\(\int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [466]

Optimal result

Integrand size = 32, antiderivative size = 591 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 f x \arctan \left (e^{c+d x}\right )}{a d}+\frac {2 b^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a \left (a^2+b^2\right ) d}+\frac {f x \arctan (\sinh (c+d x))}{a d}-\frac {(e+f x) \arctan (\sinh (c+d x))}{a d}+\frac {2 b (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {b^3 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d}-\frac {b^3 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a^2 \left (a^2+b^2\right ) d}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac {i b^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^2}+\frac {i b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a \left (a^2+b^2\right ) d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2}-\frac {b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^2 \left (a^2+b^2\right ) d^2}+\frac {b f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5708, 2701, 327, 213, 5570, 5311, 12, 4265, 2317, 2438, 3855, 5569, 4267, 5692, 5680, 2221, 6874, 3799} \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 b^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a d \left (a^2+b^2\right )}+\frac {2 b (e+f x) \text {arctanh}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac {i b^2 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {i b^2 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^2 \left (a^2+b^2\right )}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )}+\frac {b^3 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac {b^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^2 d^2 \left (a^2+b^2\right )}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a^2 d \left (a^2+b^2\right )}+\frac {b^3 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a^2 d \left (a^2+b^2\right )}-\frac {b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a^2 d \left (a^2+b^2\right )}+\frac {b f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {b f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac {(e+f x) \arctan (\sinh (c+d x))}{a d}-\frac {2 f x \arctan \left (e^{c+d x}\right )}{a d}+\frac {f x \arctan (\sinh (c+d x))}{a d}-\frac {f \text {arctanh}(\cosh (c+d x))}{a d^2}+\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^2}-\frac {(e+f x) \text {csch}(c+d x)}{a d} \]

[In]

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

[Out]

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

Rule 12

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

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

Mathematica [A] (verified)

Time = 8.95 (sec) , antiderivative size = 864, normalized size of antiderivative = 1.46 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\left (-d e \cosh \left (\frac {1}{2} (c+d x)\right )+c f \cosh \left (\frac {1}{2} (c+d x)\right )-f (c+d x) \cosh \left (\frac {1}{2} (c+d x)\right )\right ) \text {csch}\left (\frac {1}{2} (c+d x)\right )}{2 a d^2}+\frac {-\frac {b (d e-c f+f (c+d x))^2}{2 f}+(-b d e+a f+b c f-b f (c+d x)) \log \left (1-e^{-c-d x}\right )+(-b d e-a f+b c f-b f (c+d x)) \log \left (1+e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+b f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a^2 d^2}+\frac {b^3 \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^2 \left (a^2+b^2\right ) d^2}+\frac {-b d e (c+d x)+b c f (c+d x)-\frac {1}{2} b f (c+d x)^2-2 a d e \arctan \left (e^{c+d x}\right )+2 a c f \arctan \left (e^{c+d x}\right )-i a f (c+d x) \log \left (1-i e^{c+d x}\right )+i a f (c+d x) \log \left (1+i e^{c+d x}\right )+b d e \log \left (1+e^{2 (c+d x)}\right )-b c f \log \left (1+e^{2 (c+d x)}\right )+b f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+i a f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )-i a f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+\frac {1}{2} b f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{\left (a^2+b^2\right ) d^2}+\frac {\text {sech}\left (\frac {1}{2} (c+d x)\right ) \left (d e \sinh \left (\frac {1}{2} (c+d x)\right )-c f \sinh \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 a d^2} \]

[In]

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

[Out]

((-(d*e*Cosh[(c + d*x)/2]) + c*f*Cosh[(c + d*x)/2] - f*(c + d*x)*Cosh[(c + d*x)/2])*Csch[(c + d*x)/2])/(2*a*d^
2) + (-1/2*(b*(d*e - c*f + f*(c + d*x))^2)/f + (-(b*d*e) + a*f + b*c*f - b*f*(c + d*x))*Log[1 - E^(-c - d*x)]
+ (-(b*d*e) - a*f + b*c*f - b*f*(c + d*x))*Log[1 + E^(-c - d*x)] + b*f*PolyLog[2, -E^(-c - d*x)] + b*f*PolyLog
[2, E^(-c - d*x)])/(a^2*d^2) + (b^3*(-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*ArcTan
h[(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^2*(a^2 + b^2)*d^2) + (-
(b*d*e*(c + d*x)) + b*c*f*(c + d*x) - (b*f*(c + d*x)^2)/2 - 2*a*d*e*ArcTan[E^(c + d*x)] + 2*a*c*f*ArcTan[E^(c
+ d*x)] - I*a*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + I*a*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + b*d*e*Log[1 + E^(2
*(c + d*x))] - b*c*f*Log[1 + E^(2*(c + d*x))] + b*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + I*a*f*PolyLog[2, (-I)
*E^(c + d*x)] - I*a*f*PolyLog[2, I*E^(c + d*x)] + (b*f*PolyLog[2, -E^(2*(c + d*x))])/2)/((a^2 + b^2)*d^2) + (S
ech[(c + d*x)/2]*(d*e*Sinh[(c + d*x)/2] - c*f*Sinh[(c + d*x)/2] + f*(c + d*x)*Sinh[(c + d*x)/2]))/(2*a*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. 1528 vs. \(2 (556 ) = 1112\).

