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

Optimal. Leaf size=591 \[ \frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )}+\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac{b^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^2 d^2 \left (a^2+b^2\right )}-\frac{i b^2 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{i b^2 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{b f \text{PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{b f \text{PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^2}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d \left (a^2+b^2\right )}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d \left (a^2+b^2\right )}-\frac{b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a^2 d \left (a^2+b^2\right )}+\frac{2 b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d \left (a^2+b^2\right )}+\frac{2 b (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d}-\frac{(e+f x) \tan ^{-1}(\sinh (c+d x))}{a d}-\frac{2 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tan ^{-1}(\sinh (c+d x))}{a d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.900204, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 18, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {5589, 2621, 321, 207, 5462, 5203, 12, 4180, 2279, 2391, 3770, 5461, 4182, 5573, 5561, 2190, 6742, 3718} \[ \frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )}+\frac{b^3 f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac{b^3 f \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a^2 d^2 \left (a^2+b^2\right )}-\frac{i b^2 f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{i b^2 f \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{b f \text{PolyLog}\left (2,-e^{2 c+2 d x}\right )}{2 a^2 d^2}-\frac{b f \text{PolyLog}\left (2,e^{2 c+2 d x}\right )}{2 a^2 d^2}+\frac{i f \text{PolyLog}\left (2,-i e^{c+d x}\right )}{a d^2}-\frac{i f \text{PolyLog}\left (2,i e^{c+d x}\right )}{a d^2}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a^2 d \left (a^2+b^2\right )}+\frac{b^3 (e+f x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a^2 d \left (a^2+b^2\right )}-\frac{b^3 (e+f x) \log \left (e^{2 (c+d x)}+1\right )}{a^2 d \left (a^2+b^2\right )}+\frac{2 b^2 (e+f x) \tan ^{-1}\left (e^{c+d x}\right )}{a d \left (a^2+b^2\right )}+\frac{2 b (e+f x) \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a^2 d}-\frac{f \tanh ^{-1}(\cosh (c+d x))}{a d^2}-\frac{(e+f x) \text{csch}(c+d x)}{a d}-\frac{(e+f x) \tan ^{-1}(\sinh (c+d x))}{a d}-\frac{2 f x \tan ^{-1}\left (e^{c+d x}\right )}{a d}+\frac{f x \tan ^{-1}(\sinh (c+d x))}{a d} \]

Antiderivative was successfully verified.

[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 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 2621

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 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 5203

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 12

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

Rule 4180

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 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 5461

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 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 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 5561

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 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 3718

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]

Rubi steps

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

Mathematica [A]  time = 6.52814, size = 535, normalized size = 0.91 \[ \frac{\frac{2 i a f \text{PolyLog}(2,-i (\sinh (c+d x)+\cosh (c+d x)))-2 i a f \text{PolyLog}(2,i (\sinh (c+d x)+\cosh (c+d x)))+b f \text{PolyLog}(2,-\sinh (2 (c+d x))-\cosh (2 (c+d x)))-4 a d e \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))-4 a d f x \tan ^{-1}(\sinh (c+d x)+\cosh (c+d x))+b c^2 f+2 b d e \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)-2 b c d e+2 b d f x \log (\sinh (2 (c+d x))+\cosh (2 (c+d x))+1)-2 b d^2 e x-b d^2 f x^2}{a^2+b^2}+\frac{2 b^3 \left (f \text{PolyLog}\left (2,\frac{b e^{c+d x}}{\sqrt{a^2+b^2}-a}\right )+f \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )+f (c+d x) \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )+d e \log (a+b \sinh (c+d x))-c f \log (a+b \sinh (c+d x))-\frac{1}{2} f (c+d x)^2\right )}{a^2 \left (a^2+b^2\right )}+\frac{b f \left (\text{PolyLog}\left (2,e^{-2 (c+d x)}\right )-(c+d x) \left (2 \log \left (1-e^{-2 (c+d x)}\right )+c+d x\right )\right )}{a^2}-\frac{2 b d e \log (\sinh (c+d x))}{a^2}+\frac{2 b c f \log (\sinh (c+d x))}{a^2}+\frac{d (e+f x) \tanh \left (\frac{1}{2} (c+d x)\right )}{a}-\frac{d (e+f x) \coth \left (\frac{1}{2} (c+d x)\right )}{a}+\frac{2 f \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )}{a}}{2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.315, size = 1529, normalized size = 2.6 \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)^2*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (\frac{b^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + a^{2} b^{2}\right )} d} + \frac{2 \, a \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{b \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{2 \, e^{\left (-d x - c\right )}}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} - \frac{b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac{b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d}\right )} e +{\left (8 \, b d \int \frac{x}{8 \,{\left (a^{2} d e^{\left (d x + c\right )} + a^{2} d\right )}}\,{d x} - 8 \, b d \int \frac{x}{8 \,{\left (a^{2} d e^{\left (d x + c\right )} - a^{2} d\right )}}\,{d x} + a{\left (\frac{d x + c}{a^{2} d^{2}} - \frac{\log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2} d^{2}}\right )} - a{\left (\frac{d x + c}{a^{2} d^{2}} - \frac{\log \left (e^{\left (d x + c\right )} - 1\right )}{a^{2} d^{2}}\right )} - \frac{2 \, x e^{\left (d x + c\right )}}{a d e^{\left (2 \, d x + 2 \, c\right )} - a d} - 8 \, \int -\frac{a b^{3} x e^{\left (d x + c\right )} - b^{4} x}{4 \,{\left (a^{4} b + a^{2} b^{3} -{\left (a^{4} b e^{\left (2 \, c\right )} + a^{2} b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{5} e^{c} + a^{3} b^{2} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 8 \, \int \frac{a x e^{\left (d x + c\right )} + b x}{4 \,{\left (a^{2} + b^{2} +{\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x}\right )} f \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 3.03856, size = 6273, normalized size = 10.61 \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)^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 + 2*a^2*b*d*e + 2*I*a^3*c*f - 2*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 + 2*a^2*b*d*e - 2*I*a^3*c*f - 2*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 + 2*a^2*b*d*f*x + 2*I*a^3*c*f + 2*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 + 2*a^2*b*d*f*x - 2*I*a^3*c*f + 2*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)

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

[Out]

Timed out