\(\int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [446]

Optimal result

Integrand size = 32, antiderivative size = 746 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {f x}{2 a d}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac {b (e+f x) \arctan \left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {b^4 (e+f x) \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d}+\frac {b^4 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 \left (a^2+b^2\right ) d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^2}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {b f \text {sech}(c+d x)}{2 \left (a^2+b^2\right ) d^2}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac {f \tanh (c+d x)}{2 a d^2}+\frac {b^2 f \tanh (c+d x)}{2 a \left (a^2+b^2\right ) d^2}-\frac {b (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d} \]

[Out]

1/2*f*x/a/d-2*b^3*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)^2/d-b*(f*x+e)*arctan(exp(d*x+c))/(a^2+b^2)/d-2*f*x*arct
anh(exp(2*d*x+2*c))/a/d+b^4*(f*x+e)*ln(1+exp(2*d*x+2*c))/a/(a^2+b^2)^2/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a-(a^2
+b^2)^(1/2)))/a/(a^2+b^2)^2/d-b^4*(f*x+e)*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d-f*x*ln(tanh(d
*x+c))/a/d+(f*x+e)*ln(tanh(d*x+c))/a/d-1/2*I*b*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)/d^2+1/2*I*b*f*polylog(2,-I*
exp(d*x+c))/(a^2+b^2)/d^2-I*b^3*f*polylog(2,I*exp(d*x+c))/(a^2+b^2)^2/d^2+I*b^3*f*polylog(2,-I*exp(d*x+c))/(a^
2+b^2)^2/d^2+1/2*b^4*f*polylog(2,-exp(2*d*x+2*c))/a/(a^2+b^2)^2/d^2-1/2*f*polylog(2,-exp(2*d*x+2*c))/a/d^2+1/2
*f*polylog(2,exp(2*d*x+2*c))/a/d^2-b^4*f*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2-b^4*f*
polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/a/(a^2+b^2)^2/d^2-1/2*b*f*sech(d*x+c)/(a^2+b^2)/d^2-1/2*b^2*(f*x+
e)*sech(d*x+c)^2/a/(a^2+b^2)/d-1/2*f*tanh(d*x+c)/a/d^2+1/2*b^2*f*tanh(d*x+c)/a/(a^2+b^2)/d^2-1/2*b*(f*x+e)*sec
h(d*x+c)*tanh(d*x+c)/(a^2+b^2)/d-1/2*(f*x+e)*tanh(d*x+c)^2/a/d

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 746, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5708, 2700, 14, 5570, 2628, 12, 4267, 2317, 2438, 3554, 8, 5692, 5680, 2221, 6874, 4265, 3799, 4270, 5559, 3852} \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b (e+f x) \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac {2 b^3 (e+f x) \arctan \left (e^{c+d x}\right )}{d \left (a^2+b^2\right )^2}+\frac {i b f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}-\frac {i b f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 d^2 \left (a^2+b^2\right )}+\frac {b^2 f \tanh (c+d x)}{2 a d^2 \left (a^2+b^2\right )}-\frac {b f \text {sech}(c+d x)}{2 d^2 \left (a^2+b^2\right )}-\frac {b^2 (e+f x) \text {sech}^2(c+d x)}{2 a d \left (a^2+b^2\right )}-\frac {b (e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d \left (a^2+b^2\right )}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^2}-\frac {b^4 f \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^2}+\frac {b^4 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^2 \left (a^2+b^2\right )^2}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )^2}-\frac {b^4 (e+f x) \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )^2}+\frac {b^4 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )^2}+\frac {i b^3 f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {i b^3 f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )^2}-\frac {2 f x \text {arctanh}\left (e^{2 c+2 d x}\right )}{a d}-\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a d^2}+\frac {f \operatorname {PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a d^2}-\frac {f \tanh (c+d x)}{2 a d^2}-\frac {(e+f x) \tanh ^2(c+d x)}{2 a d}+\frac {(e+f x) \log (\tanh (c+d x))}{a d}-\frac {f x \log (\tanh (c+d x))}{a d}+\frac {f x}{2 a d} \]

