\(\int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [243]

Optimal result

Integrand size = 28, antiderivative size = 745 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4} \]

[Out]

-(f*x+e)^3/a/d+2*b*(f*x+e)^3*arctanh(exp(d*x+c))/a^2/d-(f*x+e)^3*coth(d*x+c)/a/d+3*f*(f*x+e)^2*ln(1-exp(2*d*x+
2*c))/a/d^2+3*b*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a^2/d^2-3*b*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a^2/d^2+3*f^2
*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^3-6*b*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a^2/d^3+6*b*f^2*(f*x+e)*polylo
g(3,exp(d*x+c))/a^2/d^3-3/2*f^3*polylog(3,exp(2*d*x+2*c))/a/d^4+6*b*f^3*polylog(4,-exp(d*x+c))/a^2/d^4-6*b*f^3
*polylog(4,exp(d*x+c))/a^2/d^4+b^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)-b^2*
(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d/(a^2+b^2)^(1/2)+3*b^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+
c)/(a-(a^2+b^2)^(1/2)))/a^2/d^2/(a^2+b^2)^(1/2)-3*b^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))
/a^2/d^2/(a^2+b^2)^(1/2)-6*b^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2
)+6*b^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^2/d^3/(a^2+b^2)^(1/2)+6*b^2*f^3*polylog(4,-
b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a^2/d^4/(a^2+b^2)^(1/2)-6*b^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)
))/a^2/d^4/(a^2+b^2)^(1/2)

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 745, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5694, 4269, 3797, 2221, 2611, 2320, 6724, 4267, 6744, 3403, 2296} \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^4 \sqrt {a^2+b^2}}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^4 \sqrt {a^2+b^2}}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^3 \sqrt {a^2+b^2}}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^3 \sqrt {a^2+b^2}}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {(e+f x)^3}{a d} \]

[In]

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

[Out]

-((e + f*x)^3/(a*d)) + (2*b*(e + f*x)^3*ArcTanh[E^(c + d*x)])/(a^2*d) - ((e + f*x)^3*Coth[c + d*x])/(a*d) + (b
^2*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^2*(e + f*x)^3*Log[
1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) + (3*f*(e + f*x)^2*Log[1 - E^(2*(c + d*x))
])/(a*d^2) + (3*b*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/(a^2*d^2) - (3*b*f*(e + f*x)^2*PolyLog[2, E^(c + d*x
)])/(a^2*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2
]*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2)
+ (3*f^2*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a*d^3) - (6*b*f^2*(e + f*x)*PolyLog[3, -E^(c + d*x)])/(a^2*d^
3) + (6*b*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/(a^2*d^3) - (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))
/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) + (6*b^2*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + S
qrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3) - (3*f^3*PolyLog[3, E^(2*(c + d*x))])/(2*a*d^4) + (6*b*f^3*PolyLo
g[4, -E^(c + d*x)])/(a^2*d^4) - (6*b*f^3*PolyLog[4, E^(c + d*x)])/(a^2*d^4) + (6*b^2*f^3*PolyLog[4, -((b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4) - (6*b^2*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt
[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^4)

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 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 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 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 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x)^3 \text {csch}^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x)^3 \coth (c+d x)}{a d}-\frac {b \int (e+f x)^3 \text {csch}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{a^2}+\frac {(3 f) \int (e+f x)^2 \coth (c+d x) \, dx}{a d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {\left (2 b^2\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a^2}-\frac {(6 f) \int \frac {e^{2 (c+d x)} (e+f x)^2}{1-e^{2 (c+d x)}} \, dx}{a d}+\frac {(3 b f) \int (e+f x)^2 \log \left (1-e^{c+d x}\right ) \, dx}{a^2 d}-\frac {(3 b f) \int (e+f x)^2 \log \left (1+e^{c+d x}\right ) \, dx}{a^2 d} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \sqrt {a^2+b^2}}-\frac {\left (2 b^3\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{a^2 \sqrt {a^2+b^2}}-\frac {\left (6 f^2\right ) \int (e+f x) \log \left (1-e^{2 (c+d x)}\right ) \, dx}{a d^2}-\frac {\left (6 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right ) \, dx}{a^2 d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right ) \, dx}{a^2 d^2} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d}+\frac {\left (3 b^2 f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d}-\frac {\left (3 f^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right ) \, dx}{a d^3}+\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,-e^{c+d x}\right ) \, dx}{a^2 d^3}-\frac {\left (6 b f^3\right ) \int \operatorname {PolyLog}\left (3,e^{c+d x}\right ) \, dx}{a^2 d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {\left (6 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d^2}+\frac {\left (6 b^2 f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d^2}-\frac {\left (3 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4}-\frac {\left (6 b f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^4} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d^3}-\frac {\left (6 b^2 f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{a^2 \sqrt {a^2+b^2} d^3} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {\left (6 b^2 f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^2 \sqrt {a^2+b^2} d^4} \\ & = -\frac {(e+f x)^3}{a d}+\frac {2 b (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a^2 d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}-\frac {b^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^2}-\frac {3 b f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^2}+\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}-\frac {3 b^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a^2 d^3}+\frac {6 b f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a^2 d^3}-\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}+\frac {6 b^2 f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 b f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a^2 d^4}-\frac {6 b f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a^2 d^4}+\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4}-\frac {6 b^2 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^4} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1493\) vs. \(2(745)=1490\).

