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

Optimal. Leaf size=734 \[ -\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}-\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{b^2 f (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{2 i b f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac{2 i b f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \left (a^2+b^2\right )}-\frac{b^2 f^2 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a d^3 \left (a^2+b^2\right )}-\frac{2 i b f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac{2 i b f^2 \text{PolyLog}\left (3,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac{f (e+f x) \text{PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \text{PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )}-\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )}+\frac{b^2 (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )}-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d} \]

[Out]

(-2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) - (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2
*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)^2*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b
^2)*d) + ((2*I)*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - ((2*I)*b*f*(e + f*x)*PolyLog[2
, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])
/(a*(a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)
*d^2) + (b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^2) - (f*(e + f*x)*PolyLog[2, -E^(2*c +
2*d*x)])/(a*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) - ((2*I)*b*f^2*PolyLog[3, (-I)*E^(c + d*
x)])/((a^2 + b^2)*d^3) + ((2*I)*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) - (b^2*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^3) + (f^2*Pol
yLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) - (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3)

________________________________________________________________________________________

Rubi [A]  time = 1.09728, antiderivative size = 734, normalized size of antiderivative = 1., number of steps used = 33, number of rules used = 12, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5589, 5461, 4182, 2531, 2282, 6589, 5573, 5561, 2190, 6742, 4180, 3718} \[ -\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^2 \left (a^2+b^2\right )}-\frac{2 b^2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{b^2 f (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right )}{a d^2 \left (a^2+b^2\right )}+\frac{2 i b f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}-\frac{2 i b f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right )}{d^2 \left (a^2+b^2\right )}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a d^3 \left (a^2+b^2\right )}+\frac{2 b^2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}\right )}{a d^3 \left (a^2+b^2\right )}-\frac{b^2 f^2 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 a d^3 \left (a^2+b^2\right )}-\frac{2 i b f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}+\frac{2 i b f^2 \text{PolyLog}\left (3,i e^{c+d x}\right )}{d^3 \left (a^2+b^2\right )}-\frac{f (e+f x) \text{PolyLog}\left (2,-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{PolyLog}\left (2,e^{2 c+2 d x}\right )}{a d^2}+\frac{f^2 \text{PolyLog}\left (3,-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \text{PolyLog}\left (3,e^{2 c+2 d x}\right )}{2 a d^3}-\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}+1\right )}{a d \left (a^2+b^2\right )}-\frac{b^2 (e+f x)^2 \log \left (\frac{b e^{c+d x}}{\sqrt{a^2+b^2}+a}+1\right )}{a d \left (a^2+b^2\right )}+\frac{b^2 (e+f x)^2 \log \left (e^{2 (c+d x)}+1\right )}{a d \left (a^2+b^2\right )}-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{d \left (a^2+b^2\right )}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) - (2*(e + f*x)^2*ArcTanh[E^(2*c + 2*d*x)])/(a*d) - (b^2
*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) - (b^2*(e + f*x)^2*Log[1 + (b*E
^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)*d) + (b^2*(e + f*x)^2*Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b
^2)*d) + ((2*I)*b*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)*d^2) - ((2*I)*b*f*(e + f*x)*PolyLog[2
, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])
/(a*(a^2 + b^2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)
*d^2) + (b^2*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2 + b^2)*d^2) - (f*(e + f*x)*PolyLog[2, -E^(2*c +
2*d*x)])/(a*d^2) + (f*(e + f*x)*PolyLog[2, E^(2*c + 2*d*x)])/(a*d^2) - ((2*I)*b*f^2*PolyLog[3, (-I)*E^(c + d*
x)])/((a^2 + b^2)*d^3) + ((2*I)*b*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((
b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) + (2*b^2*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqr
t[a^2 + b^2]))])/(a*(a^2 + b^2)*d^3) - (b^2*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)*d^3) + (f^2*Pol
yLog[3, -E^(2*c + 2*d*x)])/(2*a*d^3) - (f^2*PolyLog[3, E^(2*c + 2*d*x)])/(2*a*d^3)

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 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