[In]

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

[Out]

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

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 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 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 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

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 4270

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

Rule 5559

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

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

Mathematica [A] (verified)

Time = 10.91 (sec) , antiderivative size = 1080, normalized size of antiderivative = 1.45 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {1}{2} d^2 f x^2+d e (c+d x)+(d e-c f+f (c+d x)) \log \left (1-e^{-c-d x}\right )+(d e-c f+f (c+d x)) \log \left (1+e^{-c-d x}\right )-f \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )}{a 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 \left (a^2+b^2\right )^2 d^2}-\frac {-2 a^3 d e (c+d x)-4 a b^2 d e (c+d x)+2 a^3 c f (c+d x)+4 a b^2 c f (c+d x)-a^3 f (c+d x)^2-2 a b^2 f (c+d x)^2+2 a^2 b d e \arctan \left (e^{c+d x}\right )+6 b^3 d e \arctan \left (e^{c+d x}\right )-2 a^2 b c f \arctan \left (e^{c+d x}\right )-6 b^3 c f \arctan \left (e^{c+d x}\right )+i a^2 b f (c+d x) \log \left (1-i e^{c+d x}\right )+3 i b^3 f (c+d x) \log \left (1-i e^{c+d x}\right )-i a^2 b f (c+d x) \log \left (1+i e^{c+d x}\right )-3 i b^3 f (c+d x) \log \left (1+i e^{c+d x}\right )+2 a^3 d e \log \left (1+e^{2 (c+d x)}\right )+4 a b^2 d e \log \left (1+e^{2 (c+d x)}\right )-2 a^3 c f \log \left (1+e^{2 (c+d x)}\right )-4 a b^2 c f \log \left (1+e^{2 (c+d x)}\right )+2 a^3 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )+4 a b^2 f (c+d x) \log \left (1+e^{2 (c+d x)}\right )-i b \left (a^2+3 b^2\right ) f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )+i b \left (a^2+3 b^2\right ) f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )+a^3 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )+2 a b^2 f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 \left (a^2+b^2\right )^2 d^2}+\frac {\text {sech}(c+d x) (-b f-a f \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2}+\frac {\text {sech}^2(c+d x) (a d e-a c f+a f (c+d x)-b d e \sinh (c+d x)+b c f \sinh (c+d x)-b f (c+d x) \sinh (c+d x))}{2 \left (a^2+b^2\right ) d^2} \]

[In]

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

[Out]

