\(\int \frac {(e+f x)^2 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) [477]

Optimal result

Integrand size = 34, antiderivative size = 502 \[ \int \frac {(e+f x)^2 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}-\frac {f (e+f x) \coth (c+d x)}{a d^2}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x)^2 \text {csch}^2(c+d x)}{2 a d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {b^2 (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b^2 (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}+\frac {2 b f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^3 d^2}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5706, 5560, 4269, 3556, 4267, 2317, 2438, 5688, 3797, 2221, 2611, 2320, 6724, 5680} \[ \int \frac {(e+f x)^2 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^2 f^2 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a^3 d^3}+\frac {b^2 f (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a^3 d^2}+\frac {b^2 (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a^3 d}+\frac {4 b f (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a^2 d^2}+\frac {2 b f^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a^2 d^3}-\frac {2 b f^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a^2 d^3}+\frac {b (e+f x)^2 \text {csch}(c+d x)}{a^2 d}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^3}+\frac {2 b^2 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^3}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {2 b^2 f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d}-\frac {b^2 (e+f x)^2 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d}+\frac {f^2 \log (\sinh (c+d x))}{a d^3}-\frac {f (e+f x) \coth (c+d x)}{a d^2}-\frac {(e+f x)^2 \text {csch}^2(c+d x)}{2 a d} \]

[In]

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

[Out]

(4*b*f*(e + f*x)*ArcTanh[E^(c + d*x)])/(a^2*d^2) - (f*(e + f*x)*Coth[c + d*x])/(a*d^2) + (b*(e + f*x)^2*Csch[c
+ d*x])/(a^2*d) - ((e + f*x)^2*Csch[c + d*x]^2)/(2*a*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[
a^2 + b^2])])/(a^3*d) - (b^2*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a^3*d) + (b^2*(e + f
*x)^2*Log[1 - E^(2*(c + d*x))])/(a^3*d) + (f^2*Log[Sinh[c + d*x]])/(a*d^3) + (2*b*f^2*PolyLog[2, -E^(c + d*x)]
)/(a^2*d^3) - (2*b*f^2*PolyLog[2, E^(c + d*x)])/(a^2*d^3) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2]))])/(a^3*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a
^3*d^2) + (b^2*f*(e + f*x)*PolyLog[2, E^(2*(c + d*x))])/(a^3*d^2) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a
- Sqrt[a^2 + b^2]))])/(a^3*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a^3*d^3)
- (b^2*f^2*PolyLog[3, E^(2*(c + d*x))])/(2*a^3*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 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 3556

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

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 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 5680

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

Rule 5688

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 5706

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1816\) vs. \(2(502)=1004\).

Time = 10.32 (sec) , antiderivative size = 1816, normalized size of antiderivative = 3.62 \[ \int \frac {(e+f x)^2 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b (e+f x)^2 \text {csch}(c)}{a^2 d}+\frac {\left (-e^2-2 e f x-f^2 x^2\right ) \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 a d}-\frac {12 d e^{2 c} \left (b^2 d^2 e^2+a^2 f^2\right ) x-12 d \left (-1+e^{2 c}\right ) \left (b^2 d^2 e^2+a^2 f^2\right ) x+12 b^2 d^3 e f x^2+4 b^2 d^3 f^2 x^3-24 a b d e \left (-1+e^{2 c}\right ) f \text {arctanh}\left (e^{c+d x}\right )+6 b^2 d^2 e^2 \left (-1+e^{2 c}\right ) \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )+6 a^2 \left (-1+e^{2 c}\right ) f^2 \left (2 d x-\log \left (1-e^{2 (c+d x)}\right )\right )+12 a b \left (-1+e^{2 c}\right ) f^2 \left (d x \left (\log \left (1-e^{c+d x}\right )-\log \left (1+e^{c+d x}\right )\right )-\operatorname {PolyLog}\left (2,-e^{c+d x}\right )+\operatorname {PolyLog}\left (2,e^{c+d x}\right )\right )+6 b^2 d e \left (-1+e^{2 c}\right ) f \left (2 d x \left (d x-\log \left (1-e^{2 (c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )\right )+b^2 \left (-1+e^{2 c}\right ) f^2 \left (2 d^2 x^2 \left (2 d x-3 \log \left (1-e^{2 (c+d x)}\right )\right )-6 d x \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )+3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )\right )}{6 a^3 d^3 \left (-1+e^{2 c}\right )}+\frac {b^2 \left (6 e^2 e^{2 c} x+6 e e^{2 c} f x^2+2 e^{2 c} f^2 x^3+\frac {6 a \sqrt {a^2+b^2} e^2 \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\left (a^2+b^2\right )^{3/2} d}-\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {6 a \sqrt {-\left (a^2+b^2\right )^2} e^2 e^{2 c} \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2} d}+\frac {3 e^2 \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}-\frac {3 e^2 e^{2 c} \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )}{d}+\frac {6 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 e e^{2 c} f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {3 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {3 e^{2 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {6 e f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 e e^{2 c} f x \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}+\frac {3 f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {3 e^{2 c} f^2 x^2 \log \left (1+\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 \left (-1+e^{2 c}\right ) f (e+f x) \operatorname {PolyLog}\left (2,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^2}-\frac {6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c-\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}-\frac {6 f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}+\frac {6 e^{2 c} f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{2 c+d x}}{a e^c+\sqrt {\left (a^2+b^2\right ) e^{2 c}}}\right )}{d^3}\right )}{3 a^3 \left (-1+e^{2 c}\right )}+\frac {\left (e^2+2 e f x+f^2 x^2\right ) \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{8 a d}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-b d e^2 \sinh \left (\frac {d x}{2}\right )-a e f \sinh \left (\frac {d x}{2}\right )-2 b d e f x \sinh \left (\frac {d x}{2}\right )-a f^2 x \sinh \left (\frac {d x}{2}\right )-b d f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a^2 d^2}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-b d e^2 \sinh \left (\frac {d x}{2}\right )+a e f \sinh \left (\frac {d x}{2}\right )-2 b d e f x \sinh \left (\frac {d x}{2}\right )+a f^2 x \sinh \left (\frac {d x}{2}\right )-b d f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a^2 d^2} \]

