\(\int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [249]

Optimal result

Integrand size = 28, antiderivative size = 725 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2}{a^2 d}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}+\frac {b^3 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \sqrt {a^2+b^2} d^3} \]

[Out]

b*(f*x+e)^2/a^2/d+(f*x+e)^2*arctanh(exp(d*x+c))/a/d-2*b^2*(f*x+e)^2*arctanh(exp(d*x+c))/a^3/d-f^2*arctanh(cosh
(d*x+c))/a/d^3+b*(f*x+e)^2*coth(d*x+c)/a^2/d-f*(f*x+e)*csch(d*x+c)/a/d^2-1/2*(f*x+e)^2*coth(d*x+c)*csch(d*x+c)
/a/d-2*b*f*(f*x+e)*ln(1-exp(2*d*x+2*c))/a^2/d^2+f*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^2-2*b^2*f*(f*x+e)*polylog
(2,-exp(d*x+c))/a^3/d^2-f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2+2*b^2*f*(f*x+e)*polylog(2,exp(d*x+c))/a^3/d^2-b*
f^2*polylog(2,exp(2*d*x+2*c))/a^2/d^3-f^2*polylog(3,-exp(d*x+c))/a/d^3+2*b^2*f^2*polylog(3,-exp(d*x+c))/a^3/d^
3+f^2*polylog(3,exp(d*x+c))/a/d^3-2*b^2*f^2*polylog(3,exp(d*x+c))/a^3/d^3-b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a-(
a^2+b^2)^(1/2)))/a^3/d/(a^2+b^2)^(1/2)+b^3*(f*x+e)^2*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d/(a^2+b^2)^(1
/2)-2*b^3*f*(f*x+e)*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/d^2/(a^2+b^2)^(1/2)+2*b^3*f*(f*x+e)*polyl
og(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^2/(a^2+b^2)^(1/2)+2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)
^(1/2)))/a^3/d^3/(a^2+b^2)^(1/2)-2*b^3*f^2*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/d^3/(a^2+b^2)^(1/2
)

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 725, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5694, 4271, 3855, 4267, 2611, 2320, 6724, 4269, 3797, 2221, 2317, 2438, 3403, 2296} \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b^2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a^3 d}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^3 d^3}-\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^3 d^3}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^3 d^2}+\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^3 d^2}-\frac {b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^2 d^3}-\frac {2 b f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a^2 d^2}+\frac {b (e+f x)^2 \coth (c+d x)}{a^2 d}+\frac {b (e+f x)^2}{a^2 d}+\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3 \sqrt {a^2+b^2}}-\frac {2 b^3 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3 \sqrt {a^2+b^2}}-\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}+\frac {2 b^3 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2 \sqrt {a^2+b^2}}-\frac {b^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d \sqrt {a^2+b^2}}+\frac {b^3 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d \sqrt {a^2+b^2}}-\frac {f^2 \text {arctanh}(\cosh (c+d x))}{a d^3}+\frac {(e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}-\frac {f (e+f x) \text {csch}(c+d x)}{a d^2}-\frac {(e+f x)^2 \coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[In]

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2320

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 2438

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 2611

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 3403

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 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3855

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

Rule 4267

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 4269

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

Rule 4271

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]) /; Free
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 5694

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

Rule 6724

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]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1532\) vs. \(2(725)=1450\).

