3.445 \(\int \frac{(e+f x)^2 \text{csch}(c+d x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=1185 \[ \text{result too large to display} \]

[Out]

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

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Rubi [A]  time = 2.2249, antiderivative size = 1185, normalized size of antiderivative = 1., number of steps used = 57, number of rules used = 23, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.676, Rules used = {5589, 2620, 14, 5462, 6741, 12, 6742, 2551, 4182, 2531, 2282, 6589, 3720, 3475, 5573, 5561, 2190, 4180, 3718, 4186, 3770, 5451, 4184} \[ -\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a-\sqrt{a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac{(e+f x)^2 \log \left (\frac{e^{c+d x} b}{a+\sqrt{a^2+b^2}}+1\right ) b^4}{a \left (a^2+b^2\right )^2 d}+\frac{(e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}-\frac{2 f (e+f x) \text{PolyLog}\left (2,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac{f (e+f x) \text{PolyLog}\left (2,-e^{2 (c+d x)}\right ) b^4}{a \left (a^2+b^2\right )^2 d^2}+\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a-\sqrt{a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}+\frac{2 f^2 \text{PolyLog}\left (3,-\frac{b e^{c+d x}}{a+\sqrt{a^2+b^2}}\right ) b^4}{a \left (a^2+b^2\right )^2 d^3}-\frac{f^2 \text{PolyLog}\left (3,-e^{2 (c+d x)}\right ) b^4}{2 a \left (a^2+b^2\right )^2 d^3}-\frac{2 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d}+\frac{2 i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac{2 i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^2}-\frac{2 i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}+\frac{2 i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right ) b^3}{\left (a^2+b^2\right )^2 d^3}-\frac{(e+f x)^2 \text{sech}^2(c+d x) b^2}{2 a \left (a^2+b^2\right ) d}-\frac{f^2 \log (\cosh (c+d x)) b^2}{a \left (a^2+b^2\right ) d^3}+\frac{f (e+f x) \tanh (c+d x) b^2}{a \left (a^2+b^2\right ) d^2}-\frac{(e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d}+\frac{f^2 \tan ^{-1}(\sinh (c+d x)) b}{\left (a^2+b^2\right ) d^3}+\frac{i f (e+f x) \text{PolyLog}\left (2,-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac{i f (e+f x) \text{PolyLog}\left (2,i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^2}-\frac{i f^2 \text{PolyLog}\left (3,-i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}+\frac{i f^2 \text{PolyLog}\left (3,i e^{c+d x}\right ) b}{\left (a^2+b^2\right ) d^3}-\frac{f (e+f x) \text{sech}(c+d x) b}{\left (a^2+b^2\right ) d^2}-\frac{(e+f x)^2 \text{sech}(c+d x) \tanh (c+d x) b}{2 \left (a^2+b^2\right ) d}+\frac{f^2 x^2}{2 a d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac{e f x}{a d}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}+\frac{f^2 \log (\cosh (c+d x))}{a d^3}-\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{f (e+f x) \tanh (c+d x)}{a d^2} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

