3.83 \(\int \frac{(e x)^{-1+3 n}}{(a+b \text{csch}(c+d x^n))^2} \, dx\)

Optimal. Leaf size=1218 \[ \text{result too large to display} \]

[Out]

(e*x)^(3*n)/(3*a^2*e*n) - (b^2*(e*x)^(3*n))/(a^2*(a^2 + b^2)*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c +
d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n)
)/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b -
Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^
2 + b^2])])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2
])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) + (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a
^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^
2*(a^2 + b^2)*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a
^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))
])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b
^2]))])/(a^2*(a^2 + b^2)*d^3*e*n*x^(3*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 +
b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a
^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - S
qrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) + (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(
b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^
n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) - (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c +
d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) - (b^2*(e*x)^(3*n)*Cosh[c + d*x^n])/(a
*(a^2 + b^2)*d*e*n*x^n*(b + a*Sinh[c + d*x^n]))

________________________________________________________________________________________

Rubi [A]  time = 2.12118, antiderivative size = 1218, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {5441, 5437, 4191, 3324, 3322, 2264, 2190, 2531, 2282, 6589, 5561, 2279, 2391} \[ \frac{2 b^2 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{d x^n+c}}{b-\sqrt{a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac{2 b^2 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{d x^n+c}}{b+\sqrt{a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac{4 b (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{d x^n+c}}{b-\sqrt{a^2+b^2}}\right ) x^{-3 n}}{a^2 \sqrt{a^2+b^2} d^3 e n}-\frac{2 b^3 (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{d x^n+c}}{b-\sqrt{a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}-\frac{4 b (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{d x^n+c}}{b+\sqrt{a^2+b^2}}\right ) x^{-3 n}}{a^2 \sqrt{a^2+b^2} d^3 e n}+\frac{2 b^3 (e x)^{3 n} \text{PolyLog}\left (3,-\frac{a e^{d x^n+c}}{b+\sqrt{a^2+b^2}}\right ) x^{-3 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac{2 b^2 (e x)^{3 n} \log \left (\frac{e^{d x^n+c} a}{b-\sqrt{a^2+b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{2 b^2 (e x)^{3 n} \log \left (\frac{e^{d x^n+c} a}{b+\sqrt{a^2+b^2}}+1\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{4 b (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{d x^n+c}}{b-\sqrt{a^2+b^2}}\right ) x^{-2 n}}{a^2 \sqrt{a^2+b^2} d^2 e n}+\frac{2 b^3 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{d x^n+c}}{b-\sqrt{a^2+b^2}}\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{d x^n+c}}{b+\sqrt{a^2+b^2}}\right ) x^{-2 n}}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{2 b^3 (e x)^{3 n} \text{PolyLog}\left (2,-\frac{a e^{d x^n+c}}{b+\sqrt{a^2+b^2}}\right ) x^{-2 n}}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac{b^2 (e x)^{3 n} x^{-n}}{a^2 \left (a^2+b^2\right ) d e n}-\frac{2 b (e x)^{3 n} \log \left (\frac{e^{d x^n+c} a}{b-\sqrt{a^2+b^2}}+1\right ) x^{-n}}{a^2 \sqrt{a^2+b^2} d e n}+\frac{b^3 (e x)^{3 n} \log \left (\frac{e^{d x^n+c} a}{b-\sqrt{a^2+b^2}}+1\right ) x^{-n}}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac{2 b (e x)^{3 n} \log \left (\frac{e^{d x^n+c} a}{b+\sqrt{a^2+b^2}}+1\right ) x^{-n}}{a^2 \sqrt{a^2+b^2} d e n}-\frac{b^3 (e x)^{3 n} \log \left (\frac{e^{d x^n+c} a}{b+\sqrt{a^2+b^2}}+1\right ) x^{-n}}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac{b^2 (e x)^{3 n} \cosh \left (d x^n+c\right ) x^{-n}}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (d x^n+c\right )\right )}+\frac{(e x)^{3 n}}{3 a^2 e n} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2,x]

[Out]

