^2)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(a^2*d^2*f^2*x^2 + a^2*
d^2*e^2 + a^2*d*e*f - 2*a^2*c*f^2 + (2*a^2*d^2*e*f - a^2*d*f^2)*x)*cosh(d*x + c)^2 - 2*(a^2*d^2*f^2*x^2 + a^2*
d^2*e^2 + a^2*d*e*f - 2*a^2*c*f^2 + 6*(a^2*d*f^2*x + a^2*c*f^2)*cosh(d*x + c)^2 + (2*a^2*d^2*e*f - a^2*d*f^2)*
x - 3*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + a*b*d^2*e^2)*cosh(d*x + c))*sinh(d*x + c)^2 - 2*(a*b*d^2*f^2*x^2 +
2*a*b*d^2*e*f*x + a*b*d^2*e^2)*cosh(d*x + c) - 2*((a^2 + b^2)*d*f^2*x + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e
*f)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)
*d*f^2*x + (a^2 + b^2)*d*e*f)*sinh(d*x + c)^4 + (a^2 + b^2)*d*e*f - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f
)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - 3*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*c
osh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)^3 - ((a^2 + b^2)*
d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(
d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - 2*((a^2 + b^2)*d*f^2*x + ((a^2 + b^2)*d*f^2*x
+ (a^2 + b^2)*d*e*f)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)*sinh(d*x + c)
^3 + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*sinh(d*x + c)^4 + (a^2 + b^2)*d*e*f - 2*((a^2 + b^2)*d*f^2*x +
(a^2 + b^2)*d*e*f)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - 3*((a^2 + b^2)*d*f^2*x + (a^
2 + b^2)*d*e*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c)^
3 - ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*cosh(d*x + c))*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*
x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 2*((a^2 + b^2)*d*f^2*x + ((a^
2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f -
a*b*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2)*sinh(d*x + c)^4
+ (a^2 + b^2)*d*e*f - a*b*f^2 - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2)*cosh(d*x + c)^2 - 2*((a^
2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2 - 3*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2)*cosh(d*x + c)^3 - ((a^2 + b
^2)*d*f^2*x + (a^2 + b^2)*d*e*f - a*b*f^2)*cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c))
+ 2*((a^2 + b^2)*d*f^2*x + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f + a*b*f^2)*cosh(d*x + c)^4 + 4*((a^2 + b^2
)*d*f^2*x + (a^2 + b^2)*d*e*f + a*b*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*
e*f + a*b*f^2)*sinh(d*x + c)^4 + (a^2 + b^2)*d*e*f + a*b*f^2 - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f + a*
b*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f + a*b*f^2 - 3*((a^2 + b^2)*d*f^2*x + (a^2
+ b^2)*d*e*f + a*b*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f + a*b*f
^2)*cosh(d*x + c)^3 - ((a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f + a*b*f^2)*cosh(d*x + c))*sinh(d*x + c))*dilog(
-cosh(d*x + c) - sinh(d*x + c)) - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2 + ((a^2 +
b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d^2*e^2 - 2*(a^2
+ b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*
d*e*f + (a^2 + b^2)*c^2*f^2)*sinh(d*x + c)^4 - 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^
2*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2 - 3*((a^2 + b^2)
*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d^2
*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^3 - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d
*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt
((a^2 + b^2)/b^2) + 2*a) - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2 + ((a^2 + b^2)*d
^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)
*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f +
(a^2 + b^2)*c^2*f^2)*sinh(d*x + c)^4 - 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*
cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2 - 3*((a^2 + b^2)*d^2*e^
2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d^2*e^2 -
2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^3 - ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f +
(a^2 + b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 +
b^2)/b^2) + 2*a) - ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c
^2*f^2 + ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cos
h(d*x + c)^4 + 4*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*
f^2)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*
f - (a^2 + b^2)*c^2*f^2)*sinh(d*x + c)^4 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2
)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a
^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2 - 3*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)
*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)
*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^3 - ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2
+ b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*cosh(d*x
+ c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) - ((a^2 + b^2)*d^2
*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2 + ((a^2 + b^2)*d^2*f^2*x^2 +
2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d^2*f^
2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c)*sinh(d*x + c)^3 +
((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*sinh(d*x +
c)^4 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cos
h(d*x + c)^2 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*
f^2 - 3*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*cosh
(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f -