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 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 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)^2 \text{csch}(c+d x) \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \text{csch}(c+d x) \text{sech}(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^2 \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac{2 \int (e+f x)^2 \text{csch}(2 c+2 d x) \, dx}{a}-\frac{b \int (e+f x)^2 \text{sech}(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)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}\\ &=\frac{b^2 (e+f x)^3}{3 a \left (a^2+b^2\right ) f}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b \int \left (a (e+f x)^2 \text{sech}(c+d x)-b (e+f x)^2 \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int \frac{e^{c+d x} (e+f x)^2}{a-\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int \frac{e^{c+d x} (e+f x)^2}{a+\sqrt{a^2+b^2}+b e^{c+d x}} \, dx}{a \left (a^2+b^2\right )}-\frac{(2 f) \int (e+f x) \log \left (1-e^{2 c+2 d x}\right ) \, dx}{a d}+\frac{(2 f) \int (e+f x) \log \left (1+e^{2 c+2 d x}\right ) \, dx}{a d}\\ &=\frac{b^2 (e+f x)^3}{3 a \left (a^2+b^2\right ) f}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac{b \int (e+f x)^2 \text{sech}(c+d x) \, dx}{a^2+b^2}+\frac{b^2 \int (e+f x)^2 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}+\frac{\left (2 b^2 f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac{\left (2 b^2 f\right ) \int (e+f x) \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac{f^2 \int \text{Li}_2\left (-e^{2 c+2 d x}\right ) \, dx}{a d^2}-\frac{f^2 \int \text{Li}_2\left (e^{2 c+2 d x}\right ) \, dx}{a d^2}\\ &=-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{f (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac{\left (2 b^2\right ) \int \frac{e^{2 (c+d x)} (e+f x)^2}{1+e^{2 (c+d x)}} \, dx}{a \left (a^2+b^2\right )}+\frac{(2 i b f) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}-\frac{(2 i b f) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d}+\frac{f^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 c+2 d x}\right )}{2 a d^3}+\frac{\left (2 b^2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}+\frac{\left (2 b^2 f^2\right ) \int \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{2 i b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 i b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{f (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac{f^2 \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac{\left (2 b^2 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d}+\frac{\left (2 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\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 ) d^3}+\frac{\left (2 b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\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 ) d^3}-\frac{\left (2 i b f^2\right ) \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}+\frac{\left (2 i b f^2\right ) \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{2 i b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 i b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{b^2 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{f (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}+\frac{2 b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac{\left (2 i b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{\left (2 i b f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}-\frac{\left (b^2 f^2\right ) \int \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{2 i b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 i b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{b^2 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{f (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac{2 i b f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}\\ &=-\frac{2 b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}-\frac{b^2 (e+f x)^2 \log \left (1+\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d}+\frac{b^2 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d}+\frac{2 i b f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 i b f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{2 b^2 f (e+f x) \text{Li}_2\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^2}+\frac{b^2 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right ) d^2}-\frac{f (e+f x) \text{Li}_2\left (-e^{2 c+2 d x}\right )}{a d^2}+\frac{f (e+f x) \text{Li}_2\left (e^{2 c+2 d x}\right )}{a d^2}-\frac{2 i b f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{2 i b f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right ) d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}+\frac{2 b^2 f^2 \text{Li}_3\left (-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right )}{a \left (a^2+b^2\right ) d^3}-\frac{b^2 f^2 \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right ) d^3}+\frac{f^2 \text{Li}_3\left (-e^{2 c+2 d x}\right )}{2 a d^3}-\frac{f^2 \text{Li}_3\left (e^{2 c+2 d x}\right )}{2 a d^3}\\ \end{align*}

Mathematica [B]  time = 33.66, size = 3268, normalized size = 4.45 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

