\(\int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [493]

Optimal result

Integrand size = 32, antiderivative size = 762 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f x}{2 a d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2} \]

[Out]

1/2*b^2*f*polylog(2,exp(2*d*x+2*c))/a^3/d^2+1/2*f*x/a/d-1/2*f*coth(d*x+c)/a/d^2+f*x*ln(tanh(d*x+c))/a/d+1/2*f*
polylog(2,-exp(2*d*x+2*c))/a/d^2-I*b^3*f*polylog(2,I*exp(d*x+c))/a^2/(a^2+b^2)/d^2-2*b^2*(f*x+e)*arctanh(exp(2
*d*x+2*c))/a^3/d-1/2*b^2*f*polylog(2,-exp(2*d*x+2*c))/a^3/d^2-1/2*f*polylog(2,exp(2*d*x+2*c))/a/d^2-(f*x+e)*ln
(tanh(d*x+c))/a/d-b*f*x*arctan(sinh(d*x+c))/a^2/d+I*b*f*polylog(2,I*exp(d*x+c))/a^2/d^2+b*(f*x+e)*arctan(sinh(
d*x+c))/a^2/d+b^4*(f*x+e)*ln(1+exp(2*d*x+2*c))/a^3/(a^2+b^2)/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2
)))/a^3/(a^2+b^2)/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d-b^4*f*polylog(2,-b*exp(
d*x+c)/(a-(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)/d^2-b^4*f*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a^3/(a^2+b^2)
/d^2+b*f*arctanh(cosh(d*x+c))/a^2/d^2+b*(f*x+e)*csch(d*x+c)/a^2/d-1/2*(f*x+e)*coth(d*x+c)^2/a/d+I*b^3*f*polylo
g(2,-I*exp(d*x+c))/a^2/(a^2+b^2)/d^2+2*b*f*x*arctan(exp(d*x+c))/a^2/d-2*b^3*(f*x+e)*arctan(exp(d*x+c))/a^2/(a^
2+b^2)/d+1/2*b^4*f*polylog(2,-exp(2*d*x+2*c))/a^3/(a^2+b^2)/d^2-I*b*f*polylog(2,-I*exp(d*x+c))/a^2/d^2+2*f*x*a
rctanh(exp(2*d*x+2*c))/a/d

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.00, number of steps used = 49, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.719, Rules used = {5708, 2700, 14, 5570, 3554, 8, 2628, 12, 4267, 2317, 2438, 2701, 327, 213, 5311, 4265, 3855, 5569, 5692, 5680, 2221, 6874, 3799} \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 d \left (a^2+b^2\right )}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 d^2 \left (a^2+b^2\right )}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 d^2 \left (a^2+b^2\right )}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 d^2 \left (a^2+b^2\right )}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^3 d \left (a^2+b^2\right )}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^3 d \left (a^2+b^2\right )}+\frac {b^4 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a^3 d \left (a^2+b^2\right )}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {f x \log (\tanh (c+d x))}{a d}+\frac {f x}{2 a d} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 327

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

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 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 2628

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/u), x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 2701

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && 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 4265

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c +
d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^(I*k*Pi)]/(f*fz*I)), x] + (-Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*
Log[1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[d*(m/(f*fz*I)), Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*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 5311

Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[x*(D[u, x]/(1 + u^2)), x], x] /; Inv
erseFunctionFreeQ[u, x]

Rule 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5570

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

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 5692

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[(e + f*x)^m*(Sech[c + d*x]^(n - 2)/(a + b*Sinh[c + d*x])), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5708

Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(p_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Sech[c + d*x]^p*Csch[c + d*x]^n, x], x] - Dis
t[b/a, Int[(e + f*x)^m*Sech[c + d*x]^p*(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] && IGtQ[p, 0]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a} \\ & = -\frac {(e+f x) \coth ^2(c+d x)}{2 a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b \int (e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^2}-\frac {f \int \left (-\frac {\coth ^2(c+d x)}{2 d}-\frac {\log (\tanh (c+d x))}{d}\right ) \, dx}{a} \\ & = \frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {b^2 \int (e+f x) \text {csch}(c+d x) \text {sech}(c+d x) \, dx}{a^3}-\frac {b^3 \int \frac {(e+f x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {(b f) \int \left (-\frac {\arctan (\sinh (c+d x))}{d}-\frac {\text {csch}(c+d x)}{d}\right ) \, dx}{a^2}+\frac {f \int \coth ^2(c+d x) \, dx}{2 a d}+\frac {f \int \log (\tanh (c+d x)) \, dx}{a d} \\ & = \frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {\left (2 b^2\right ) \int (e+f x) \text {csch}(2 c+2 d x) \, dx}{a^3}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {(e+f x) \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \int 1 \, dx}{2 a d}-\frac {f \int 2 d x \text {csch}(2 c+2 d x) \, dx}{a d}-\frac {(b f) \int \arctan (\sinh (c+d x)) \, dx}{a^2 d}-\frac {(b f) \int \text {csch}(c+d x) \, dx}{a^2 d} \\ & = \frac {f x}{2 a d}+\frac {b^4 (e+f x)^2}{2 a^3 \left (a^2+b^2\right ) f}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^3 \int (a (e+f x) \text {sech}(c+d x)-b (e+f x) \tanh (c+d x)) \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {b^5 \int \frac {e^{c+d x} (e+f x)}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{a^3 \left (a^2+b^2\right )}-\frac {(2 f) \int x \text {csch}(2 c+2 d x) \, dx}{a}+\frac {(b f) \int d x \text {sech}(c+d x) \, dx}{a^2 d}-\frac {\left (b^2 f\right ) \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a^3 d}+\frac {\left (b^2 f\right ) \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a^3 d} \\ & = \frac {f x}{2 a d}+\frac {b^4 (e+f x)^2}{2 a^3 \left (a^2+b^2\right ) f}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^3 \int (e+f x) \text {sech}(c+d x) \, dx}{a^2 \left (a^2+b^2\right )}+\frac {b^4 \int (e+f x) \tanh (c+d x) \, dx}{a^3 \left (a^2+b^2\right )}+\frac {(b f) \int x \text {sech}(c+d x) \, dx}{a^2}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {f \int \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}-\frac {f \int \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}+\frac {\left (b^4 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d}+\frac {\left (b^4 f\right ) \int \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d} \\ & = \frac {f x}{2 a d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}+\frac {\left (2 b^4\right ) \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}} \, dx}{a^3 \left (a^2+b^2\right )}+\frac {f \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a-\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {(i b f) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {(i b f) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 d}+\frac {\left (i b^3 f\right ) \int \log \left (1-i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d}-\frac {\left (i b^3 f\right ) \int \log \left (1+i e^{c+d x}\right ) \, dx}{a^2 \left (a^2+b^2\right ) d} \\ & = \frac {f x}{2 a d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}+\frac {(i b f) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 d^2}+\frac {\left (i b^3 f\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (i b^3 f\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {\left (b^4 f\right ) \int \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a^3 \left (a^2+b^2\right ) d} \\ & = \frac {f x}{2 a d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {\left (b^4 f\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2} \\ & = \frac {f x}{2 a d}+\frac {2 b f x \arctan \left (e^{c+d x}\right )}{a^2 d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b f x \arctan (\sinh (c+d x))}{a^2 d}+\frac {b (e+f x) \arctan (\sinh (c+d x))}{a^2 d}+\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {2 b^2 (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^3 d}+\frac {b f \text {arctanh}(\cosh (c+d x))}{a^2 d^2}-\frac {f \coth (c+d x)}{2 a d^2}-\frac {(e+f x) \coth ^2(c+d x)}{2 a d}+\frac {b (e+f x) \text {csch}(c+d x)}{a^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^3 \left (a^2+b^2\right ) d}+\frac {f x \log (\tanh (c+d x))}{a d}-\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 d^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^3 \left (a^2+b^2\right ) d^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b^2 f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^3 d^2}-\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}+\frac {b^2 f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^3 d^2} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 9.43 (sec) , antiderivative size = 1009, normalized size of antiderivative = 1.32 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (2 b d e \cosh \left (\frac {1}{2} (c+d x)\right )-a f \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 ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{4 a^2 d^2}+\frac {(-d e+c f-f (c+d x)) \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {-\frac {\left (a^2-b^2\right ) (d e+d f x)^2}{2 f}-\left (a b f+a^2 (d e+d f x)-b^2 (d e+d f x)\right ) \log \left (1-e^{-c-d x}\right )+\left (a b f-a^2 (d e+d f x)+b^2 (d e+d f x)\right ) \log \left (1+e^{-c-d x}\right )+\left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+\left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a^3 d^2}-\frac {b^4 \left (-2 d e (c+d x)+2 c f (c+d x)-f (c+d x)^2+\frac {4 a \sqrt {a^2+b^2} d e \arctan \left (\frac {a+b e^{c+d x}}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-\left (a^2+b^2\right )^2}}-\frac {4 a \sqrt {-\left (a^2+b^2\right )^2} d e \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+2 f (c+d x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-2 c f \log \left (b-2 a e^{c+d x}-b e^{2 (c+d x)}\right )+2 d e \log \left (2 a e^{c+d x}+b \left (-1+e^{2 (c+d x)}\right )\right )+2 f \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+2 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )\right )}{2 a^3 \left (a^2+b^2\right ) d^2}+\frac {-a d e (c+d x)+a c f (c+d x)-\frac {1}{2} a f (c+d x)^2+2 b d e \arctan \left (e^{c+d x}\right )-2 b c f \arctan \left (e^{c+d x}\right )+i b f (c+d x) \log \left (1-i e^{c+d x}\right )-i b f (c+d x) \log \left (1+i e^{c+d x}\right )+a d e \log \left (1+e^{2 (c+d x)}\right )-a c f \log \left (1+e^{2 (c+d x)}\right )+a f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+\frac {1}{2} a f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {(d e-c f+f (c+d x)) \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 a d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (-2 b d e \sinh \left (\frac {1}{2} (c+d x)\right )-a f \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} \]