(e*f*x)/(a*d) + (f^2*x^2)/(2*a*d) - (2*b^3*(e + f*x)^2*ArcTan[E^(c + d*x)])/((a^2 + b^2)^2*d) - (b*(e + f*x)^2
*ArcTan[E^(c + d*x)])/((a^2 + b^2)*d) + (b*f^2*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d^3) - (2*(e + f*x)^2*ArcTa
nh[E^(2*c + 2*d*x)])/(a*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2
*d) - (b^4*(e + f*x)^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(a*(a^2 + b^2)^2*d) + (b^4*(e + f*x)^2*
Log[1 + E^(2*(c + d*x))])/(a*(a^2 + b^2)^2*d) + (f^2*Log[Cosh[c + d*x]])/(a*d^3) - (b^2*f^2*Log[Cosh[c + d*x]]
)/(a*(a^2 + b^2)*d^3) + ((2*I)*b^3*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/((a^2 + b^2)^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^3*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^
2 + b^2)^2*d^2) - (I*b*f*(e + f*x)*PolyLog[2, I*E^(c + d*x)])/((a^2 + b^2)*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2
, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) - (2*b^4*f*(e + f*x)*PolyLog[2, -((b*E^(c +
d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^2) + (b^4*f*(e + f*x)*PolyLog[2, -E^(2*(c + d*x))])/(a*(a^2
+ b^2)^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^3*f^2*PolyLog[3, (-I)*E^(c + d*x)])/((a^2 + b^2)^2*d^3) - (I*b*f^2*PolyLog[3, (-I)*E^(c +
d*x)])/((a^2 + b^2)*d^3) + ((2*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)])/((a^2 + b^2)^2*d^3) + (I*b*f^2*PolyLog[3
, I*E^(c + d*x)])/((a^2 + b^2)*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(a*(a^2
+ b^2)^2*d^3) + (2*b^4*f^2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(a*(a^2 + b^2)^2*d^3) - (b^4
*f^2*PolyLog[3, -E^(2*(c + d*x))])/(2*a*(a^2 + b^2)^2*d^3) + (f^2*PolyLog[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) - (b*f*(e + f*x)*Sech[c + d*x])/((a^2 + b^2)*d^2) - (b^2*(e + f*x)^2
*Sech[c + d*x]^2)/(2*a*(a^2 + b^2)*d) - (f*(e + f*x)*Tanh[c + d*x])/(a*d^2) + (b^2*f*(e + f*x)*Tanh[c + d*x])/
(a*(a^2 + b^2)*d^2) - (b*(e + f*x)^2*Sech[c + d*x]*Tanh[c + d*x])/(2*(a^2 + b^2)*d) - ((e + f*x)^2*Tanh[c + d*
x]^2)/(2*a*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 2620

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 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 6741

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

Rule 12

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

Rule 6742

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

Rule 2551

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

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 3720

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

Rule 4186

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*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]

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]