(e*x)^(3*n)/(3*a^2*e*n) - (b^2*(e*x)^(3*n))/(a^2*(a^2 + b^2)*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c +
d*x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n)
)/(b - Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) - (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b -
Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^
2 + b^2])])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) - (b^3*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2
])])/(a^2*(a^2 + b^2)^(3/2)*d*e*n*x^n) + (2*b*(e*x)^(3*n)*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a
^2*Sqrt[a^2 + b^2]*d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^
2*(a^2 + b^2)*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a
^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) - (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))
])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) + (2*b^2*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b
^2]))])/(a^2*(a^2 + b^2)*d^3*e*n*x^(3*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 +
b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2*e*n*x^(2*n)) + (4*b*(e*x)^(3*n)*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a
^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2*e*n*x^(2*n)) - (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(b - S
qrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) + (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^n))/(
b - Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) + (2*b^3*(e*x)^(3*n)*PolyLog[3, -((a*E^(c + d*x^
n))/(b + Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^3*e*n*x^(3*n)) - (4*b*(e*x)^(3*n)*PolyLog[3, -((a*E^(c +
d*x^n))/(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^3*e*n*x^(3*n)) - (b^2*(e*x)^(3*n)*Cosh[c + d*x^n])/(a
*(a^2 + b^2)*d*e*n*x^n*(b + a*Sinh[c + d*x^n]))

Rule 5441

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Dist[(e^IntPart[m]*(e*
x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Csch[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]

Rule 5437

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Csch[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 4191

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3322

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

Rule 2264

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

Rule 2190

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

Rule 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 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 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]

Rubi steps

\begin{align*} \int \frac{(e x)^{-1+3 n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \int \frac{x^{-1+3 n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{(a+b \text{csch}(c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \left (\frac{x^2}{a^2}+\frac{b^2 x^2}{a^2 (b+a \sinh (c+d x))^2}-\frac{2 b x^2}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{\left (2 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{(b+a \sinh (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 e n}+\frac{\left (b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x^2}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}+\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{x \cosh (c+d x)}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{-a+2 b e^{c+d x}+a e^{2 (c+d x)}} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b-2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt{a^2+b^2} e n}+\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b+2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \sqrt{a^2+b^2} e n}+\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{b-\sqrt{a^2+b^2}+a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}+\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{b+\sqrt{a^2+b^2}+a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b-2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right )^{3/2} e n}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x^2}{2 b+2 \sqrt{a^2+b^2}+2 a e^{c+d x}} \, dx,x,x^n\right )}{a \left (a^2+b^2\right )^{3/2} e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{c+d x}}{b-\sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{a e^{c+d x}}{b+\sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{a^2+b^2} d e n}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{a^2+b^2} d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{b-\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{a x}{b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}+\frac{2 b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}+\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt{a^2+b^2} d^3 e n}-\frac{\left (4 b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt{a^2+b^2} d^3 e n}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b-2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-\frac{2 a e^{c+d x}}{2 b+2 \sqrt{a^2+b^2}}\right ) \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}+\frac{2 b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}+\frac{4 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^3 e n}-\frac{4 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^3 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}-\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{-b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac{\left (2 b^3 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{a x}{b+\sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}\\ &=\frac{(e x)^{3 n}}{3 a^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{a^2 \left (a^2+b^2\right ) d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{b^3 x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac{2 b x^{-n} (e x)^{3 n} \log \left (1+\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d e n}+\frac{2 b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}+\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}+\frac{2 b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right ) d^3 e n}-\frac{2 b^3 x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac{4 b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{2 b^3 x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}+\frac{4 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^3 e n}+\frac{2 b^3 x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^3 e n}-\frac{4 b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-\frac{a e^{c+d x^n}}{b+\sqrt{a^2+b^2}}\right )}{a^2 \sqrt{a^2+b^2} d^3 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )}\\ \end{align*}

Mathematica [F]  time = 118.006, size = 0, normalized size = 0. \[ \int \frac{(e x)^{-1+3 n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2,x]

[Out]

Integrate[(e*x)^(-1 + 3*n)/(a + b*Csch[c + d*x^n])^2, x]

