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

Optimal. Leaf size=413 \[ -\frac{b^2 f \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac{b^2 f \text{PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}-\frac{f \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac{f \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}-\frac{2 b^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{b f \log (\sinh (c+d x))}{a^2 d^2}+\frac{b (e+f x) \coth (c+d x)}{a^2 d}-\frac{f \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{(e+f x) \coth (c+d x) \text{csch}(c+d x)}{2 a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.928764, antiderivative size = 413, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 17, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.531, Rules used = {5587, 5457, 4182, 2279, 2391, 4185, 5569, 3720, 3475, 5585, 5450, 3296, 2637, 5565, 3322, 2264, 2190} \[ -\frac{b^2 f \text{PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac{b^2 f \text{PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^3 d^2}+\frac{b f \sqrt{a^2+b^2} \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^3 d^2}-\frac{f \text{PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}+\frac{f \text{PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^3 d}+\frac{b \sqrt{a^2+b^2} (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^3 d}-\frac{2 b^2 (e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a^3 d}-\frac{b f \log (\sinh (c+d x))}{a^2 d^2}+\frac{b (e+f x) \coth (c+d x)}{a^2 d}-\frac{f \text{csch}(c+d x)}{2 a d^2}-\frac{(e+f x) \tanh ^{-1}\left (e^{c+d x}\right )}{a d}-\frac{(e+f x) \coth (c+d x) \text{csch}(c+d x)}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 4185

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

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 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 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]

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]

Rubi steps

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

Mathematica [C]  time = 8.17093, size = 734, normalized size = 1.78 \[ \frac{b \sqrt{a^2+b^2} \left (-f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+2 d e \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )-f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )-2 c f \tanh ^{-1}\left (\frac{a+b e^{c+d x}}{\sqrt{a^2+b^2}}\right )\right )}{a^3 d^2}-\frac{i b^2 f \left (i \left (\text{PolyLog}\left (2,-e^{-c-d x}\right )-\text{PolyLog}\left (2,e^{-c-d x}\right )\right )+i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )}{a^3 d^2}-\frac{i f \left (i \left (\text{PolyLog}\left (2,-e^{-c-d x}\right )-\text{PolyLog}\left (2,e^{-c-d x}\right )\right )+i (c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )}{2 a d^2}-\frac{b^2 c f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 d^2}+\frac{b^2 e \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a^3 d}+\frac{\text{csch}\left (\frac{1}{2} (c+d x)\right ) \left (-a f \cosh \left (\frac{1}{2} (c+d x)\right )+2 b d e \cosh \left (\frac{1}{2} (c+d x)\right )-2 b c f \cosh \left (\frac{1}{2} (c+d x)\right )+2 b f (c+d x) \cosh \left (\frac{1}{2} (c+d x)\right )\right )}{4 a^2 d^2}+\frac{\text{sech}\left (\frac{1}{2} (c+d x)\right ) \left (a f \sinh \left (\frac{1}{2} (c+d x)\right )+2 b d e \sinh \left (\frac{1}{2} (c+d x)\right )-2 b c f \sinh \left (\frac{1}{2} (c+d x)\right )+2 b f (c+d x) \sinh \left (\frac{1}{2} (c+d x)\right )\right )}{4 a^2 d^2}-\frac{b f \log (\sinh (c+d x))}{a^2 d^2}+\frac{\text{csch}^2\left (\frac{1}{2} (c+d x)\right ) (-f (c+d x)+c f-d e)}{8 a d^2}+\frac{\text{sech}^2\left (\frac{1}{2} (c+d x)\right ) (-f (c+d x)+c f-d e)}{8 a d^2}-\frac{c f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 a d^2}+\frac{e \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*d*e*Cosh[(c + d*x)/2] - a*f*Cosh[(c + d*x)/2] - 2*b*c*f*Cosh[(c + d*x)/2] + 2*b*f*(c + d*x)*Cosh[(c + d*
x)/2])*Csch[(c + d*x)/2])/(4*a^2*d^2) + ((-(d*e) + c*f - f*(c + d*x))*Csch[(c + d*x)/2]^2)/(8*a*d^2) - (b*f*Lo
g[Sinh[c + d*x]])/(a^2*d^2) + (e*Log[Tanh[(c + d*x)/2]])/(2*a*d) + (b^2*e*Log[Tanh[(c + d*x)/2]])/(a^3*d) - (c
*f*Log[Tanh[(c + d*x)/2]])/(2*a*d^2) - (b^2*c*f*Log[Tanh[(c + d*x)/2]])/(a^3*d^2) - ((I/2)*f*(I*(c + d*x)*(Log
[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + I*(PolyLog[2, -E^(-c - d*x)] - PolyLog[2, E^(-c - d*x)])))/(a*d^
2) - (I*b^2*f*(I*(c + d*x)*(Log[1 - E^(-c - d*x)] - Log[1 + E^(-c - d*x)]) + I*(PolyLog[2, -E^(-c - d*x)] - Po
lyLog[2, E^(-c - d*x)])))/(a^3*d^2) + (b*Sqrt[a^2 + b^2]*(2*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] -
2*c*f*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] - f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]
)] + f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - f*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2
+ b^2])] + f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(a^3*d^2) + ((-(d*e) + c*f - f*(c + d*x))*
Sech[(c + d*x)/2]^2)/(8*a*d^2) + (Sech[(c + d*x)/2]*(2*b*d*e*Sinh[(c + d*x)/2] + a*f*Sinh[(c + d*x)/2] - 2*b*c
*f*Sinh[(c + d*x)/2] + 2*b*f*(c + d*x)*Sinh[(c + d*x)/2]))/(4*a^2*d^2)