Rubi steps

\begin{align*} \int \frac{(e+f x)^2 \text{csch}(c+d x) \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac{\int (e+f x)^2 \text{csch}(c+d x) \text{sech}^3(c+d x) \, dx}{a}-\frac{b \int \frac{(e+f x)^2 \text{sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx}{a}\\ &=\frac{(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac{b \int (e+f x)^2 \text{sech}^3(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )}-\frac{b^3 \int \frac{(e+f x)^2 \text{sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )}-\frac{(2 f) \int (e+f x) \left (\frac{\log (\tanh (c+d x))}{d}-\frac{\tanh ^2(c+d x)}{2 d}\right ) \, dx}{a}\\ &=\frac{(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac{b^3 \int (e+f x)^2 \text{sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{a \left (a^2+b^2\right )^2}-\frac{b^5 \int \frac{(e+f x)^2 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{a \left (a^2+b^2\right )^2}-\frac{b \int \left (a (e+f x)^2 \text{sech}^3(c+d x)-b (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x)\right ) \, dx}{a \left (a^2+b^2\right )}-\frac{(2 f) \int \frac{(e+f x) \left (2 \log (\tanh (c+d x))-\tanh ^2(c+d x)\right )}{2 d} \, dx}{a}\\ &=\frac{b^4 (e+f x)^3}{3 a \left (a^2+b^2\right )^2 f}+\frac{(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac{b^3 \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 )^2}-\frac{b^5 \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 )^2}-\frac{b^5 \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 )^2}-\frac{b \int (e+f x)^2 \text{sech}^3(c+d x) \, dx}{a^2+b^2}+\frac{b^2 \int (e+f x)^2 \text{sech}^2(c+d x) \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )}-\frac{f \int (e+f x) \left (2 \log (\tanh (c+d x))-\tanh ^2(c+d x)\right ) \, dx}{a d}\\ &=\frac{b^4 (e+f x)^3}{3 a \left (a^2+b^2\right )^2 f}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\frac{b^3 \int (e+f x)^2 \text{sech}(c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac{b^4 \int (e+f x)^2 \tanh (c+d x) \, dx}{a \left (a^2+b^2\right )^2}-\frac{b \int (e+f x)^2 \text{sech}(c+d x) \, dx}{2 \left (a^2+b^2\right )}-\frac{f \int \left (2 (e+f x) \log (\tanh (c+d x))-(e+f x) \tanh ^2(c+d x)\right ) \, dx}{a d}+\frac{\left (2 b^4 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 )^2 d}+\frac{\left (2 b^4 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 )^2 d}+\frac{\left (b^2 f\right ) \int (e+f x) \text{sech}^2(c+d x) \, dx}{a \left (a^2+b^2\right ) d}+\frac{\left (b f^2\right ) \int \text{sech}(c+d x) \, dx}{\left (a^2+b^2\right ) d^2}\\ &=-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{(e+f x)^2 \log (\tanh (c+d x))}{a d}-\frac{2 b^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac{\left (2 b^4\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 )^2}+\frac{f \int (e+f x) \tanh ^2(c+d x) \, dx}{a d}-\frac{(2 f) \int (e+f x) \log (\tanh (c+d x)) \, dx}{a d}+\frac{\left (2 i b^3 f\right ) \int (e+f x) \log \left (1-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}-\frac{\left (2 i b^3 f\right ) \int (e+f x) \log \left (1+i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d}+\frac{(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{(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{\left (2 b^4 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 )^2 d^2}+\frac{\left (2 b^4 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 )^2 d^2}-\frac{\left (b^2 f^2\right ) \int \tanh (c+d x) \, dx}{a \left (a^2+b^2\right ) d^2}\\ &=-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}-\frac{b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac{2 i b^3 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{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^3 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{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^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \tanh (c+d x)}{a d^2}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac{\int 2 d (e+f x)^2 \text{csch}(2 c+2 d x) \, dx}{a d}+\frac{f \int (e+f x) \, dx}{a d}-\frac{\left (2 b^4 f\right ) \int (e+f x) \log \left (1+e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d}+\frac{\left (2 b^4 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 )^2 d^3}+\frac{\left (2 b^4 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 )^2 d^3}+\frac{f^2 \int \tanh (c+d x) \, dx}{a d^2}-\frac{\left (2 i b^3 f^2\right ) \int \text{Li}_2\left (-i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}+\frac{\left (2 i b^3 f^2\right ) \int \text{Li}_2\left (i e^{c+d x}\right ) \, dx}{\left (a^2+b^2\right )^2 d^2}-\frac{\left (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 (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{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{f^2 \log (\cosh (c+d x))}{a d^3}-\frac{b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac{2 i b^3 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{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^3 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{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^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}+\frac{b^4 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}+\frac{2 b^4 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 )^2 d^3}+\frac{2 b^4 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 )^2 d^3}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \tanh (c+d x)}{a d^2}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}+\frac{2 \int (e+f x)^2 \text{csch}(2 c+2 d x) \, dx}{a}-\frac{\left (2 i b^3 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 )^2 d^3}+\frac{\left (2 i b^3 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 )^2 d^3}-\frac{\left (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 (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^4 f^2\right ) \int \text{Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{a \left (a^2+b^2\right )^2 d^2}\\ &=\frac{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{f^2 \log (\cosh (c+d x))}{a d^3}-\frac{b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac{2 i b^3 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{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^3 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{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^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}+\frac{b^4 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d^2}-\frac{2 i b^3 f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{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^3 f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{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^4 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 )^2 d^3}+\frac{2 b^4 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 )^2 d^3}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \tanh (c+d x)}{a d^2}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}-\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{\left (b^4 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 )^2 d^3}\\ &=\frac{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{f^2 \log (\cosh (c+d x))}{a d^3}-\frac{b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac{2 i b^3 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{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^3 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{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^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}+\frac{b^4 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 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^3 f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{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^3 f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{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^4 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 )^2 d^3}+\frac{2 b^4 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 )^2 d^3}-\frac{b^4 f^2 \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \tanh (c+d x)}{a d^2}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a 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{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{f^2 \log (\cosh (c+d x))}{a d^3}-\frac{b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac{2 i b^3 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{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^3 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{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^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}+\frac{b^4 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 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^3 f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{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^3 f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{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^4 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 )^2 d^3}+\frac{2 b^4 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 )^2 d^3}-\frac{b^4 f^2 \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 d^3}-\frac{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \tanh (c+d x)}{a d^2}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a 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{e f x}{a d}+\frac{f^2 x^2}{2 a d}-\frac{2 b^3 (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d}-\frac{b (e+f x)^2 \tan ^{-1}\left (e^{c+d x}\right )}{\left (a^2+b^2\right ) d}+\frac{b f^2 \tan ^{-1}(\sinh (c+d x))}{\left (a^2+b^2\right ) d^3}-\frac{2 (e+f x)^2 \tanh ^{-1}\left (e^{2 c+2 d x}\right )}{a d}-\frac{b^4 (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 )^2 d}-\frac{b^4 (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 )^2 d}+\frac{b^4 (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 d}+\frac{f^2 \log (\cosh (c+d x))}{a d^3}-\frac{b^2 f^2 \log (\cosh (c+d x))}{a \left (a^2+b^2\right ) d^3}+\frac{2 i b^3 f (e+f x) \text{Li}_2\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}+\frac{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^3 f (e+f x) \text{Li}_2\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^2}-\frac{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^4 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 )^2 d^2}-\frac{2 b^4 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 )^2 d^2}+\frac{b^4 f (e+f x) \text{Li}_2\left (-e^{2 (c+d x)}\right )}{a \left (a^2+b^2\right )^2 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^3 f^2 \text{Li}_3\left (-i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}-\frac{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^3 f^2 \text{Li}_3\left (i e^{c+d x}\right )}{\left (a^2+b^2\right )^2 d^3}+\frac{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^4 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 )^2 d^3}+\frac{2 b^4 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 )^2 d^3}-\frac{b^4 f^2 \text{Li}_3\left (-e^{2 (c+d x)}\right )}{2 a \left (a^2+b^2\right )^2 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{b f (e+f x) \text{sech}(c+d x)}{\left (a^2+b^2\right ) d^2}-\frac{b^2 (e+f x)^2 \text{sech}^2(c+d x)}{2 a \left (a^2+b^2\right ) d}-\frac{f (e+f x) \tanh (c+d x)}{a d^2}+\frac{b^2 f (e+f x) \tanh (c+d x)}{a \left (a^2+b^2\right ) d^2}-\frac{b (e+f x)^2 \text{sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right ) d}-\frac{(e+f x)^2 \tanh ^2(c+d x)}{2 a d}\\ \end{align*}

