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

Optimal. Leaf size=442 \[ -\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^{3/2}}+\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )^{3/2}}-\frac{f \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{f \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac{b^2 f \tan ^{-1}(\sinh (c+d x))}{a d^2 \left (a^2+b^2\right )}+\frac{b f \log (\cosh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )^{3/2}}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )^{3/2}}-\frac{b (e+f x) \tanh (c+d x)}{d \left (a^2+b^2\right )}-\frac{b^2 (e+f x) \text{sech}(c+d x)}{a d \left (a^2+b^2\right )}-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.805706, antiderivative size = 442, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 19, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.594, Rules used = {5589, 2622, 321, 207, 5462, 6271, 12, 4182, 2279, 2391, 3770, 5573, 3322, 2264, 2190, 6742, 4184, 3475, 5451} \[ -\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )^{3/2}}+\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )^{3/2}}-\frac{f \text{PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac{f \text{PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac{b^2 f \tan ^{-1}(\sinh (c+d x))}{a d^2 \left (a^2+b^2\right )}+\frac{b f \log (\cosh (c+d x))}{d^2 \left (a^2+b^2\right )}-\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )^{3/2}}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )^{3/2}}-\frac{b (e+f x) \tanh (c+d x)}{d \left (a^2+b^2\right )}-\frac{b^2 (e+f x) \text{sech}(c+d x)}{a d \left (a^2+b^2\right )}-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5589

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 2622

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

Rule 321

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 207

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

Rule 5462

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 6271

Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 - u^2), x], x] /; I
nverseFunctionFreeQ[u, 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 4182

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 2279

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 2391

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 3770

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

Rule 5573

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 3322

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(-(I*b) + 2*a*E^(-(I*e) + f*fz*x) + I*b*E^(2*(-(I*e) + f*fz*x))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 6742

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

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

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

Rule 5451

Int[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> -Si
mp[((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]