2*(-(b^2*x*(3*e^2 + 3*e*f*x + f^2*x^2)*Csch[2*c])/(6*a*(a^2 + b^2)) + (a*E^c*((e + f*x)^3/(3*E^c*f) + ((1 + E^
(-c))*(e + f*x)^2*Log[1 + E^(-c - d*x)])/d - (2*(1 + E^c)*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] + f*PolyLog
[3, -E^(-c - d*x)]))/(d^3*E^c)))/(2*(a^2 + b^2)*(1 + E^c)) + (d^2*(d*x*((-3*I)*b*e*f*x + a*((-3*I)*e^2*E^c + 3
*e*f*x + f^2*x^2)) + 3*(1 + I*E^c)*f*x*(2*a*e - (2*I)*b*e + a*f*x)*Log[1 - I*E^(-c - d*x)] + 3*a*e^2*(1 + I*E^
c)*Log[I - E^(c + d*x)]) - (6*I)*d*(-I + E^c)*f*((-I)*b*e + a*(e + f*x))*PolyLog[2, I*E^(-c - d*x)] - (6*I)*a*
(-I + E^c)*f^2*PolyLog[3, I*E^(-c - d*x)])/(6*(a - I*b)*((-I)*a + b)*d^3*(-I + E^c)) - (b^2*E^(2*c)*((2*(e + f
*x)^3)/(E^(2*c)*f) - (3*(1 - E^(-2*c))*(e + f*x)^2*Log[1 - E^(-c - d*x)])/d - (3*(1 - E^(-2*c))*(e + f*x)^2*Lo
g[1 + E^(-c - d*x)])/d + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] + f*PolyLog[3, -E^(-c - d*
x)]))/(d^3*E^(2*c)) + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, E^(-c - d*x)] + f*PolyLog[3, E^(-c - d*x)]))
/(d^3*E^(2*c))))/(6*a*(a^2 + b^2)*(-1 + E^(2*c))) - ((I/2)*b*((-2*I)*d^2*e^2*ArcTan[E^(c + d*x)] + d^2*f^2*x^2
*Log[1 - I*E^(c + d*x)] - d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - 2*d*f^2*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*f^
2*x*PolyLog[2, I*E^(c + d*x)] + 2*f^2*PolyLog[3, (-I)*E^(c + d*x)] - 2*f^2*PolyLog[3, I*E^(c + d*x)]))/((a^2 +
b^2)*d^3) - ((-I)*b*d^3*e*E^(2*c)*f*x^2 + 2*a*d^2*e^2*ArcTan[1 - (1 + I)*E^(c + d*x)] + (2*I)*a*d^2*e^2*E^(2*
c)*ArcTan[1 - (1 + I)*E^(c + d*x)] + (2*I)*a*d^2*e*f*x*Log[1 - E^(c + d*x)] - 2*a*d^2*e*E^(2*c)*f*x*Log[1 - E^
(c + d*x)] + I*a*d^2*f^2*x^2*Log[1 - E^(c + d*x)] - a*d^2*E^(2*c)*f^2*x^2*Log[1 - E^(c + d*x)] - (2*I)*a*d^2*e
*f*x*Log[1 - I*E^(c + d*x)] + 2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + 2*a*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*
x)] + (2*I)*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] - I*a*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + a*d^2*E^(2*c
)*f^2*x^2*Log[1 - I*E^(c + d*x)] + 2*d*(-I + E^(2*c))*f*(I*b*e + a*(e + f*x))*PolyLog[2, I*E^(c + d*x)] - 2*a*
d*(-I + E^(2*c))*f*(e + f*x)*PolyLog[2, E^(c + d*x)] + (2*I)*a*f^2*PolyLog[3, I*E^(c + d*x)] - 2*a*E^(2*c)*f^2
*PolyLog[3, I*E^(c + d*x)] - (2*I)*a*f^2*PolyLog[3, E^(c + d*x)] + 2*a*E^(2*c)*f^2*PolyLog[3, E^(c + d*x)])/(2
*(a^2 + b^2)*d^3*(-I + E^(2*c))) + (b^2*(6*e^2*E^(2*c)*x + 6*e*E^(2*c)*f*x^2 + 2*E^(2*c)*f^2*x^3 + (6*a*Sqrt[a
^2 + b^2]*e^2*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/(Sqrt[-(a^2 + b^2)^2]*d) + (6*a*Sqrt[-(a^2 + b^2)^
2]*e^2*E^(2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) - (6*a*Sqrt[-(a^2 + b^2)^2]
*e^2*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (6*a*Sqrt[-(a^2 + b^2)^2]*e^2*E^(2
*c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (3*e^2*Log[2*a*E^(c + d*x) + b*(-1
+ E^(2*(c + d*x)))])/d - (3*e^2*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e*f*x*Log[1 +
(b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c
- Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/
d - (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f*x*Log[1 + (b
*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c +
Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (3*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d
- (3*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1 + E^(2*c))*f*(
e + f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 + E^(2*c))*f*(e +
f*x)*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*f^2*PolyLog[3, -((b*E^(2*c
+ d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - S
qrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (6*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))]
)/d^3 + (6*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3))/(6*a*(a^2 +
b^2)*(-1 + E^(2*c))) + (b^2*e*f*x^2)/(2*a*(a^2 + b^2)*(Cosh[c] + Sinh[c])^2) + (b^2*f^2*x^3)/(6*a*(a^2 + b^2)
*(Cosh[c] + Sinh[c])^2) + (b^2*f^2*x^3*Coth[2*c])/(6*a*(a^2 + b^2)*(Cosh[c] + Sinh[c])^2) + (b^2*e*f*x^2*Cosh[
2*c]*Csch[c]*Sech[c])/(4*a*(a^2 + b^2)*(Cosh[c] + Sinh[c])^2) + (e^2*x*Csch[c]^2*(-(a^2*Coth[c]) + Csch[c]*(a^
2 + b^2 - I*a^2*Sinh[c])))/(a*(a^2 + b^2)*(Csch[c/2] - I*Sech[c/2])*(Csch[c/2] + I*Sech[c/2])) - ((1/4 - I/4)*
a*e*f*x^2*Cosh[c])/((a^2 + b^2)*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) + (b*e*f*x^2*Cos
h[c])/(4*(a^2 + b^2)*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) - ((1/12 - I/12)*a*f^2*x^3*
Cosh[c])/((a^2 + b^2)*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) - (b*e*f*x^2*Cosh[3*c])/(4
*(a^2 + b^2)*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) - ((1/4 - I/4)*a*e*f*x^2*Sinh[c])/(
(a^2 + b^2)*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) + (b*e*f*x^2*Sinh[c])/(4*(a^2 + b^2)
*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) - ((1/12 - I/12)*a*f^2*x^3*Sinh[c])/((a^2 + b^2
)*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) + ((1/4 - I/4)*a*e*f*x^2*Cosh[3*c])/((a^2 + b^
2)*(Cosh[c] + Sinh[c])^2*(Cosh[c] + I*(-I + Sinh[c] + Sinh[2*c]))) + ((1/12 - I/12)*a*f^2*x^3*Cosh[3*c])/((a^2
+ b^2)*(Cosh[c] + Sinh[c])^2*(Cosh[c] + I*(-I + Sinh[c] + Sinh[2*c]))) - (b*e*f*x^2*Sinh[3*c])/(4*(a^2 + b^2)
*(Cosh[c] + Sinh[c])^2*(-I - I*Cosh[c] + Sinh[c] + Sinh[2*c])) + ((1/4 - I/4)*a*e*f*x^2*Sinh[3*c])/((a^2 + b^2
)*(Cosh[c] + Sinh[c])^2*(Cosh[c] + I*(-I + Sinh[c] + Sinh[2*c]))) + ((1/12 - I/12)*a*f^2*x^3*Sinh[3*c])/((a^2
+ b^2)*(Cosh[c] + Sinh[c])^2*(Cosh[c] + I*(-I + Sinh[c] + Sinh[2*c]))))