Mathematica [B]  time = 36.4513, size = 4072, normalized size = 3.44 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((e + f*x)^2*Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-(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*Log[1 + E^(-c - d*x)])/d + (6*(-1 + E^(2*c))*f*(d*(e + f*x)*PolyLog[2, -E^(-c - d*x)] + f*Po
lyLog[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*PolyLo
g[3, E^(-c - d*x)]))/(d^3*E^(2*c))))/(3*a*(-1 + E^(2*c))) - (-12*a^3*d^3*e^2*E^(2*c)*x - 24*a*b^2*d^3*e^2*E^(2
*c)*x + 12*a^3*d*E^(2*c)*f^2*x + 12*a*b^2*d*E^(2*c)*f^2*x - 12*a^3*d^3*e*E^(2*c)*f*x^2 - 24*a*b^2*d^3*e*E^(2*c
)*f*x^2 - 4*a^3*d^3*E^(2*c)*f^2*x^3 - 8*a*b^2*d^3*E^(2*c)*f^2*x^3 + 6*a^2*b*d^2*e^2*ArcTan[E^(c + d*x)] + 18*b
^3*d^2*e^2*ArcTan[E^(c + d*x)] + 6*a^2*b*d^2*e^2*E^(2*c)*ArcTan[E^(c + d*x)] + 18*b^3*d^2*e^2*E^(2*c)*ArcTan[E
^(c + d*x)] - 12*a^2*b*f^2*ArcTan[E^(c + d*x)] - 12*b^3*f^2*ArcTan[E^(c + d*x)] - 12*a^2*b*E^(2*c)*f^2*ArcTan[
E^(c + d*x)] - 12*b^3*E^(2*c)*f^2*ArcTan[E^(c + d*x)] + (6*I)*a^2*b*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*
b^3*d^2*e*f*x*Log[1 - I*E^(c + d*x)] + (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (18*I)*b^3*d^2*e
*E^(2*c)*f*x*Log[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*f^2*x^2*L
og[1 - I*E^(c + d*x)] + (3*I)*a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 - I*E^(c + d*x)] + (9*I)*b^3*d^2*E^(2*c)*f^2*x^2
*Log[1 - I*E^(c + d*x)] - (6*I)*a^2*b*d^2*e*f*x*Log[1 + I*E^(c + d*x)] - (18*I)*b^3*d^2*e*f*x*Log[1 + I*E^(c +
d*x)] - (6*I)*a^2*b*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c + d*x)] - (18*I)*b^3*d^2*e*E^(2*c)*f*x*Log[1 + I*E^(c +
d*x)] - (3*I)*a^2*b*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (9*I)*b^3*d^2*f^2*x^2*Log[1 + I*E^(c + d*x)] - (3*I)*
a^2*b*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] - (9*I)*b^3*d^2*E^(2*c)*f^2*x^2*Log[1 + I*E^(c + d*x)] + 6*a^
3*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*e^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*e^2*E^(2*c)*Log[1 +
E^(2*(c + d*x))] + 12*a*b^2*d^2*e^2*E^(2*c)*Log[1 + E^(2*(c + d*x))] - 6*a^3*f^2*Log[1 + E^(2*(c + d*x))] - 6
*a*b^2*f^2*Log[1 + E^(2*(c + d*x))] - 6*a^3*E^(2*c)*f^2*Log[1 + E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*Log[1 +
E^(2*(c + d*x))] + 12*a^3*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] + 24*a*b^2*d^2*e*f*x*Log[1 + E^(2*(c + d*x))] +
12*a^3*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 24*a*b^2*d^2*e*E^(2*c)*f*x*Log[1 + E^(2*(c + d*x))] + 6*a^
3*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*f^2*x^2*Log[1 + E^(2*(c + d*x))] + 6*a^3*d^2*E^(2*c)*f^2
*x^2*Log[1 + E^(2*(c + d*x))] + 12*a*b^2*d^2*E^(2*c)*f^2*x^2*Log[1 + E^(2*(c + d*x))] - (6*I)*b*(a^2 + 3*b^2)*
d*(1 + E^(2*c))*f*(e + f*x)*PolyLog[2, (-I)*E^(c + d*x)] + (6*I)*b*(a^2 + 3*b^2)*d*(1 + E^(2*c))*f*(e + f*x)*P