[In]

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

[Out]

(b*(e + f*x)^2*Csch[c])/(a^2*d) + ((-e^2 - 2*e*f*x - f^2*x^2)*Csch[c/2 + (d*x)/2]^2)/(8*a*d) - (12*d*E^(2*c)*(
b^2*d^2*e^2 + a^2*f^2)*x - 12*d*(-1 + E^(2*c))*(b^2*d^2*e^2 + a^2*f^2)*x + 12*b^2*d^3*e*f*x^2 + 4*b^2*d^3*f^2*
x^3 - 24*a*b*d*e*(-1 + E^(2*c))*f*ArcTanh[E^(c + d*x)] + 6*b^2*d^2*e^2*(-1 + E^(2*c))*(2*d*x - Log[1 - E^(2*(c
+ d*x))]) + 6*a^2*(-1 + E^(2*c))*f^2*(2*d*x - Log[1 - E^(2*(c + d*x))]) + 12*a*b*(-1 + E^(2*c))*f^2*(d*x*(Log
[1 - E^(c + d*x)] - Log[1 + E^(c + d*x)]) - PolyLog[2, -E^(c + d*x)] + PolyLog[2, E^(c + d*x)]) + 6*b^2*d*e*(-
1 + E^(2*c))*f*(2*d*x*(d*x - Log[1 - E^(2*(c + d*x))]) - PolyLog[2, E^(2*(c + d*x))]) + b^2*(-1 + E^(2*c))*f^2
*(2*d^2*x^2*(2*d*x - 3*Log[1 - E^(2*(c + d*x))]) - 6*d*x*PolyLog[2, E^(2*(c + d*x))] + 3*PolyLog[3, E^(2*(c +
d*x))]))/(6*a^3*d^3*(-1 + E^(2*c))) + (b^2*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqr
t[a^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^
2)^2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)
^2]*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E
^(2*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(
-1 + E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1
+ (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E
^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])
])/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 +
(b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c
+ Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])
/d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*
f*(e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e
+ f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(
2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c
- Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]
))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(3*a^3*(
-1 + E^(2*c))) + ((e^2 + 2*e*f*x + f^2*x^2)*Sech[c/2 + (d*x)/2]^2)/(8*a*d) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-
(b*d*e^2*Sinh[(d*x)/2]) - a*e*f*Sinh[(d*x)/2] - 2*b*d*e*f*x*Sinh[(d*x)/2] - a*f^2*x*Sinh[(d*x)/2] - b*d*f^2*x^
2*Sinh[(d*x)/2]))/(2*a^2*d^2) + (Csch[c/2]*Csch[c/2 + (d*x)/2]*(-(b*d*e^2*Sinh[(d*x)/2]) + a*e*f*Sinh[(d*x)/2]
- 2*b*d*e*f*x*Sinh[(d*x)/2] + a*f^2*x*Sinh[(d*x)/2] - b*d*f^2*x^2*Sinh[(d*x)/2]))/(2*a^2*d^2)

Maple [F]

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

[In]

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

[Out]

int((f*x+e)^2*coth(d*x+c)*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. 6479 vs. \(2 (472) = 944\).

Time = 0.34 (sec) , antiderivative size = 6479, normalized size of antiderivative = 12.91 \[ \int \frac {(e+f x)^2 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-e^2*(2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x -
4*c) - a^2)*d) + b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) - b^2*log(e^(-d*x - c) + 1)/(a^3*
d) - b^2*log(e^(-d*x - c) - 1)/(a^3*d)) + 2*(a*f^2*x + a*e*f + (b*d*f^2*x^2*e^(3*c) + 2*b*d*e*f*x*e^(3*c))*e^(
3*d*x) - (a*d*f^2*x^2*e^(2*c) + a*e*f*e^(2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) - (b*d*f^2*x^2*e^c + 2*
b*d*e*f*x*e^c)*e^(d*x))/(a^2*d^2*e^(4*d*x + 4*c) - 2*a^2*d^2*e^(2*d*x + 2*c) + a^2*d^2) + (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)))*b^2*f^2/(a^3*d^3) + (d^2*x^2*log(-e^(d*x +
c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*b^2*f^2/(a^3*d^3) - (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) + 2*(b^2*d*e*f + a*b*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c))
)/(a^3*d^3) + 2*(b^2*d*e*f - a*b*f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a^3*d^3) - 1/3*(b^2*d^
3*f^2*x^3 + 3*(b^2*d*e*f + a*b*f^2)*d^2*x^2)/(a^3*d^3) - 1/3*(b^2*d^3*f^2*x^3 + 3*(b^2*d*e*f - a*b*f^2)*d^2*x^
2)/(a^3*d^3) + integrate(-2*(b^3*f^2*x^2 + 2*b^3*e*f*x - (a*b^2*f^2*x^2*e^c + 2*a*b^2*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. \[ \int \frac {(e+f x)^2 \coth (c+d x) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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