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

Optimal. Leaf size=1284 \[ -\frac {2 b^2 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^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 {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^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 {Li}_3\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}-\frac {2 b^3 (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^3 e n}-\frac {4 b (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^2} d^3 e n}+\frac {2 b^3 (e x)^{3 n} \text {Li}_3\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^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 {b^2-a^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 {b^2-a^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 {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}+\frac {2 b^3 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^2\right )^{3/2} d^2 e n}+\frac {4 b (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^2} d^2 e n}-\frac {2 b^3 (e x)^{3 n} \text {Li}_2\left (-\frac {a e^{d x^n+c}}{b+\sqrt {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^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 {b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}+\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^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 {b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}-\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}+\frac {b^2 (e x)^{3 n} \sinh \left (d x^n+c\right ) x^{-n}}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (d x^n+c\right )\right )}+\frac {(e x)^{3 n}}{3 a^2 e n} \]

[Out]

1/3*(e*x)^(3*n)/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)*ln(1+a*exp(c+d*x^n)/(b-(-a
^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n/(x^(2*n))+b^3*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/
(-a^2+b^2)^(3/2)/d/e/n/(x^n)-2*b^2*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^2/e/n
/(x^(2*n))-b^3*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d/e/n/(x^n)-2*b^2*(e
*x)^(3*n)*polylog(2,-a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))+2*b^3*(e*x)^(3*n)*po
lylog(2,-a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-2*b^2*(e*x)^(3*n)*polylog
(2,-a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(a^2-b^2)/d^3/e/n/(x^(3*n))-2*b^3*(e*x)^(3*n)*polylog(2,-a*exp(c+
d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-2*b^3*(e*x)^(3*n)*polylog(3,-a*exp(c+d*x^n
)/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))+2*b^3*(e*x)^(3*n)*polylog(3,-a*exp(c+d*x^n)/(b+
(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/d^3/e/n/(x^(3*n))+b^2*(e*x)^(3*n)*sinh(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n
)/(b+a*cosh(c+d*x^n))-2*b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/
2)+2*b*(e*x)^(3*n)*ln(1+a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/d/e/n/(x^n)/(-a^2+b^2)^(1/2)-4*b*(e*x)^(3*n)*
polylog(2,-a*exp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)+4*b*(e*x)^(3*n)*polylog
(2,-a*exp(c+d*x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/d^2/e/n/(x^(2*n))/(-a^2+b^2)^(1/2)+4*b*(e*x)^(3*n)*polylog(3,-a*e
xp(c+d*x^n)/(b-(-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)-4*b*(e*x)^(3*n)*polylog(3,-a*exp(c+d*
x^n)/(b+(-a^2+b^2)^(1/2)))/a^2/d^3/e/n/(x^(3*n))/(-a^2+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 2.26, antiderivative size = 1284, normalized size of antiderivative = 1.00, 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 = {5440, 5436, 4191, 3324, 3320, 2264, 2190, 2531, 2282, 6589, 5562, 2279, 2391} \[ -\frac {2 b^2 (e x)^{3 n} \text {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {b^2-a^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 {b^2-a^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 {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^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 {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^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 {b^2-a^2}}\right ) x^{-3 n}}{a^2 \sqrt {b^2-a^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 {b^2-a^2}}\right ) x^{-3 n}}{a^2 \left (b^2-a^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 {b^2-a^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 {b^2-a^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 {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^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 {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^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 {b^2-a^2}}\right ) x^{-2 n}}{a^2 \sqrt {b^2-a^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 {b^2-a^2}}\right ) x^{-2 n}}{a^2 \left (b^2-a^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 {b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}+\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^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 {b^2-a^2}}+1\right ) x^{-n}}{a^2 \sqrt {b^2-a^2} d e n}-\frac {b^3 (e x)^{3 n} \log \left (\frac {e^{d x^n+c} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{-n}}{a^2 \left (b^2-a^2\right )^{3/2} d e n}+\frac {b^2 (e x)^{3 n} \sinh \left (d x^n+c\right ) x^{-n}}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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*Sech[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 + Sq
rt[-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 - 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)) + (2*b^3*(e*x)^(3*n)*Pol
yLog[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)*Sinh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(b + a*Cosh[c + d*x^n]))

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 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 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 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 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 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 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 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 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 5436

Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[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 5440

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

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), 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 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]

Rubi steps

\begin {align*} \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\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 {sech}\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 {sech}(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 \cosh (c+d x))^2}-\frac {2 b x^2}{a^2 (b+a \cosh (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 \cosh (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 \cosh (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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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 \cosh (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 \sinh (c+d x)}{b+a \cosh (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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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 ) \sqrt {-a^2+b^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 ) \sqrt {-a^2+b^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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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 ) \sqrt {-a^2+b^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 ) \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 {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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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 ) \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 \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 ) \sqrt {-a^2+b^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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \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 ) \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 \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 ) \sqrt {-a^2+b^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} \sinh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (b+a \cosh \left (c+d x^n\right )\right )}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 117.91, size = 0, normalized size = 0.00 \[ \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {sech}\left (c+d x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 \, n - 1}}{{\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 1.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x \right )^{3 n -1}}{\left (a +b \,\mathrm {sech}\left (c +d \,x^{n}\right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {6 \, a b^{2} e^{3 \, n} x^{2 \, n} - {\left (a^{3} d e^{3 \, n} - a b^{2} d e^{3 \, n}\right )} x^{3 \, n} - {\left (a^{3} d e^{3 \, n} e^{\left (2 \, c\right )} - a b^{2} d e^{3 \, n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + 3 \, n \log \relax (x)\right )} + 2 \, {\left (3 \, b^{3} e^{3 \, n} e^{\left (2 \, n \log \relax (x) + c\right )} - {\left (a^{2} b d e^{3 \, n} e^{c} - b^{3} d e^{3 \, n} e^{c}\right )} x^{3 \, n}\right )} e^{\left (d x^{n}\right )}}{3 \, {\left (a^{5} d e n - a^{3} b^{2} d e n + {\left (a^{5} d e n e^{\left (2 \, c\right )} - a^{3} b^{2} d e n e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n}\right )} + 2 \, {\left (a^{4} b d e n e^{c} - a^{2} b^{3} d e n e^{c}\right )} e^{\left (d x^{n}\right )}\right )}} - \int -\frac {2 \, {\left (2 \, a b^{2} e^{3 \, n} x^{2 \, n} + {\left (2 \, b^{3} e^{3 \, n} e^{\left (2 \, n \log \relax (x) + c\right )} - {\left (2 \, a^{2} b d e^{3 \, n} e^{c} - b^{3} d e^{3 \, n} e^{c}\right )} x^{3 \, n}\right )} e^{\left (d x^{n}\right )}\right )}}{{\left (a^{5} d e e^{\left (2 \, c\right )} - a^{3} b^{2} d e e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x^{n}\right )} + 2 \, {\left (a^{4} b d e e^{c} - a^{2} b^{3} d e e^{c}\right )} x e^{\left (d x^{n}\right )} + {\left (a^{5} d e - a^{3} b^{2} d e\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+3*n)/(a+b*sech(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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,x\right )}^{3\,n-1}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x\right )^{3 n - 1}}{\left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________