Time = 8.33 (sec) , antiderivative size = 1529, normalized size of antiderivative = 2.59

method	result	size
risch	\(\text {Expression too large to display}\)	\(1529\)

[In]

int((f*x+e)*csch(d*x+c)^2*sech(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

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

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. 2593 vs. \(2 (539) = 1078\).

Time = 0.35 (sec) , antiderivative size = 2593, normalized size of antiderivative = 4.39 \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-(2*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*d*e)*cosh(d*x + c) - (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c)*s
inh(d*x + c) + b^3*f*sinh(d*x + c)^2 - b^3*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*
sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) - (b^3*f*cosh(d*x + c)^2 + 2*b^3*f*cosh(d*x + c)*sinh(d*x + c
) + b^3*f*sinh(d*x + c)^2 - b^3*f)*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x +
c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + ((a^2*b + b^3)*f*cosh(d*x + c)^2 + 2*(a^2*b + b^3)*f*cosh(d*x + c)*sin
h(d*x + c) + (a^2*b + b^3)*f*sinh(d*x + c)^2 - (a^2*b + b^3)*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) - (I*a^3*
f - a^2*b*f + (-I*a^3*f + a^2*b*f)*cosh(d*x + c)^2 - 2*(I*a^3*f - a^2*b*f)*cosh(d*x + c)*sinh(d*x + c) + (-I*a
^3*f + a^2*b*f)*sinh(d*x + c)^2)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - (-I*a^3*f - a^2*b*f + (I*a^3*f + a
^2*b*f)*cosh(d*x + c)^2 - 2*(-I*a^3*f - a^2*b*f)*cosh(d*x + c)*sinh(d*x + c) + (I*a^3*f + a^2*b*f)*sinh(d*x +
c)^2)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + ((a^2*b + b^3)*f*cosh(d*x + c)^2 + 2*(a^2*b + b^3)*f*cosh(d*
x + c)*sinh(d*x + c) + (a^2*b + b^3)*f*sinh(d*x + c)^2 - (a^2*b + b^3)*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)
) + (b^3*d*e - b^3*c*f - (b^3*d*e - b^3*c*f)*cosh(d*x + c)^2 - 2*(b^3*d*e - b^3*c*f)*cosh(d*x + c)*sinh(d*x +
c) - (b^3*d*e - b^3*c*f)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) + 2*b*sqrt((a^2 + b^2)/b^2
) + 2*a) + (b^3*d*e - b^3*c*f - (b^3*d*e - b^3*c*f)*cosh(d*x + c)^2 - 2*(b^3*d*e - b^3*c*f)*cosh(d*x + c)*sinh
(d*x + c) - (b^3*d*e - b^3*c*f)*sinh(d*x + c)^2)*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b
^2)/b^2) + 2*a) + (b^3*d*f*x + b^3*c*f - (b^3*d*f*x + b^3*c*f)*cosh(d*x + c)^2 - 2*(b^3*d*f*x + b^3*c*f)*cosh(
d*x + c)*sinh(d*x + c) - (b^3*d*f*x + b^3*c*f)*sinh(d*x + c)^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*c
osh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + (b^3*d*f*x + b^3*c*f - (b^3*d*f*x + b^3*c*f)*c
osh(d*x + c)^2 - 2*(b^3*d*f*x + b^3*c*f)*cosh(d*x + c)*sinh(d*x + c) - (b^3*d*f*x + b^3*c*f)*sinh(d*x + c)^2)*
log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) -
((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^3 + a*b^2)*f)*cosh(d*
x + c)^2 - 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^3 + a*b^2)*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b
+ b^3)*d*f*x + (a^2*b + b^3)*d*e + (a^3 + a*b^2)*f)*sinh(d*x + c)^2 + (a^3 + a*b^2)*f)*log(cosh(d*x + c) + si