((d^2*f*x^2)/2 + d*e*(c + d*x) + (d*e - c*f + f*(c + d*x))*Log[1 - E^(-c - d*x)] + (d*e - c*f + f*(c + d*x))*L
og[1 + E^(-c - d*x)] - f*PolyLog[2, -E^(-c - d*x)] - f*PolyLog[2, E^(-c - d*x)])/(a*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*(a^2 + b^2)^2*d^2) - (-2*a^3*d*e*(c + d*x) - 4*a*b^2*d*e*(c + d*x) + 2*a^
3*c*f*(c + d*x) + 4*a*b^2*c*f*(c + d*x) - a^3*f*(c + d*x)^2 - 2*a*b^2*f*(c + d*x)^2 + 2*a^2*b*d*e*ArcTan[E^(c
+ d*x)] + 6*b^3*d*e*ArcTan[E^(c + d*x)] - 2*a^2*b*c*f*ArcTan[E^(c + d*x)] - 6*b^3*c*f*ArcTan[E^(c + d*x)] + I*
a^2*b*f*(c + d*x)*Log[1 - I*E^(c + d*x)] + (3*I)*b^3*f*(c + d*x)*Log[1 - I*E^(c + d*x)] - I*a^2*b*f*(c + d*x)*
Log[1 + I*E^(c + d*x)] - (3*I)*b^3*f*(c + d*x)*Log[1 + I*E^(c + d*x)] + 2*a^3*d*e*Log[1 + E^(2*(c + d*x))] + 4
*a*b^2*d*e*Log[1 + E^(2*(c + d*x))] - 2*a^3*c*f*Log[1 + E^(2*(c + d*x))] - 4*a*b^2*c*f*Log[1 + E^(2*(c + d*x))
] + 2*a^3*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] + 4*a*b^2*f*(c + d*x)*Log[1 + E^(2*(c + d*x))] - I*b*(a^2 + 3*b
^2)*f*PolyLog[2, (-I)*E^(c + d*x)] + I*b*(a^2 + 3*b^2)*f*PolyLog[2, I*E^(c + d*x)] + a^3*f*PolyLog[2, -E^(2*(c
+ d*x))] + 2*a*b^2*f*PolyLog[2, -E^(2*(c + d*x))])/(2*(a^2 + b^2)^2*d^2) + (Sech[c + d*x]*(-(b*f) - a*f*Sinh[
c + d*x]))/(2*(a^2 + b^2)*d^2) + (Sech[c + d*x]^2*(a*d*e - a*c*f + a*f*(c + d*x) - b*d*e*Sinh[c + d*x] + b*c*f
*Sinh[c + d*x] - b*f*(c + d*x)*Sinh[c + d*x]))/(2*(a^2 + b^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. 2579 vs. \(2 (693 ) = 1386\).

Time = 30.79 (sec) , antiderivative size = 2580, normalized size of antiderivative = 3.46

method	result	size
risch	\(\text {Expression too large to display}\)	\(2580\)

[In]

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

[Out]