________________________________________________________________________________________

Maple [B]  time = 0.233, size = 1284, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a^3/d^2*b^2*f*c*ln(exp(d*x+c)-1)-1/a^3/d^2*b^3*f/(a^2+b^2)^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-
a+(a^2+b^2)^(1/2)))+1/a^3/d^2*b^3*f/(a^2+b^2)^(1/2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2))
)-1/a^3/d*b^2*f*ln(exp(d*x+c)+1)*x+2/a^3/d*b^3*e/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1
/2))+1/a^3/d*b^3*f/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/a^3/d^2*b^3*f/
(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/a^3/d^2*b^3*f/(a^2+b^2)^(1/2)*l
n((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-2/a^3/d^2*b^3*f*c/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*e
xp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/a^3/d*b^3*f/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^
2)^(1/2)))*x+2/d*e*b/a/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/d^2*f*b/a/(a^2+b^2)
^(1/2)*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))+1/d^2*f*b/a/(a^2+b^2)^(1/2)*dilog((b*exp(
d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-2/d^2*f*c*b/a/(a^2+b^2)^(1/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/
(a^2+b^2)^(1/2))-1/d*f*b/a/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/d^2*
f*b/a/(a^2+b^2)^(1/2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/d*f*b/a/(a^2+b^2)^(1/2)*l
n((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/d^2*f*b/a/(a^2+b^2)^(1/2)*ln((b*exp(d*x+c)+(a^2+b^
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/a^3/d^2*b^2*f*dilog(exp(d*x+c))-1/a^3/d^2*b^2*f*dilog(exp(d*x+c)+1)+2/a^2
/d^2*b*f*ln(exp(d*x+c))-1/a^2/d^2*b*f*ln(exp(d*x+c)-1)-1/a^2/d^2*b*f*ln(exp(d*x+c)+1)+1/a^3/d*b^2*e*ln(exp(d*x
+c)-1)-1/a^3/d*b^2*e*ln(exp(d*x+c)+1)-1/2/d^2*f/a*dilog(exp(d*x+c))-1/2/d^2*f/a*dilog(exp(d*x+c)+1)+1/2/d/a*e*
ln(exp(d*x+c)-1)-1/2/d/a*e*ln(exp(d*x+c)+1)-1/2/d/a*ln(exp(d*x+c)+1)*f*x-1/2/d^2/a*f*c*ln(exp(d*x+c)-1)-(a*d*f
*x*exp(3*d*x+3*c)+a*d*e*exp(3*d*x+3*c)-2*b*d*f*x*exp(2*d*x+2*c)+a*d*f*x*exp(d*x+c)+a*f*exp(3*d*x+3*c)-2*b*d*e*
exp(2*d*x+2*c)+a*d*e*exp(d*x+c)+2*b*d*f*x-a*f*exp(d*x+c)+2*b*d*e)/d^2/a^2/(exp(2*d*x+2*c)-1)^2