nh(d*x + c) + 1) - (I*a^3*d*e - a^2*b*d*e - I*a^3*c*f + a^2*b*c*f + (-I*a^3*d*e + a^2*b*d*e + I*a^3*c*f - a^2*
b*c*f)*cosh(d*x + c)^2 - 2*(I*a^3*d*e - a^2*b*d*e - I*a^3*c*f + a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (-I*a
^3*d*e + a^2*b*d*e + I*a^3*c*f - a^2*b*c*f)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) + I) - (-I*a^3*
d*e - a^2*b*d*e + I*a^3*c*f + a^2*b*c*f + (I*a^3*d*e + a^2*b*d*e - I*a^3*c*f - a^2*b*c*f)*cosh(d*x + c)^2 - 2*
(-I*a^3*d*e - a^2*b*d*e + I*a^3*c*f + a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (I*a^3*d*e + a^2*b*d*e - I*a^3*
c*f - a^2*b*c*f)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d*x + c) - I) - ((a^2*b + b^3)*d*e - ((a^2*b + b^3)
*d*e - (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d*e - (a^3 + a*b^2 + (a^2*b + b^3
)*c)*f)*cosh(d*x + c)*sinh(d*x + c) - ((a^2*b + b^3)*d*e - (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*sinh(d*x + c)^2
- (a^3 + a*b^2 + (a^2*b + b^3)*c)*f)*log(cosh(d*x + c) + sinh(d*x + c) - 1) - (-I*a^3*d*f*x - a^2*b*d*f*x - I*
a^3*c*f - a^2*b*c*f + (I*a^3*d*f*x + a^2*b*d*f*x + I*a^3*c*f + a^2*b*c*f)*cosh(d*x + c)^2 - 2*(-I*a^3*d*f*x -
a^2*b*d*f*x - I*a^3*c*f - a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (I*a^3*d*f*x + a^2*b*d*f*x + I*a^3*c*f + a^
2*b*c*f)*sinh(d*x + c)^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) + 1) - (I*a^3*d*f*x - a^2*b*d*f*x + I*a^3*c*f
- a^2*b*c*f + (-I*a^3*d*f*x + a^2*b*d*f*x - I*a^3*c*f + a^2*b*c*f)*cosh(d*x + c)^2 - 2*(I*a^3*d*f*x - a^2*b*d*
f*x + I*a^3*c*f - a^2*b*c*f)*cosh(d*x + c)*sinh(d*x + c) + (-I*a^3*d*f*x + a^2*b*d*f*x - I*a^3*c*f + a^2*b*c*f
)*sinh(d*x + c)^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f - ((
a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*cosh(d*x
+ c)*sinh(d*x + c) - ((a^2*b + b^3)*d*f*x + (a^2*b + b^3)*c*f)*sinh(d*x + c)^2)*log(-cosh(d*x + c) - sinh(d*x
+ c) + 1) + 2*((a^3 + a*b^2)*d*f*x + (a^3 + a*b^2)*d*e)*sinh(d*x + c))/((a^4 + a^2*b^2)*d^2*cosh(d*x + c)^2 +
2*(a^4 + a^2*b^2)*d^2*cosh(d*x + c)*sinh(d*x + c) + (a^4 + a^2*b^2)*d^2*sinh(d*x + c)^2 - (a^4 + a^2*b^2)*d^2
)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

(b^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + a^2*b^2)*d) + 2*a*arctan(e^(-d*x - c))/((a^2 + b^
2)*d) + b*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*e^(-d*x - c)/((a*e^(-2*d*x - 2*c) - a)*d) - b*log(e^(-
d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d))*e + (8*b*d*integrate(1/8*x/(a^2*d*e^(d*x + c) + a^2*d
), x) - 8*b*d*integrate(1/8*x/(a^2*d*e^(d*x + c) - a^2*d), x) + a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) + 1)/
(a^2*d^2)) - a*((d*x + c)/(a^2*d^2) - log(e^(d*x + c) - 1)/(a^2*d^2)) - 2*x*e^(d*x + c)/(a*d*e^(2*d*x + 2*c) -
a*d) - 8*integrate(-1/4*(a*b^3*x*e^(d*x + c) - b^4*x)/(a^4*b + a^2*b^3 - (a^4*b*e^(2*c) + a^2*b^3*e^(2*c))*e^
(2*d*x) - 2*(a^5*e^c + a^3*b^2*e^c)*e^(d*x)), x) - 8*integrate(1/4*(a*x*e^(d*x + c) + b*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. \[ \int \frac {(e+f x) \text {csch}^2(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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