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Maple [F]  time = 0.58, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}{\rm csch} \left (dx+c\right ){\rm sech} \left (dx+c\right )}{a+b\sinh \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -e^{2}{\left (\frac{b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{3} + a b^{2}\right )} d} - \frac{2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac{a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac{\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac{\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} + \frac{2 \,{\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac{2 \,{\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac{{\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x{\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac{{\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x{\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} - \frac{2 \,{\left (d^{3} f^{2} x^{3} + 3 \, d^{3} e f x^{2}\right )}}{3 \, a d^{3}} + \int \frac{2 \,{\left (b^{3} f^{2} x^{2} + 2 \, b^{3} e f x -{\left (a b^{2} f^{2} x^{2} e^{c} + 2 \, a b^{2} e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{3} b + a b^{3} -{\left (a^{3} b e^{\left (2 \, c\right )} + a b^{3} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{4} e^{c} + a^{2} b^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int -\frac{2 \,{\left (a f^{2} x^{2} + 2 \, a e f x -{\left (b f^{2} x^{2} e^{c} + 2 \, b e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{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 )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-e^2*(b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^3 + a*b^2)*d) - 2*b*arctan(e^(-d*x - c))/((a^2 +
b^2)*d) + a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) - log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(
a*d)) + 2*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))*e*f/(a*d^2) + 2*(d*x*log(-e^(d*x + c) + 1) + dilog(
e^(d*x + c)))*e*f/(a*d^2) + (d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x +
c)))*f^2/(a*d^3) + (d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d*x + c)) - 2*polylog(3, e^(d*x + c)))*f^2
/(a*d^3) - 2/3*(d^3*f^2*x^3 + 3*d^3*e*f*x^2)/(a*d^3) + integrate(2*(b^3*f^2*x^2 + 2*b^3*e*f*x - (a*b^2*f^2*x^2
*e^c + 2*a*b^2*e*f*x*e^c)*e^(d*x))/(a^3*b + a*b^3 - (a^3*b*e^(2*c) + a*b^3*e^(2*c))*e^(2*d*x) - 2*(a^4*e^c + a
^2*b^2*e^c)*e^(d*x)), x) - integrate(-2*(a*f^2*x^2 + 2*a*e*f*x - (b*f^2*x^2*e^c + 2*b*e*f*x*e^c)*e^(d*x))/(a^2
+ b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)