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.71947, size = 8606, normalized size = 20.84 \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)*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*f*x + a*b*c*f)*cosh(d*x + c)^4 + 4*(a*b*d*f*x + a*b*c*f)*sinh(d*x + c)^4 - 4*a*b*d*e + 4*a*b*c*f
- 2*(a^2*d*f*x + a^2*d*e + a^2*f)*cosh(d*x + c)^3 - 2*(a^2*d*f*x + a^2*d*e + a^2*f - 8*(a*b*d*f*x + a*b*c*f)*
cosh(d*x + c))*sinh(d*x + c)^3 - 4*(a*b*d*f*x - a*b*d*e + 2*a*b*c*f)*cosh(d*x + c)^2 - 2*(2*a*b*d*f*x - 2*a*b*
d*e + 4*a*b*c*f - 12*(a*b*d*f*x + a*b*c*f)*cosh(d*x + c)^2 + 3*(a^2*d*f*x + a^2*d*e + a^2*f)*cosh(d*x + c))*si
nh(d*x + c)^2 - 2*(b^2*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b
^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3
- b^2*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*f*cosh(d*x + c)^4 + 4*b^2*f*cosh(d*x +
c)*sinh(d*x + c)^3 + b^2*f*sinh(d*x + c)^4 - 2*b^2*f*cosh(d*x + c)^2 + b^2*f + 2*(3*b^2*f*cosh(d*x + c)^2 - b
^2*f)*sinh(d*x + c)^2 + 4*(b^2*f*cosh(d*x + c)^3 - b^2*f*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*d
ilog((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*e - b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*e -
b^2*c*f)*sinh(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*e - b^2*c*f -
3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*e - b
^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sq
rt((a^2 + b^2)/b^2) + 2*a) - 2*((b^2*d*e - b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*e - b^2*c*f)*cosh(d*x + c)*sinh
(d*x + c)^3 + (b^2*d*e - b^2*c*f)*sinh(d*x + c)^4 + b^2*d*e - b^2*c*f - 2*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2
- 2*(b^2*d*e - b^2*c*f - 3*(b^2*d*e - b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*e - b^2*c*f)*cosh(
d*x + c)^3 - (b^2*d*e - b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*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) - 2*(b^2*d*f*x + (b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^4 + 4
*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c)^4 + b^2*c*f - 2*(b^
2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 - 2*(b^2*d*f*x + b^2*c*f - 3*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2)*sinh(d*
x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^3 - (b^2*d*f*x + b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*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^2*d*f*x + (b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^4 + 4*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)
*sinh(d*x + c)^3 + (b^2*d*f*x + b^2*c*f)*sinh(d*x + c)^4 + b^2*c*f - 2*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2 -
2*(b^2*d*f*x + b^2*c*f - 3*(b^2*d*f*x + b^2*c*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((b^2*d*f*x + b^2*c*f)*
cosh(d*x + c)^3 - (b^2*d*f*x + b^2*c*f)*cosh(d*x + c))*sinh(d*x + c))*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*(a^2*d*f*x + a^2
*d*e - a^2*f)*cosh(d*x + c) + ((a^2 + 2*b^2)*f*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*f*cosh(d*x + c)*sinh(d*x + c)
^3 + (a^2 + 2*b^2)*f*sinh(d*x + c)^4 - 2*(a^2 + 2*b^2)*f*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*b^2)*f*cosh(d*x + c)^