Time = 8.54 (sec) , antiderivative size = 1493, normalized size of antiderivative = 2.00 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {-d^3 e^2 \left (-1+e^{2 c}\right ) (b d e-3 a f) x+d^3 e^2 \left (-1+e^{2 c}\right ) (b d e+3 a f) x+2 a d^3 (e+f x)^3+3 d^2 e \left (-1+e^{2 c}\right ) f (b d e-2 a f) x \log \left (1-e^{-c-d x}\right )+3 d^2 \left (-1+e^{2 c}\right ) f^2 (b d e-a f) x^2 \log \left (1-e^{-c-d x}\right )+b d^3 \left (-1+e^{2 c}\right ) f^3 x^3 \log \left (1-e^{-c-d x}\right )-3 d^2 e \left (-1+e^{2 c}\right ) f (b d e+2 a f) x \log \left (1+e^{-c-d x}\right )-3 d^2 \left (-1+e^{2 c}\right ) f^2 (b d e+a f) x^2 \log \left (1+e^{-c-d x}\right )-b d^3 \left (-1+e^{2 c}\right ) f^3 x^3 \log \left (1+e^{-c-d x}\right )+d^2 e^2 \left (-1+e^{2 c}\right ) (b d e-3 a f) \log \left (1-e^{c+d x}\right )-d^2 e^2 \left (-1+e^{2 c}\right ) (b d e+3 a f) \log \left (1+e^{c+d x}\right )+3 d e \left (-1+e^{2 c}\right ) f (b d e+2 a f) \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+6 d \left (-1+e^{2 c}\right ) f^2 (b d e+a f) x \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+3 b d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-3 d e \left (-1+e^{2 c}\right ) f (b d e-2 a f) \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-6 d \left (-1+e^{2 c}\right ) f^2 (b d e-a f) x \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-3 b d^2 \left (-1+e^{2 c}\right ) f^3 x^2 \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+6 \left (-1+e^{2 c}\right ) f^2 (b d e+a f) \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 b d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+6 \left (-1+e^{2 c}\right ) f^2 (-b d e+a f) \operatorname {PolyLog}\left (3,e^{-c-d x}\right )-6 b d \left (-1+e^{2 c}\right ) f^3 x \operatorname {PolyLog}\left (3,e^{-c-d x}\right )+6 b \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (4,-e^{-c-d x}\right )-6 b \left (-1+e^{2 c}\right ) f^3 \operatorname {PolyLog}\left (4,e^{-c-d x}\right )}{a^2 d^4 \left (-1+e^{2 c}\right )}+\frac {b^2 \left (-2 d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )+3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-3 d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-3 d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-3 d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-6 d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+6 d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+6 f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-6 f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{a^2 \sqrt {a^2+b^2} d^4}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-e^3 \sinh \left (\frac {d x}{2}\right )-3 e^2 f x \sinh \left (\frac {d x}{2}\right )-3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )-f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^3 \sinh \left (\frac {d x}{2}\right )+3 e^2 f x \sinh \left (\frac {d x}{2}\right )+3 e f^2 x^2 \sinh \left (\frac {d x}{2}\right )+f^3 x^3 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d} \]