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Fricas [C]  time = 3.05557, size = 3794, normalized size = 5.17 \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)^2*csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

(2*b^2*f^2*polylog(3, (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) + 2*b^2*f^2*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqr
t((a^2 + b^2)/b^2))/b) - 2*(a^2 + b^2)*f^2*polylog(3, cosh(d*x + c) + sinh(d*x + c)) - 2*(a^2 + b^2)*f^2*polyl
og(3, -cosh(d*x + c) - sinh(d*x + c)) - 2*(b^2*d*f^2*x + b^2*d*e*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) - 2*(b^2*d*f^2*x + b^2*d*e*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) + 2*((
a^2 + b^2)*d*f^2*x + (a^2 + b^2)*d*e*f)*dilog(cosh(d*x + c) + sinh(d*x + c)) - (2*a^2*d*f^2*x + 2*I*a*b*d*f^2*
x + 2*a^2*d*e*f + 2*I*a*b*d*e*f)*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) - (2*a^2*d*f^2*x - 2*I*a*b*d*f^2*x +
2*a^2*d*e*f - 2*I*a*b*d*e*f)*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + 2*((a^2 + b^2)*d*f^2*x + (a^2 + b^2)
*d*e*f)*dilog(-cosh(d*x + c) - sinh(d*x + c)) - (b^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^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^2*d^2*e^2 - 2*b^2*c*d*e*f + b^2*c^2*f^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^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x +
2*b^2*c*d*e*f - b^2*c^2*f^2)*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sq
rt((a^2 + b^2)/b^2) - b)/b) - (b^2*d^2*f^2*x^2 + 2*b^2*d^2*e*f*x + 2*b^2*c*d*e*f - b^2*c^2*f^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^2)*d
^2*f^2*x^2 + 2*(a^2 + b^2)*d^2*e*f*x + (a^2 + b^2)*d^2*e^2)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (a^2*d^2*
e^2 + I*a*b*d^2*e^2 - 2*a^2*c*d*e*f - 2*I*a*b*c*d*e*f + a^2*c^2*f^2 + I*a*b*c^2*f^2)*log(cosh(d*x + c) + sinh(
d*x + c) + I) - (a^2*d^2*e^2 - I*a*b*d^2*e^2 - 2*a^2*c*d*e*f + 2*I*a*b*c*d*e*f + a^2*c^2*f^2 - I*a*b*c^2*f^2)*
log(cosh(d*x + c) + sinh(d*x + c) - I) + ((a^2 + b^2)*d^2*e^2 - 2*(a^2 + b^2)*c*d*e*f + (a^2 + b^2)*c^2*f^2)*l
og(cosh(d*x + c) + sinh(d*x + c) - 1) - (a^2*d^2*f^2*x^2 - I*a*b*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x - 2*I*a*b*d^2*e
*f*x + 2*a^2*c*d*e*f - 2*I*a*b*c*d*e*f - a^2*c^2*f^2 + I*a*b*c^2*f^2)*log(I*cosh(d*x + c) + I*sinh(d*x + c) +
1) - (a^2*d^2*f^2*x^2 + I*a*b*d^2*f^2*x^2 + 2*a^2*d^2*e*f*x + 2*I*a*b*d^2*e*f*x + 2*a^2*c*d*e*f + 2*I*a*b*c*d*
e*f - a^2*c^2*f^2 - I*a*b*c^2*f^2)*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) + ((a^2 + b^2)*d^2*f^2*x^2 + 2*
(a^2 + b^2)*d^2*e*f*x + 2*(a^2 + b^2)*c*d*e*f - (a^2 + b^2)*c^2*f^2)*log(-cosh(d*x + c) - sinh(d*x + c) + 1) +
(2*a^2*f^2 + 2*I*a*b*f^2)*polylog(3, I*cosh(d*x + c) + I*sinh(d*x + c)) + (2*a^2*f^2 - 2*I*a*b*f^2)*polylog(3
, -I*cosh(d*x + c) - I*sinh(d*x + c)))/((a^3 + a*b^2)*d^3)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{2} \operatorname{csch}\left (d x + c\right ) \operatorname{sech}\left (d x + c\right )}{b \sinh \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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