Rubi steps

\begin{align*} \int \frac{(e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x) \text{csch}(c+d x) \text{sech}^2(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x) \text{sech}^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=-\frac{(e+f x) \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{b \int (e+f x) \text{sech}^2(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}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac{f \int \left (-\frac{\tanh ^{-1}(\cosh (c+d x))}{d}+\frac{\text{sech}(c+d x)}{d}\right ) \, dx}{a}\\ &=-\frac{(e+f x) \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{b \int \left (a (e+f x) \text{sech}^2(c+d x)-b (e+f x) \text{sech}(c+d x) \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac{\left (2 b^3\right ) \int \frac{e^{c+d x} (e+f x)}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}+\frac{f \int \tanh ^{-1}(\cosh (c+d x)) \, dx}{a d}-\frac{f \int \text{sech}(c+d x) \, dx}{a d}\\ &=-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (c+d x))}{a d}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{\left (2 b^4\right ) \int \frac{e^{c+d x} (e+f x)}{2 a-2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^{3/2}}+\frac{\left (2 b^4\right ) \int \frac{e^{c+d x} (e+f x)}{2 a+2 \sqrt{a^2+b^2}+2 b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )^{3/2}}-\frac{b \int (e+f x) \text{sech}^2(c+d x) \, dx}{a^2+b^2}+\frac{b^2 \int (e+f x) \text{sech}(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac{f \int d x \text{csch}(c+d x) \, dx}{a d}\\ &=-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (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 \left (a^2+b^2\right )^{3/2} d}+\frac{b^3 (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 )^{3/2} d}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{b^2 (e+f x) \text{sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac{b (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac{f \int x \text{csch}(c+d x) \, dx}{a}+\frac{\left (b^3 f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a-2 \sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d}-\frac{\left (b^3 f\right ) \int \log \left (1+\frac{2 b e^{c+d x}}{2 a+2 \sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right )^{3/2} d}+\frac{(b f) \int \tanh (c+d x) \, dx}{\left (a^2+b^2\right ) d}+\frac{\left (b^2 f\right ) \int \text{sech}(c+d x) \, dx}{a \left (a^2+b^2\right ) d}\\ &=-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{b^2 f \tan ^{-1}(\sinh (c+d x))}{a \left (a^2+b^2\right ) d^2}-\frac{2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (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 \left (a^2+b^2\right )^{3/2} d}+\frac{b^3 (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 )^{3/2} d}+\frac{b f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{b^2 (e+f x) \text{sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac{b (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d}+\frac{\left (b^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac{\left (b^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 b x}{2 a+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}-\frac{f \int \log \left (1-e^{c+d x}\right ) \, dx}{a d}+\frac{f \int \log \left (1+e^{c+d x}\right ) \, dx}{a d}\\ &=-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{b^2 f \tan ^{-1}(\sinh (c+d x))}{a \left (a^2+b^2\right ) d^2}-\frac{2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (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 \left (a^2+b^2\right )^{3/2} d}+\frac{b^3 (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 )^{3/2} d}+\frac{b f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}-\frac{b^3 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{b^2 (e+f x) \text{sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac{b (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d}-\frac{f \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}+\frac{f \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{c+d x}\right )}{a d^2}\\ &=-\frac{f \tan ^{-1}(\sinh (c+d x))}{a d^2}+\frac{b^2 f \tan ^{-1}(\sinh (c+d x))}{a \left (a^2+b^2\right ) d^2}-\frac{2 f x \tanh ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tanh ^{-1}(\cosh (c+d x))}{a d}-\frac{(e+f x) \tanh ^{-1}(\cosh (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 \left (a^2+b^2\right )^{3/2} d}+\frac{b^3 (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 )^{3/2} d}+\frac{b f \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d^2}-\frac{f \text{Li}_2\left (-e^{c+d x}\right )}{a d^2}+\frac{f \text{Li}_2\left (e^{c+d x}\right )}{a d^2}-\frac{b^3 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac{b^3 f \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right )^{3/2} d^2}+\frac{(e+f x) \text{sech}(c+d x)}{a d}-\frac{b^2 (e+f x) \text{sech}(c+d x)}{a \left (a^2+b^2\right ) d}-\frac{b (e+f x) \tanh (c+d x)}{\left (a^2+b^2\right ) d}\\ \end{align*}