[In]

Integrate[((e + f*x)*Csch[c + d*x]^3*Sech[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) + (-1/2*(
(a^2 - b^2)*(d*e + d*f*x)^2)/f - (a*b*f + a^2*(d*e + d*f*x) - b^2*(d*e + d*f*x))*Log[1 - E^(-c - d*x)] + (a*b*
f - a^2*(d*e + d*f*x) + b^2*(d*e + d*f*x))*Log[1 + E^(-c - d*x)] + (a^2 - b^2)*f*PolyLog[2, -E^(-c - d*x)] + (
a^2 - b^2)*f*PolyLog[2, E^(-c - d*x)])/(a^3*d^2) - (b^4*(-2*d*e*(c + d*x) + 2*c*f*(c + d*x) - f*(c + d*x)^2 +
(4*a*Sqrt[a^2 + b^2]*d*e*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/Sqrt[-(a^2 + b^2)^2] - (4*a*Sqrt[-(a^2
+ b^2)^2]*d*e*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/(-a^2 - b^2)^(3/2) + 2*f*(c + d*x)*Log[1 + (b*E^(c
+ d*x))/(a - Sqrt[a^2 + b^2])] + 2*f*(c + d*x)*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] - 2*c*f*Log[b -
2*a*E^(c + d*x) - b*E^(2*(c + d*x))] + 2*d*e*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))] + 2*f*PolyLog[2,
(b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*f*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(2*a^3*(
a^2 + b^2)*d^2) + (-(a*d*e*(c + d*x)) + a*c*f*(c + d*x) - (a*f*(c + d*x)^2)/2 + 2*b*d*e*ArcTan[E^(c + d*x)] -
2*b*c*f*ArcTan[E^(c + d*x)] + I*b*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - I*b*f*(c + d*x)*Log[1 + I*E^(c + d*x)]
+ a*d*e*Log[1 + E^(2*(c + d*x))] - a*c*f*Log[1 + E^(2*(c + d*x))] + a*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] - I
*b*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*f*PolyLog[2, I*E^(c + d*x)] + (a*f*PolyLog[2, -E^(2*(c + d*x))])/2)/((
a^2 + b^2)*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*Sin
h[(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] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1477 vs. \(2 (714 ) = 1428\).