Time = 8.24 (sec) , antiderivative size = 1532, normalized size of antiderivative = 2.11 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {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 \text {arctanh}\left (e^{c+d x}\right )+4 b^2 d^2 e^2 \text {arctanh}\left (e^{c+d x}\right )+2 a^2 d^2 e^2 e^{2 c} \text {arctanh}\left (e^{c+d x}\right )-4 b^2 d^2 e^2 e^{2 c} \text {arctanh}\left (e^{c+d x}\right )+4 a^2 f^2 \text {arctanh}\left (e^{c+d x}\right )-4 a^2 e^{2 c} f^2 \text {arctanh}\left (e^{c+d x}\right )+2 a^2 d^2 e f x \log \left (1-e^{c+d x}\right )-4 b^2 d^2 e f x \log \left (1-e^{c+d x}\right )-2 a^2 d^2 e e^{2 c} f x \log \left (1-e^{c+d x}\right )+4 b^2 d^2 e e^{2 c} f x \log \left (1-e^{c+d x}\right )+a^2 d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )-2 b^2 d^2 f^2 x^2 \log \left (1-e^{c+d x}\right )-a^2 d^2 e^{2 c} f^2 x^2 \log \left (1-e^{c+d x}\right )+2 b^2 d^2 e^{2 c} f^2 x^2 \log \left (1-e^{c+d x}\right )-2 a^2 d^2 e f x \log \left (1+e^{c+d x}\right )+4 b^2 d^2 e f x \log \left (1+e^{c+d x}\right )+2 a^2 d^2 e e^{2 c} f x \log \left (1+e^{c+d x}\right )-4 b^2 d^2 e e^{2 c} f x \log \left (1+e^{c+d x}\right )-a^2 d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )+2 b^2 d^2 f^2 x^2 \log \left (1+e^{c+d x}\right )+a^2 d^2 e^{2 c} f^2 x^2 \log \left (1+e^{c+d x}\right )-2 b^2 d^2 e^{2 c} f^2 x^2 \log \left (1+e^{c+d x}\right )+4 a b d e f \log \left (1-e^{2 (c+d x)}\right )-4 a b d e e^{2 c} f \log \left (1-e^{2 (c+d x)}\right )+4 a b d f^2 x \log \left (1-e^{2 (c+d x)}\right )-4 a b d e^{2 c} f^2 x \log \left (1-e^{2 (c+d x)}\right )+2 \left (a^2-2 b^2\right ) d \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-2 \left (a^2-2 b^2\right ) d \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )+2 a b f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )-2 a b e^{2 c} f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )+2 a^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )-4 b^2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )-2 a^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )+4 b^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )-2 a^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )+4 b^2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )+2 a^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )-4 b^2 e^{2 c} f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{2 a^3 d^3 \left (-1+e^{2 c}\right )}-\frac {b^3 \left (-2 d^2 e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-2 d^2 e f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^2 f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+2 d f (e+f x) \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^3 \sqrt {a^2+b^2} d^3}+\frac {\text {csch}(c) \text {csch}^2(c+d x) \left (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)\right )}{4 a^2 d^2} \]

[In]

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

Maple [F]

\[\int \frac {\left (f x +e \right )^{2} \operatorname {csch}\left (d x +c \right )^{3}}{a +b \sinh \left (d x +c \right )}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10341 vs. \(2 (676) = 1352\).

Time = 0.40 (sec) , antiderivative size = 10341, normalized size of antiderivative = 14.26 \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\left (e + f x\right )^{2} \operatorname {csch}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[In]

integrate((f*x+e)**2*csch(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**2*csch(c + d*x)**3/(a + b*sinh(c + d*x)), x)

Maxima [F]

\[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{3}}{b \sinh \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*e^2*(2*b^3*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 +
b^2)*a^3*d) - 2*(a*e^(-d*x - c) + 2*b*e^(-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) -
a^2*e^(-4*d*x - 4*c) - a^2)*d) - (a^2 - 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + (a^2 - 2*b^2)*log(e^(-d*x - c)
- 1)/(a^3*d)) - (2*b*d*f^2*x^2 + 4*b*d*e*f*x + (a*d*f^2*x^2*e^(3*c) + 2*a*e*f*e^(3*c) + 2*(d*e*f + f^2)*a*x*e
^(3*c))*e^(3*d*x) - 2*(b*d*f^2*x^2*e^(2*c) + 2*b*d*e*f*x*e^(2*c))*e^(2*d*x) + (a*d*f^2*x^2*e^c - 2*a*e*f*e^c +
2*(d*e*f - f^2)*a*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) + 1/2*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x
+ c)) - 2*polylog(3, -e^(d*x + c)))*(a^2*f^2 - 2*b^2*f^2)/(a^3*d^3) - 1/2*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*
d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*(a^2*f^2 - 2*b^2*f^2)/(a^3*d^3) + (a^2*d*e*f - 2*b^2*d*e*f
- 2*a*b*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a^3*d^3) - (a^2*d*e*f - 2*b^2*d*e*f + 2*a*b*f^
2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) + 1/6*((a^2*f^2 - 2*b^2*f^2)*d^3*x^3 + 3*(a^2*d*
e*f - 2*b^2*d*e*f + 2*a*b*f^2)*d^2*x^2)/(a^3*d^3) - 1/6*((a^2*f^2 - 2*b^2*f^2)*d^3*x^3 + 3*(a^2*d*e*f - 2*b^2*
d*e*f - 2*a*b*f^2)*d^2*x^2)/(a^3*d^3) - integrate(2*(b^3*f^2*x^2*e^c + 2*b^3*e*f*x*e^c)*e^(d*x)/(a^3*b*e^(2*d*
x + 2*c) + 2*a^4*e^(d*x + c) - a^3*b), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((f*x+e)^2*csch(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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