Mathematica [A]  time = 7.2462, size = 459, normalized size = 1.04 \[ \frac{\text{csch}(c+d x) (a+b \sinh (c+d x)) \left (\frac{b^3 \left (-f \text{PolyLog}\left (2,\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}+a}\right )+2 d e \tanh ^{-1}\left (\frac{a+b \sinh (c+d x)+b \cosh (c+d x)}{\sqrt{a^2+b^2}}\right )-f (c+d x) \log \left (\frac{b (\sinh (c+d x)+\cosh (c+d x))}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b (\sinh (c+d x)+\cosh (c+d x))}{\sqrt{a^2+b^2}+a}+1\right )-2 c f \tanh ^{-1}\left (\frac{a+b \sinh (c+d x)+b \cosh (c+d x)}{\sqrt{a^2+b^2}}\right )\right )}{a \left (a^2+b^2\right )^{3/2}}+\frac{f \left (\text{PolyLog}\left (2,-e^{-c-d x}\right )-\text{PolyLog}\left (2,e^{-c-d x}\right )+(c+d x) \left (\log \left (1-e^{-c-d x}\right )-\log \left (e^{-c-d x}+1\right )\right )\right )}{a}+\frac{d (e+f x) \text{sech}(c+d x) (a-b \sinh (c+d x))}{a^2+b^2}-\frac{2 a f \tan ^{-1}\left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a^2+b^2}+\frac{b f \log (\cosh (c+d x))}{a^2+b^2}+\frac{d e \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a}-\frac{c f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a}\right )}{d^2 (a \text{csch}(c+d x)+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.279, size = 1815, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(a^2+b^2)^(5/2)/d^2*f*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a^3-2/(a^2+b^2)^(5/2)/d^2*f*b^3*a
rctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a+1/(a^2+b^2)/d*a*e*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d*a*e*ln(exp
(d*x+c)+1)-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)^(5/2)/d^2
*a*b^3*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/(a^2+b^2)^(5/2)/d^2*a*b^3*f*ln((-b*exp(d
*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c+1/(a^2+b^2)^(5/2)/d*f*b^5/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+
a)/(a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(3/2)/d^2*b^3*f*c/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/
(a^2+b^2)^(5/2)/d^2*f*a^3*b*c*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/2)/d^2*b^5*f*c/
a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/2)/d*a*b^3*f*ln((-b*exp(d*x+c)+(a^2+b^2)^(1
/2)-a)/(-a+(a^2+b^2)^(1/2)))*x-1/(a^2+b^2)^(5/2)/d*f*b^5/a*ln((-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^
(1/2)))*x+1/(a^2+b^2)^(5/2)/d*a*b^3*f*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*x+1/(a^2+b^2)^(
5/2)/d^2*f*b^5/a*ln((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))*c-1/(a^2+b^2)^(5/2)/d^2*f*b^5/a*ln((
-b*exp(d*x+c)+(a^2+b^2)^(1/2)-a)/(-a+(a^2+b^2)^(1/2)))*c-2/(a^2+b^2)/d^2*b*f*ln(exp(d*x+c))-1/(a^2+b^2)/d*ln(e
xp(d*x+c)+1)*a*f*x-8/(a^2+b^2)/d^2*a^3*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+4/(a^2+b^2)/d^2*f*b^3/(4*a^2+4*b^2)*
ln(1+exp(2*d*x+2*c))-1/(a^2+b^2)/d^2*a*f*c*ln(exp(d*x+c)-1)-1/(a^2+b^2)/d^2*b^2*f/a*dilog(exp(d*x+c))-1/(a^2+b
^2)/d^2*b^2*f/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)+b/d*e/(a^2+b^2)^(3/2)*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a+2*(f*x+e)*(a*exp(d*x+c)+b)/d/(a^
2+b^2)/(1+exp(2*d*x+2*c))-1/(a^2+b^2)/d^2*b^2*f*c/a*ln(exp(d*x+c)-1)+4/(a^2+b^2)/d^2*a^2*b*f/(4*a^2+4*b^2)*ln(
1+exp(2*d*x+2*c))-1/(a^2+b^2)^(5/2)/d^2*a*b^3*f*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)^(5/2)/d^2*a*b^3*f*dilog((b*exp(d*x+c)+(a^2+b^2)^(1/2)+a)/(a+(a^2+b^2)^(1/2)))-8/(a^2+b^2)/d^2*a*b^
2*f/(4*a^2+4*b^2)*arctan(exp(d*x+c))+1/(a^2+b^2)^(3/2)/d^2*f*b^3/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^
(1/2))-1/(a^2+b^2)^(5/2)/d^2*f*b^5/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)
^(5/2)/d^2*f*b^5/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)^(5/2)/d^2*f*b^5/a*a
rctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(3/2)/d^2*a*f*b*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/
(a^2+b^2)^(1/2))+1/(a^2+b^2)^(3/2)/d*b^3*e/a*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)^(5/
2)/d*a^3*b*e*arctanh(1/2*(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))+1/(a^2+b^2)^(5/2)/d*b^5*e/a*arctanh(1/2*(2*b*ex
p(d*x+c)+2*a)/(a^2+b^2)^(1/2))-1/(a^2+b^2)/d*b^2*f/a*ln(exp(d*x+c)+1)*x-b/d^2*f*c/(a^2+b^2)^(3/2)*arctanh(1/2*
(2*b*exp(d*x+c)+2*a)/(a^2+b^2)^(1/2))*a