[In]

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

[Out]

-((-(d^3*e^2*(-1 + E^(2*c))*(b*d*e - 3*a*f)*x) + d^3*e^2*(-1 + E^(2*c))*(b*d*e + 3*a*f)*x + 2*a*d^3*(e + f*x)^
3 + 3*d^2*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*x*Log[1 - E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*(b*d*e - a*f)*
x^2*Log[1 - E^(-c - d*x)] + b*d^3*(-1 + E^(2*c))*f^3*x^3*Log[1 - E^(-c - d*x)] - 3*d^2*e*(-1 + E^(2*c))*f*(b*d
*e + 2*a*f)*x*Log[1 + E^(-c - d*x)] - 3*d^2*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*x^2*Log[1 + E^(-c - d*x)] - b*d^3
*(-1 + E^(2*c))*f^3*x^3*Log[1 + E^(-c - d*x)] + d^2*e^2*(-1 + E^(2*c))*(b*d*e - 3*a*f)*Log[1 - E^(c + d*x)] -
d^2*e^2*(-1 + E^(2*c))*(b*d*e + 3*a*f)*Log[1 + E^(c + d*x)] + 3*d*e*(-1 + E^(2*c))*f*(b*d*e + 2*a*f)*PolyLog[2
, -E^(-c - d*x)] + 6*d*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*x*PolyLog[2, -E^(-c - d*x)] + 3*b*d^2*(-1 + E^(2*c))*f
^3*x^2*PolyLog[2, -E^(-c - d*x)] - 3*d*e*(-1 + E^(2*c))*f*(b*d*e - 2*a*f)*PolyLog[2, E^(-c - d*x)] - 6*d*(-1 +
E^(2*c))*f^2*(b*d*e - a*f)*x*PolyLog[2, E^(-c - d*x)] - 3*b*d^2*(-1 + E^(2*c))*f^3*x^2*PolyLog[2, E^(-c - d*x
)] + 6*(-1 + E^(2*c))*f^2*(b*d*e + a*f)*PolyLog[3, -E^(-c - d*x)] + 6*b*d*(-1 + E^(2*c))*f^3*x*PolyLog[3, -E^(
-c - d*x)] + 6*(-1 + E^(2*c))*f^2*(-(b*d*e) + a*f)*PolyLog[3, E^(-c - d*x)] - 6*b*d*(-1 + E^(2*c))*f^3*x*PolyL
og[3, E^(-c - d*x)] + 6*b*(-1 + E^(2*c))*f^3*PolyLog[4, -E^(-c - d*x)] - 6*b*(-1 + E^(2*c))*f^3*PolyLog[4, E^(
-c - d*x)])/(a^2*d^4*(-1 + E^(2*c)))) + (b^2*(-2*d^3*e^3*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]] + 3*d^3*
e^2*f*x*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] + 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2])] + d^3*f^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - 3*d^3*e^2*f*x*Log[1 + (b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2])] - 3*d^3*e*f^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - d^3*f^3*x^3*Log[1 +
(b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*f*(e + f*x)^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2]
)] - 3*d^2*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*e*f^2*PolyLog[3, (b*E^(c +
d*x))/(-a + Sqrt[a^2 + b^2])] - 6*d*f^3*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*e*f^2*Poly
Log[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*d*f^3*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]
))] + 6*f^3*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[
a^2 + b^2]))]))/(a^2*Sqrt[a^2 + b^2]*d^4) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-(e^3*Sinh[(d*x)/2]) - 3*e^2*f*x*S
inh[(d*x)/2] - 3*e*f^2*x^2*Sinh[(d*x)/2] - f^3*x^3*Sinh[(d*x)/2]))/(2*a*d) + (Csch[c/2]*Csch[c/2 + (d*x)/2]*(e
^3*Sinh[(d*x)/2] + 3*e^2*f*x*Sinh[(d*x)/2] + 3*e*f^2*x^2*Sinh[(d*x)/2] + f^3*x^3*Sinh[(d*x)/2]))/(2*a*d)

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^3*csch(d*x+c)^2/(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. 6416 vs. \(2 (691) = 1382\).

Time = 0.39 (sec) , antiderivative size = 6416, normalized size of antiderivative = 8.61 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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