1/(a^2+b^2)/d*f*b^2/a*ln(exp(d*x+c)+1)*x-1/(a^2+b^2)/d^2*b^2*c*f/a*ln(exp(d*x+c)-1)-4/d/(a^2+b^2)*a^3*f/(4*a^2
+4*b^2)*ln(1-I*exp(d*x+c))*x-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+1/d^2/(a^2+b^2)^2*c*f*b^
4/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)+1/2/d^2/(a^2+b^2)^(3/2)*c*b^2*f*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^
2+b^2)^(1/2))-1/d/(a^2+b^2)^2*f*b^4/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*x+2*I/d^2/(a^
2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*c-2*I/d^2/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*c
+1/2/d/(a^2+b^2)^(5/2)*b^4*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/2/d/(a^2+b^2)^(3/2)*e*b^2*arc
tanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/d/(a^2+b^2)*a^3*e/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-1/d/(a^2
+b^2)^2*b^4*e/a*ln(b*exp(2*d*x+2*c)+2*a*exp(d*x+c)-b)-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c)
)-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-1/d^2/(a^2+b^2)^2*b^4*f/a*dilog((-b*exp(d*x+c)+(a^
2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))-1/(a^2+b^2)/d^2*a*f*dilog(exp(d*x+c))+1/(a^2+b^2)/d^2*a*f*dilog(exp(d*x+
c)+1)+1/(a^2+b^2)/d*e*a*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d*e*a*ln(exp(d*x+c)+1)+(-b*d*f*x*exp(3*d*x+3*c)+2*a*d*f*x
*exp(2*d*x+2*c)-b*d*e*exp(3*d*x+3*c)+2*a*d*e*exp(2*d*x+2*c)+b*d*f*x*exp(d*x+c)-b*f*exp(3*d*x+3*c)+a*f*exp(2*d*
x+2*c)+b*d*e*exp(d*x+c)-f*b*exp(d*x+c)+a*f)/d^2/(a^2+b^2)/(1+exp(2*d*x+2*c))^2+4/d^2/(a^2+b^2)*c*a^2*f/(4*a^2+
4*b^2)*b*arctan(exp(d*x+c))+8/d^2/(a^2+b^2)*c*f*b^2/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))-1/2/d^2/(a^2+b^2)^(5/
2)*c*f*b^2*a^2*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-8/d/(a^2+b^2)*f*b^2/(4*a^2+4*b^2)*ln(1+I*exp(
d*x+c))*a*x-8/d^2/(a^2+b^2)*f*b^2/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*a*c-8/d/(a^2+b^2)*f*b^2/(4*a^2+4*b^2)*ln(1-
I*exp(d*x+c))*a*x-8/d^2/(a^2+b^2)*f*b^2/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*a*c+6*I/d^2/(a^2+b^2)*f*b^3/(4*a^2+4*
b^2)*ln(1+I*exp(d*x+c))*c-6*I/d^2/(a^2+b^2)*f*b^3/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*c+2*I/d^2/(a^2+b^2)*a^2*f/(
4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))*b-2*I/d^2/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*b-6*I/d/(a^2+
b^2)*f*b^3/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*x+6*I/d/(a^2+b^2)*f*b^3/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*x-1/d^2/(
a^2+b^2)^2*b^4*f/a*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-12/d/(a^2+b^2)*b^3*e/(4*a^2+4*b
^2)*arctan(exp(d*x+c))-1/(a^2+b^2)/d^2*c*a*f*ln(exp(d*x+c)-1)+1/(a^2+b^2)/d*ln(exp(d*x+c)+1)*a*f*x-1/(a^2+b^2)
/d^2*f*b^2/a*dilog(exp(d*x+c))+1/(a^2+b^2)/d^2*f*b^2/a*dilog(exp(d*x+c)+1)+1/(a^2+b^2)/d*b^2*e/a*ln(exp(d*x+c)
-1)+1/(a^2+b^2)/d*b^2*e/a*ln(exp(d*x+c)+1)-1/2/d^2/(a^2+b^2)^(5/2)*c*f*b^4*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a
^2+b^2)^(1/2))+4/d^2/(a^2+b^2)*c*a^3*f/(4*a^2+4*b^2)*ln(1+exp(2*d*x+2*c))-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*
dilog(1+I*exp(d*x+c))*a-8/d^2/(a^2+b^2)*b^2*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))*a-4/d/(a^2+b^2)*a^3*f/(4*a^2
+4*b^2)*ln(1+I*exp(d*x+c))*x-4/d^2/(a^2+b^2)*a^3*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*c-1/d^2/(a^2+b^2)^2*f*b^4/
a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-1/d/(a^2+b^2)^2*f*b^4/a*ln((b*exp(d*x+c)+(a^2+b
^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x-1/d^2/(a^2+b^2)^2*f*b^4/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)
^(1/2)))*c-8/d/(a^2+b^2)*b^2*e/(4*a^2+4*b^2)*a*ln(1+exp(2*d*x+2*c))+1/2/d/(a^2+b^2)^(5/2)*b^2*e*a^2*arctanh(1/
2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-4/d/(a^2+b^2)*e*a^2*b/(4*a^2+4*b^2)*arctan(exp(d*x+c))+12/d^2/(a^2+b^2
)*c*f*b^3/(4*a^2+4*b^2)*arctan(exp(d*x+c))+6*I/d^2/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog(1+I*exp(d*x+c))-6*I/d^2
/(a^2+b^2)*b^3*f/(4*a^2+4*b^2)*dilog(1-I*exp(d*x+c))-2*I/d/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1-I*exp(d*x+c))*b*
x+2*I/d/(a^2+b^2)*a^2*f/(4*a^2+4*b^2)*ln(1+I*exp(d*x+c))*b*x

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. 7645 vs. \(2 (676) = 1352\).

Time = 0.49 (sec) , antiderivative size = 7645, normalized size of antiderivative = 10.25 \[ \int \frac {(e+f x) \text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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