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.63917, size = 5233, normalized size = 11.84 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*(a^3*b + a*b^3)*d*f*x*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*d*f*x*sinh(d*x + c)^2 - 2*(a^3*b + a*b^3)*d*e +
(b^4*f*cosh(d*x + c)^2 + 2*b^4*f*cosh(d*x + c)*sinh(d*x + c) + b^4*f*sinh(d*x + c)^2 + b^4*f)*sqrt((a^2 + b^2)
/b^2)*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^4*f*cosh(d*x + c)^2 + 2*b^4*f*cosh(d*x + c)*sinh(d*x + c) + b^4*f*sinh(d*x + c)^2 + b^4*f)*sqrt(
(a^2 + b^2)/b^2)*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^4*d*e - b^4*c*f + (b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*e - b^4*c*f)*cosh(d*
x + c)*sinh(d*x + c) + (b^4*d*e - b^4*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^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^4*d*e - b^4*c*f + (b^4*d*e - b^4*c*f)*cosh(d*x + c)^2 +
2*(b^4*d*e - b^4*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d*e - b^4*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^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^4*d*f*x + b^4*c*f + (b^4*d*
f*x + b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d*f*x + b^4*c*f)*s
inh(d*x + c)^2)*sqrt((a^2 + b^2)/b^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) - (b^4*d*f*x + b^4*c*f + (b^4*d*f*x + b^4*c*f)*cosh(d*x + c)^2 + 2*(b^4*d*
f*x + b^4*c*f)*cosh(d*x + c)*sinh(d*x + c) + (b^4*d*f*x + b^4*c*f)*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^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) + 2*((
a^4 + a^2*b^2)*f*cosh(d*x + c)^2 + 2*(a^4 + a^2*b^2)*f*cosh(d*x + c)*sinh(d*x + c) + (a^4 + a^2*b^2)*f*sinh(d*
x + c)^2 + (a^4 + a^2*b^2)*f)*arctan(cosh(d*x + c) + sinh(d*x + c)) - 2*((a^4 + a^2*b^2)*d*f*x + (a^4 + a^2*b^
2)*d*e)*cosh(d*x + c) - ((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)
*sinh(d*x + c) + (a^4 + 2*a^2*b^2 + b^4)*f*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f)*dilog(cosh(d*x + c) +
sinh(d*x + c)) + ((a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4)*f*cosh(d*x + c)*sinh(d
*x + c) + (a^4 + 2*a^2*b^2 + b^4)*f*sinh(d*x + c)^2 + (a^4 + 2*a^2*b^2 + b^4)*f)*dilog(-cosh(d*x + c) - sinh(d
*x + c)) - ((a^3*b + a*b^3)*f*cosh(d*x + c)^2 + 2*(a^3*b + a*b^3)*f*cosh(d*x + c)*sinh(d*x + c) + (a^3*b + a*b
^3)*f*sinh(d*x + c)^2 + (a^3*b + a*b^3)*f)*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + ((a^4 + 2*a^
2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e + ((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*
e)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)*cosh(d*x + c)*sinh(d*x +
c) + ((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*d*e)*sinh(d*x + c)^2)*log(cosh(d*x + c) + sinh(d
*x + c) + 1) - ((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f + ((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^
4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh
(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d*e - (a^4 + 2*a^2*b^2 + b^4)*c*f)*sinh(d*x + c)^2)*log(cos
h(d*x + c) + sinh(d*x + c) - 1) - ((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f + ((a^4 + 2*a^2
*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)^2 + 2*((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2
*a^2*b^2 + b^4)*c*f)*cosh(d*x + c)*sinh(d*x + c) + ((a^4 + 2*a^2*b^2 + b^4)*d*f*x + (a^4 + 2*a^2*b^2 + b^4)*c*
f)*sinh(d*x + c)^2)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) + 2*(2*(a^3*b + a*b^3)*d*f*x*cosh(d*x + c) - (a^4
+ a^2*b^2)*d*f*x - (a^4 + a^2*b^2)*d*e)*sinh(d*x + c))/((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c)^2 + 2*(a^5
+ 2*a^3*b^2 + a*b^4)*d^2*cosh(d*x + c)*sinh(d*x + c) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh(d*x + c)^2 + (a^5 +
2*a^3*b^2 + a*b^4)*d^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out