(a^2 + b^2)*c^2*f^2)*cosh(d*x + c)^3 - ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d
*e*f - (a^2 + b^2)*c^2*f^2)*cosh(d*x + c))*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*
x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + ((a^2 + b^2)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a
*b*d*e*f + ((a^2 + b^2)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((a^2 + b^2)*d^2*e*f + a
*b*d*f^2)*x)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((
a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*
e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x)*sinh(d*x + c)^4 + a^2*f^2 - 2*((a^2 + b^2
)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x)*cosh(d*x
+ c)^2 - 2*((a^2 + b^2)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 - 3*((a^2 + b^2)*d^2*f^2*x^2
+ (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x)*cosh(d*x + c)^2 + 2*((
a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x)*sinh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x + 4*(((a^2 + b^2)*d
^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((a^2 + b^2)*d^2*e*f + a*b*d*f^2)*x)*cosh(d*x + c
)^3 - ((a^2 + b^2)*d^2*f^2*x^2 + (a^2 + b^2)*d^2*e^2 + 2*a*b*d*e*f + a^2*f^2 + 2*((a^2 + b^2)*d^2*e*f + 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 + b^2)*d^2*e^2 + ((a^2 +
b^2)*d^2*e^2 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*cosh(d*x + c)^4 + 4*((a^
2 + b^2)*d^2*e^2 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*cosh(d*x + c)*sinh(d
*x + c)^3 + ((a^2 + b^2)*d^2*e^2 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*sinh
(d*x + c)^4 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2 - 2*((a^2 + b^2)*d^2*e^2 -
2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*e
^2 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2 - 3*((a^2 + b^2)*d^2*e^2 - 2*(a*b +
(a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + b
^2)*d^2*e^2 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*cosh(d*x + c)^3 - ((a^2 +
b^2)*d^2*e^2 - 2*(a*b + (a^2 + b^2)*c)*d*e*f + (2*a*b*c + (a^2 + b^2)*c^2 + a^2)*f^2)*cosh(d*x + c))*sinh(d*x
+ c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f + ((a^2 + b^2
)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f - (2*a*b*c + (a^2 + b^2)*c^2)*f^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*
x)*cosh(d*x + c)^4 + 4*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f - (2*a*b*c + (a^2 + b^2)*c^2)*f^2 + 2*
((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*
c*d*e*f - (2*a*b*c + (a^2 + b^2)*c^2)*f^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*sinh(d*x + c)^4 - (2*a*b*c
+ (a^2 + b^2)*c^2)*f^2 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f - (2*a*b*c + (a^2 + b^2)*c^2)*f^2
+ 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 - 2*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f
- (2*a*b*c + (a^2 + b^2)*c^2)*f^2 - 3*((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f - (2*a*b*c + (a^2 + b^2
)*c^2)*f^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*s
inh(d*x + c)^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x + 4*(((a^2 + b^2)*d^2*f^2*x^2 + 2*(a^2 + b^2)*c*d*e*f -
(2*a*b*c + (a^2 + b^2)*c^2)*f^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*cosh(d*x + c)^3 - ((a^2 + b^2)*d^2*f
^2*x^2 + 2*(a^2 + b^2)*c*d*e*f - (2*a*b*c + (a^2 + b^2)*c^2)*f^2 + 2*((a^2 + b^2)*d^2*e*f - a*b*d*f^2)*x)*cosh
(d*x + c))*sinh(d*x + c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 2*((a^2 + b^2)*f^2*cosh(d*x + c)^4 + 4*(a^
2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)*f^2*sinh(d*x + c)^4 - 2*(a^2 + b^2)*f^2*cosh(d*x + c)
^2 + (a^2 + b^2)*f^2 + 2*(3*(a^2 + b^2)*f^2*cosh(d*x + c)^2 - (a^2 + b^2)*f^2)*sinh(d*x + c)^2 + 4*((a^2 + b^2
)*f^2*cosh(d*x + c)^3 - (a^2 + b^2)*f^2*cosh(d*x + c))*sinh(d*x + c))*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x
+ c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + 2*((a^2 + b^2)*f^2*cosh(d*x + c)^4 + 4
*(a^2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)*f^2*sinh(d*x + c)^4 - 2*(a^2 + b^2)*f^2*cosh(d*x
+ c)^2 + (a^2 + b^2)*f^2 + 2*(3*(a^2 + b^2)*f^2*cosh(d*x + c)^2 - (a^2 + b^2)*f^2)*sinh(d*x + c)^2 + 4*((a^2 +
b^2)*f^2*cosh(d*x + c)^3 - (a^2 + b^2)*f^2*cosh(d*x + c))*sinh(d*x + c))*polylog(3, (a*cosh(d*x + c) + a*sinh
(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) - 2*((a^2 + b^2)*f^2*cosh(d*x + c)^4
+ 4*(a^2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)*f^2*sinh(d*x + c)^4 - 2*(a^2 + b^2)*f^2*cosh(
d*x + c)^2 + (a^2 + b^2)*f^2 + 2*(3*(a^2 + b^2)*f^2*cosh(d*x + c)^2 - (a^2 + b^2)*f^2)*sinh(d*x + c)^2 + 4*((a
^2 + b^2)*f^2*cosh(d*x + c)^3 - (a^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 + b^2)*f^2*cosh(d*x + c)^4 + 4*(a^2 + b^2)*f^2*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + b^2)
*f^2*sinh(d*x + c)^4 - 2*(a^2 + b^2)*f^2*cosh(d*x + c)^2 + (a^2 + b^2)*f^2 + 2*(3*(a^2 + b^2)*f^2*cosh(d*x + c
)^2 - (a^2 + b^2)*f^2)*sinh(d*x + c)^2 + 4*((a^2 + b^2)*f^2*cosh(d*x + c)^3 - (a^2 + b^2)*f^2*cosh(d*x + c))*s
inh(d*x + c))*polylog(3, -cosh(d*x + c) - sinh(d*x + c)) - 2*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + a*b*d^2*e^2
+ 4*(a^2*d*f^2*x + a^2*c*f^2)*cosh(d*x + c)^3 - 3*(a*b*d^2*f^2*x^2 + 2*a*b*d^2*e*f*x + a*b*d^2*e^2)*cosh(d*x +
c)^2 + 2*(a^2*d^2*f^2*x^2 + a^2*d^2*e^2 + a^2*d*e*f - 2*a^2*c*f^2 + (2*a^2*d^2*e*f - a^2*d*f^2)*x)*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)**3/(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)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out