olyLog[2, I*E^(c + d*x)] + 6*a^3*d*e*f*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*e*f*PolyLog[2, -E^(2*(c + d*x
))] + 6*a^3*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*e*E^(2*c)*f*PolyLog[2, -E^(2*(c + d*x))] +
6*a^3*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 6*a^3*d*E^(2*c)*
f^2*x*PolyLog[2, -E^(2*(c + d*x))] + 12*a*b^2*d*E^(2*c)*f^2*x*PolyLog[2, -E^(2*(c + d*x))] + (6*I)*a^2*b*f^2*P
olyLog[3, (-I)*E^(c + d*x)] + (18*I)*b^3*f^2*PolyLog[3, (-I)*E^(c + d*x)] + (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[3,
(-I)*E^(c + d*x)] + (18*I)*b^3*E^(2*c)*f^2*PolyLog[3, (-I)*E^(c + d*x)] - (6*I)*a^2*b*f^2*PolyLog[3, I*E^(c +
d*x)] - (18*I)*b^3*f^2*PolyLog[3, I*E^(c + d*x)] - (6*I)*a^2*b*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - (18*I)
*b^3*E^(2*c)*f^2*PolyLog[3, I*E^(c + d*x)] - 3*a^3*f^2*PolyLog[3, -E^(2*(c + d*x))] - 6*a*b^2*f^2*PolyLog[3, -
E^(2*(c + d*x))] - 3*a^3*E^(2*c)*f^2*PolyLog[3, -E^(2*(c + d*x))] - 6*a*b^2*E^(2*c)*f^2*PolyLog[3, -E^(2*(c +
d*x))])/(6*(a^2 + b^2)^2*d^3*(1 + E^(2*c))) + (b^4*(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 - Sqrt[(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))/
(3*a*(a^2 + b^2)^2*(-1 + E^(2*c))) + (Csch[c]*Sech[c]*Sech[c + d*x]^2*(-6*a^3*e*f - 6*a*b^2*e*f + 12*a^3*d^2*e
^2*x + 24*a*b^2*d^2*e^2*x - 6*a^3*f^2*x - 6*a*b^2*f^2*x + 12*a^3*d^2*e*f*x^2 + 24*a*b^2*d^2*e*f*x^2 + 4*a^3*d^
2*f^2*x^3 + 8*a*b^2*d^2*f^2*x^3 + 6*a^3*e*f*Cosh[2*c] + 6*a*b^2*e*f*Cosh[2*c] + 6*a^3*f^2*x*Cosh[2*c] + 6*a*b^
2*f^2*x*Cosh[2*c] + 6*a^3*e*f*Cosh[2*d*x] + 6*a*b^2*e*f*Cosh[2*d*x] + 6*a^3*f^2*x*Cosh[2*d*x] + 6*a*b^2*f^2*x*
Cosh[2*d*x] + 3*a^2*b*d*e^2*Cosh[c - d*x] + 3*b^3*d*e^2*Cosh[c - d*x] + 6*a^2*b*d*e*f*x*Cosh[c - d*x] + 6*b^3*
d*e*f*x*Cosh[c - d*x] + 3*a^2*b*d*f^2*x^2*Cosh[c - d*x] + 3*b^3*d*f^2*x^2*Cosh[c - d*x] - 3*a^2*b*d*e^2*Cosh[3
*c + d*x] - 3*b^3*d*e^2*Cosh[3*c + d*x] - 6*a^2*b*d*e*f*x*Cosh[3*c + d*x] - 6*b^3*d*e*f*x*Cosh[3*c + d*x] - 3*
a^2*b*d*f^2*x^2*Cosh[3*c + d*x] - 3*b^3*d*f^2*x^2*Cosh[3*c + d*x] - 6*a^3*e*f*Cosh[2*c + 2*d*x] - 6*a*b^2*e*f*
Cosh[2*c + 2*d*x] + 12*a^3*d^2*e^2*x*Cosh[2*c + 2*d*x] + 24*a*b^2*d^2*e^2*x*Cosh[2*c + 2*d*x] - 6*a^3*f^2*x*Co
sh[2*c + 2*d*x] - 6*a*b^2*f^2*x*Cosh[2*c + 2*d*x] + 12*a^3*d^2*e*f*x^2*Cosh[2*c + 2*d*x] + 24*a*b^2*d^2*e*f*x^
2*Cosh[2*c + 2*d*x] + 4*a^3*d^2*f^2*x^3*Cosh[2*c + 2*d*x] + 8*a*b^2*d^2*f^2*x^3*Cosh[2*c + 2*d*x] + 6*a^3*d*e^
2*Sinh[2*c] + 6*a*b^2*d*e^2*Sinh[2*c] + 12*a^3*d*e*f*x*Sinh[2*c] + 12*a*b^2*d*e*f*x*Sinh[2*c] + 6*a^3*d*f^2*x^
2*Sinh[2*c] + 6*a*b^2*d*f^2*x^2*Sinh[2*c] - 6*a^2*b*e*f*Sinh[c - d*x] - 6*b^3*e*f*Sinh[c - d*x] - 6*a^2*b*f^2*
x*Sinh[c - d*x] - 6*b^3*f^2*x*Sinh[c - d*x] - 6*a^2*b*e*f*Sinh[3*c + d*x] - 6*b^3*e*f*Sinh[3*c + d*x] - 6*a^2*
b*f^2*x*Sinh[3*c + d*x] - 6*b^3*f^2*x*Sinh[3*c + d*x]))/(24*(a^2 + b^2)^2*d^2)