2 - (a^2 + 2*b^2)*f)*sinh(d*x + c)^2 + (a^2 + 2*b^2)*f + 4*((a^2 + 2*b^2)*f*cosh(d*x + c)^3 - (a^2 + 2*b^2)*f*
cosh(d*x + c))*sinh(d*x + c))*dilog(cosh(d*x + c) + sinh(d*x + c)) - ((a^2 + 2*b^2)*f*cosh(d*x + c)^4 + 4*(a^2
+ 2*b^2)*f*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*b^2)*f*sinh(d*x + c)^4 - 2*(a^2 + 2*b^2)*f*cosh(d*x + c)^
2 + 2*(3*(a^2 + 2*b^2)*f*cosh(d*x + c)^2 - (a^2 + 2*b^2)*f)*sinh(d*x + c)^2 + (a^2 + 2*b^2)*f + 4*((a^2 + 2*b^
2)*f*cosh(d*x + c)^3 - (a^2 + 2*b^2)*f*cosh(d*x + c))*sinh(d*x + c))*dilog(-cosh(d*x + c) - sinh(d*x + c)) - (
((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f)*cosh(d*x + c)^4 + 4*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d
*e + 2*a*b*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f)*sinh(d*x + c
)^4 + (a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f - 2*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f
)*cosh(d*x + c)^2 - 2*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f - 3*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b
^2)*d*e + 2*a*b*f)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f)*c
osh(d*x + c)^3 - ((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*d*e + 2*a*b*f)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*
x + c) + sinh(d*x + c) + 1) + (((a^2 + 2*b^2)*d*e - (2*a*b + (a^2 + 2*b^2)*c)*f)*cosh(d*x + c)^4 + 4*((a^2 + 2
*b^2)*d*e - (2*a*b + (a^2 + 2*b^2)*c)*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d*e - (2*a*b + (a^2 +
2*b^2)*c)*f)*sinh(d*x + c)^4 + (a^2 + 2*b^2)*d*e - 2*((a^2 + 2*b^2)*d*e - (2*a*b + (a^2 + 2*b^2)*c)*f)*cosh(d*
x + c)^2 - 2*((a^2 + 2*b^2)*d*e - 3*((a^2 + 2*b^2)*d*e - (2*a*b + (a^2 + 2*b^2)*c)*f)*cosh(d*x + c)^2 - (2*a*b
+ (a^2 + 2*b^2)*c)*f)*sinh(d*x + c)^2 - (2*a*b + (a^2 + 2*b^2)*c)*f + 4*(((a^2 + 2*b^2)*d*e - (2*a*b + (a^2 +
2*b^2)*c)*f)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d*e - (2*a*b + (a^2 + 2*b^2)*c)*f)*cosh(d*x + c))*sinh(d*x + c)
)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + (((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f)*cosh(d*x + c)^4 + 4*((a^
2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f)*cosh(d*x + c)*sinh(d*x + c)^3 + ((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f
)*sinh(d*x + c)^4 + (a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f - 2*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f)*cosh
(d*x + c)^2 - 2*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f - 3*((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f)*cosh(d*
x + c)^2)*sinh(d*x + c)^2 + 4*(((a^2 + 2*b^2)*d*f*x + (a^2 + 2*b^2)*c*f)*cosh(d*x + c)^3 - ((a^2 + 2*b^2)*d*f*
x + (a^2 + 2*b^2)*c*f)*cosh(d*x + c))*sinh(d*x + c))*log(-cosh(d*x + c) - sinh(d*x + c) + 1) - 2*(a^2*d*f*x +
a^2*d*e - 8*(a*b*d*f*x + a*b*c*f)*cosh(d*x + c)^3 - a^2*f + 3*(a^2*d*f*x + a^2*d*e + a^2*f)*cosh(d*x + c)^2 +
4*(a*b*d*f*x - a*b*d*e + 2*a*b*c*f)*cosh(d*x + c))*sinh(d*x + c))/(a^3*d^2*cosh(d*x + c)^4 + 4*a^3*d^2*cosh(d*
x + c)*sinh(d*x + c)^3 + a^3*d^2*sinh(d*x + c)^4 - 2*a^3*d^2*cosh(d*x + c)^2 + a^3*d^2 + 2*(3*a^3*d^2*cosh(d*x
+ c)^2 - a^3*d^2)*sinh(d*x + c)^2 + 4*(a^3*d^2*cosh(d*x + c)^3 - a^3*d^2*cosh(d*x + c))*sinh(d*x + c))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out