Time = 15.11 (sec) , antiderivative size = 1478, normalized size of antiderivative = 1.94

method	result	size
risch	\(\text {Expression too large to display}\)	\(1478\)

[In]

int((f*x+e)*csch(d*x+c)^3*sech(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-(-2*b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x*exp(2*d*x+2*c)-2*b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2*c)+2*b*d*f*x*e
xp(d*x+c)+a*f*exp(2*d*x+2*c)+2*b*d*e*exp(d*x+c)-a*f)/a^2/d^2/(exp(2*d*x+2*c)-1)^2-1/d*f/a*ln(exp(d*x+c)+1)*x+1
/d^2*c*f/a*ln(exp(d*x+c)-1)-1/d/a^3*f*b^4/(a^2+b^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*b^2/a^3*f*dilog(exp(d*x+c))+1/d^2*b^2/a^3*f*dilog(exp(d*x+c)+1)+1/d*b^2/a^3*e*ln(exp(d*x+c)-1)+1/d*b^2/
a^3*e*ln(exp(d*x+c)+1)-1/d^2*b/a^2*f*ln(exp(d*x+c)-1)+1/d^2*b/a^2*f*ln(exp(d*x+c)+1)+1/d^2*f/a*dilog(exp(d*x+c
))-1/d*e/a*ln(exp(d*x+c)-1)-1/d*e/a*ln(exp(d*x+c)+1)-1/d^2*f/a*dilog(exp(d*x+c)+1)-1/d^2/a^3*f*b^4/(a^2+b^2)*l
n((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d^2/a^3*f*b^4/(a^2+b^2)*ln((b*exp(d*x+c)+(a^2+b^
2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c+4*I/d*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*x-4*I/d*f/(4*a^2+4*b^2)*ln(1+I*e
xp(d*x+c))*b*x-4*I/d^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c+4*I/d^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c-1
/d^2/a^3*f*b^4/(a^2+b^2)*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-1/d^2/a^3*f*b^4/(a^2+b^2)
*dilog((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/d/a^3*b^4*e/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2*a
*exp(d*x+c)-b)-4/d^2*a*c*f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))+4/d^2*a*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c+4/d
^2*a*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+4/d*a*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x+4/d*a*f/(4*a^2+4*b^2)*ln(
1-I*exp(d*x+c))*x-1/d^2/a^2*b^2*f/(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/a^2*
f*b^4/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-8/d^2*c*f/(4*a^2+4*b^2)*b*arctan(exp(d
*x+c))+4*I/d^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b-4*I/d^2*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b+1/d*b^2
/a^3*f*ln(exp(d*x+c)+1)*x-1/d^2*b^2/a^3*c*f*ln(exp(d*x+c)-1)+1/d^2/a^3*b^4*c*f/(a^2+b^2)*ln(b*exp(2*d*x+2*c)+2
*a*exp(d*x+c)-b)-1/d/a^3*f*b^4/(a^2+b^2)*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+8/d*e/(4
*a^2+4*b^2)*b*arctan(exp(d*x+c))+1/d^2*f*b^2/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))
+4/d^2*a*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))+4/d^2*a*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))+4/d*a*e/(4*a^2+4*
b^2)*ln(1+exp(2*d*x+2*c))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 5731 vs. \(2 (695) = 1390\).

Time = 0.42 (sec) , antiderivative size = 5731, normalized size of antiderivative = 7.52 \[ \int \frac {(e+f x) \text {csch}^3(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-(b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + a^3*b^2)*d) + 2*b*arctan(e^(-d*x - c))/((a^2 + b
^2)*d) - a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x -
3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c) - a^2)*d) + (a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) +
(a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d))*e + (16*a^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16
*b^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) + a^3*d), x) - 16*a^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) - a^3*d
), x) + 16*b^2*d*integrate(1/16*x/(a^3*d*e^(d*x + c) - a^3*d), x) - a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c)
+ 1)/(a^3*d^2)) + a*b*((d*x + c)/(a^3*d^2) - log(e^(d*x + c) - 1)/(a^3*d^2)) + (2*b*d*x*e^(3*d*x + 3*c) - 2*b
*d*x*e^(d*x + c) - (2*a*d*x*e^(2*c) + a*e^(2*c))*e^(2*d*x) + a)/(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) + 16*integrate(-1/8*(a*b^4*x*e^(d*x + c) - b^5*x)/(a^5*b + a^3*b^3 - (a^5*b*e^(2*c) + a^3*b^
3*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + a^4*b^2*e^c)*e^(d*x)), x) + 16*integrate(1/8*(b*x*e^(d*x + c) - a*x)/(a^2
+ b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x))*f

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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