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Maple [F]  time = 1.372, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{2}{\rm csch} \left (dx+c\right ) \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{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)^3/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \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)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-a^2*b*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^
(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 3*b^3*d^2*f^2*integrate(x^2*e^(d*x + c)/(a^4*d^2*e^(2
*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x)
+ 2*a^3*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2
*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*a*b^2*d^2*f^2*integrate(x^2/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*
b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 2*a^2*b*d^2*e*f*i
ntegrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^
4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - 6*b^3*d^2*e*f*integrate(x*e^(d*x + c)/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*
b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) + 4*a^3*d^2*e*f*int
egrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c) + b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*
b^2*d^2 + b^4*d^2), x) + 8*a*b^2*d^2*e*f*integrate(x/(a^4*d^2*e^(2*d*x + 2*c) + 2*a^2*b^2*d^2*e^(2*d*x + 2*c)
+ b^4*d^2*e^(2*d*x + 2*c) + a^4*d^2 + 2*a^2*b^2*d^2 + b^4*d^2), x) - a^3*f^2*(2*(d*x + c)/((a^4 + 2*a^2*b^2 +
b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - a*b^2*f^2*(2*(d*x + c)/((a^4 + 2*a^2*b^2
+ b^4)*d^3) - log(e^(2*d*x + 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d^3)) - (b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*
d*x - 2*c) - b)/((a^5 + 2*a^3*b^2 + a*b^4)*d) - (a^2*b + 3*b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*
d) + (a^3 + 2*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) + (b*e^(-d*x - c) - 2*a*e^(-2*d*x -
2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d) -
log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d))*e^2 + 2*a^2*b*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2
*b^2 + b^4)*d^3) + 2*b^3*f^2*arctan(e^(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d^3) + (2*a*f^2*x + 2*a*e*f - (b*d*f
^2*x^2*e^(3*c) + 2*b*e*f*e^(3*c) + 2*(d*e*f + f^2)*b*x*e^(3*c))*e^(3*d*x) + 2*(a*d*f^2*x^2*e^(2*c) + a*e*f*e^(
2*c) + (2*d*e*f + f^2)*a*x*e^(2*c))*e^(2*d*x) + (b*d*f^2*x^2*e^c - 2*b*e*f*e^c + 2*(d*e*f - f^2)*b*x*e^c)*e^(d
*x))/(a^2*d^2 + b^2*d^2 + (a^2*d^2*e^(4*c) + b^2*d^2*e^(4*c))*e^(4*d*x) + 2*(a^2*d^2*e^(2*c) + b^2*d^2*e^(2*c)
)*e^(2*d*x)) + 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^5*f^2*x^2 + 2*b^5*e*f*x - (a*b^4*
f^2*x^2*e^c + 2*a*b^4*e*f*x*e^c)*e^(d*x))/(a^5*b + 2*a^3*b^3 + a*b^5 - (a^5*b*e^(2*c) + 2*a^3*b^3*e^(2*c) + a*
b^5*e^(2*c))*e^(2*d*x) - 2*(a^6*e^c + 2*a^4*b^2*e^c + a^2*b^4*e^c)*e^(d*x)), x)

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Fricas [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)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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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)**3/(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)^2*csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out