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Maple [F]  time = 0.408, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{-1+3\,n}}{ \left ( a+b{\rm csch} \left (c+d{x}^{n}\right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x)

[Out]

int((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{6 \, a b^{2} e^{3 \, n} x^{2 \, n} +{\left (a^{3} d e^{3 \, n} + a b^{2} d e^{3 \, n}\right )} x^{3 \, n} -{\left (a^{3} d e^{3 \, n} e^{\left (2 \, c\right )} + a b^{2} d e^{3 \, n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + 3 \, n \log \left (x\right )\right )} - 2 \,{\left (3 \, b^{3} e^{3 \, n} e^{\left (2 \, n \log \left (x\right ) + c\right )} +{\left (a^{2} b d e^{3 \, n} e^{c} + b^{3} d e^{3 \, n} e^{c}\right )} x^{3 \, n}\right )} e^{\left (d x^{n}\right )}}{3 \,{\left (a^{5} d e n + a^{3} b^{2} d e n -{\left (a^{5} d e n e^{\left (2 \, c\right )} + a^{3} b^{2} d e n e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n}\right )} - 2 \,{\left (a^{4} b d e n e^{c} + a^{2} b^{3} d e n e^{c}\right )} e^{\left (d x^{n}\right )}\right )}} - \int -\frac{2 \,{\left (2 \, a b^{2} e^{3 \, n} x^{2 \, n} -{\left (2 \, b^{3} e^{3 \, n} e^{\left (2 \, n \log \left (x\right ) + c\right )} +{\left (2 \, a^{2} b d e^{3 \, n} e^{c} + b^{3} d e^{3 \, n} e^{c}\right )} x^{3 \, n}\right )} e^{\left (d x^{n}\right )}\right )}}{{\left (a^{5} d e e^{\left (2 \, c\right )} + a^{3} b^{2} d e e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x^{n}\right )} + 2 \,{\left (a^{4} b d e e^{c} + a^{2} b^{3} d e e^{c}\right )} x e^{\left (d x^{n}\right )} -{\left (a^{5} d e + a^{3} b^{2} d e\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")

[Out]

1/3*(6*a*b^2*e^(3*n)*x^(2*n) + (a^3*d*e^(3*n) + a*b^2*d*e^(3*n))*x^(3*n) - (a^3*d*e^(3*n)*e^(2*c) + a*b^2*d*e^
(3*n)*e^(2*c))*e^(2*d*x^n + 3*n*log(x)) - 2*(3*b^3*e^(3*n)*e^(2*n*log(x) + c) + (a^2*b*d*e^(3*n)*e^c + b^3*d*e
^(3*n)*e^c)*x^(3*n))*e^(d*x^n))/(a^5*d*e*n + a^3*b^2*d*e*n - (a^5*d*e*n*e^(2*c) + a^3*b^2*d*e*n*e^(2*c))*e^(2*
d*x^n) - 2*(a^4*b*d*e*n*e^c + a^2*b^3*d*e*n*e^c)*e^(d*x^n)) - integrate(-2*(2*a*b^2*e^(3*n)*x^(2*n) - (2*b^3*e
^(3*n)*e^(2*n*log(x) + c) + (2*a^2*b*d*e^(3*n)*e^c + b^3*d*e^(3*n)*e^c)*x^(3*n))*e^(d*x^n))/((a^5*d*e*e^(2*c)
+ a^3*b^2*d*e*e^(2*c))*x*e^(2*d*x^n) + 2*(a^4*b*d*e*e^c + a^2*b^3*d*e*e^c)*x*e^(d*x^n) - (a^5*d*e + a^3*b^2*d*
e)*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((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{3 n - 1}}{\left (a + b \operatorname{csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**(-1+3*n)/(a+b*csch(c+d*x**n))**2,x)

[Out]

Integral((e*x)**(3*n - 1)/(a + b*csch(c + d*x**n))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname{csch}\left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="giac")

[Out]

integrate((e*x)^(3*n - 1)/(b*csch(d*x^n + c) + a)^2, x)