3.1.83 \(\int \frac {(e x)^{-1+3 n}}{(a+b \text {csch}(c+d x^n))^2} \, dx\) [83]

Optimal. Leaf size=1218 \[ \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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (2,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 {PolyLog}\left (3,-\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 )} \]

[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)*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^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)*cosh(c+d*x^n)/a/(a^2+b^2)/d/e/n/(x^n)/(b+a*sinh(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*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)-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 = 1.47, antiderivative size = 1218, 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 = {5549, 5545, 4276, 3405, 3403, 2296, 2221, 2611, 2320, 6724, 5680, 2317, 2438} \begin {gather*} \frac {2 b^2 (e x)^{3 n} \text {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_3\left (-\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 {Li}_3\left (-\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 {Li}_3\left (-\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 {Li}_3\left (-\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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_2\left (-\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 {Li}_2\left (-\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} \end {gather*}

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2320

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 2438

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 2611

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 3403

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 3405

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 4276

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 5545

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 5549

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 5680

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 6724

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 {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [F]
time = 94.54, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(e x)^{-1+3 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx \end {gather*}

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 = 1.90, size = 0, normalized size = 0.00 \[\int \frac {\left (e x \right )^{3 n -1}}{\left (a +b \,\mathrm {csch}\left (c +d \,x^{n}\right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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^(2*n*log(x) + 3*n) + (a^3*d*e^(3*n) + a*b^2*d*e^(3*n))*x^(3*n) - (a^3*d*e^(2*c + 3*n) + a*b^2*d
*e^(2*c + 3*n))*e^(2*d*x^n + 3*n*log(x)) - 2*(3*b^3*e^(2*n*log(x) + c + 3*n) + (a^2*b*d*e^(c + 3*n) + b^3*d*e^
(c + 3*n))*x^(3*n))*e^(d*x^n))/((a^5*d*n + a^3*b^2*d*n)*e - (a^5*d*n*e^(2*c) + a^3*b^2*d*n*e^(2*c))*e^(2*d*x^n
+ 1) - 2*(a^4*b*d*n*e^c + a^2*b^3*d*n*e^c)*e^(d*x^n + 1)) - integrate(2*(2*a*b^2*e^(2*n*log(x) + 3*n) - (2*b^
3*e^(2*n*log(x) + c + 3*n) + (2*a^2*b*d*e^(c + 3*n) + b^3*d*e^(c + 3*n))*x^(3*n))*e^(d*x^n))/((a^5*d + a^3*b^2
*d)*x*e - (a^5*d*e^(2*c) + a^3*b^2*d*e^(2*c))*x*e^(2*d*x^n + 1) - 2*(a^4*b*d*e^c + a^2*b^3*d*e^c)*x*e^(d*x^n +
1)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 13355 vs. \(2 (1162) = 2324\).
time = 0.57, size = 13355, normalized size = 10.96 \begin {gather*} \text {Too large to display} \end {gather*}

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]

-1/3*(((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x
))^3 + 6*(a^3*b^2 + a*b^4)*c^2*cosh(3*n - 1) + 3*((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^5 + 2*a^3*b
^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x))^2*sinh(n*log(x)) + 3*((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1
) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x))*sinh(n*log(x))^2 + ((a^5 + 2*a^3*b^2 + a*b^4)*
d^3*cosh(3*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*sinh(n*log(x))^3 + 6*(a^3*b^2 + a*b^4)*c^2*si
nh(3*n - 1) - (((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cos
h(n*log(x))^3 + 6*(a^3*b^2 + a*b^4)*c^2*cosh(3*n - 1) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^5 +
2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*sinh(n*log(x))^3 + 6*(a^3*b^2 + a*b^4)*c^2*sinh(3*n - 1) - 6*((a^3*b^2 +
a*b^4)*d^2*cosh(3*n - 1) + (a^3*b^2 + a*b^4)*d^2*sinh(3*n - 1))*cosh(n*log(x))^2 - 3*(2*(a^3*b^2 + a*b^4)*d^2
*cosh(3*n - 1) + 2*(a^3*b^2 + a*b^4)*d^2*sinh(3*n - 1) - ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^5 +
2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x)))*sinh(n*log(x))^2 + 3*(((a^5 + 2*a^3*b^2 + a*b^4)*d^3*co
sh(3*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x))^2 - 4*((a^3*b^2 + a*b^4)*d^2*cosh(3*
n - 1) + (a^3*b^2 + a*b^4)*d^2*sinh(3*n - 1))*cosh(n*log(x)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n
*log(x)) + c)^2 - (((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))
*cosh(n*log(x))^3 + 6*(a^3*b^2 + a*b^4)*c^2*cosh(3*n - 1) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a^
5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*sinh(n*log(x))^3 + 6*(a^3*b^2 + a*b^4)*c^2*sinh(3*n - 1) - 6*((a^3*b
^2 + a*b^4)*d^2*cosh(3*n - 1) + (a^3*b^2 + a*b^4)*d^2*sinh(3*n - 1))*cosh(n*log(x))^2 - 3*(2*(a^3*b^2 + a*b^4)
*d^2*cosh(3*n - 1) + 2*(a^3*b^2 + a*b^4)*d^2*sinh(3*n - 1) - ((a^5 + 2*a^3*b^2 + a*b^4)*d^3*cosh(3*n - 1) + (a
^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x)))*sinh(n*log(x))^2 + 3*(((a^5 + 2*a^3*b^2 + a*b^4)*d^
3*cosh(3*n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d^3*sinh(3*n - 1))*cosh(n*log(x))^2 - 4*((a^3*b^2 + a*b^4)*d^2*cos
h(3*n - 1) + (a^3*b^2 + a*b^4)*d^2*sinh(3*n - 1))*cosh(n*log(x)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c)^2 - 2*(((a^4*b + 2*a^2*b^3 + b^5)*d^3*cosh(3*n - 1) + (a^4*b + 2*a^2*b^3 + b^5)*d^3*sinh(3*n
- 1))*cosh(n*log(x))^3 + 6*(a^2*b^3 + b^5)*c^2*cosh(3*n - 1) + ((a^4*b + 2*a^2*b^3 + b^5)*d^3*cosh(3*n - 1) +
(a^4*b + 2*a^2*b^3 + b^5)*d^3*sinh(3*n - 1))*sinh(n*log(x))^3 + 6*(a^2*b^3 + b^5)*c^2*sinh(3*n - 1) - 3*((a^2
*b^3 + b^5)*d^2*cosh(3*n - 1) + (a^2*b^3 + b^5)*d^2*sinh(3*n - 1))*cosh(n*log(x))^2 - 3*((a^2*b^3 + b^5)*d^2*c
osh(3*n - 1) + (a^2*b^3 + b^5)*d^2*sinh(3*n - 1) - ((a^4*b + 2*a^2*b^3 + b^5)*d^3*cosh(3*n - 1) + (a^4*b + 2*a
^2*b^3 + b^5)*d^3*sinh(3*n - 1))*cosh(n*log(x)))*sinh(n*log(x))^2 + 3*(((a^4*b + 2*a^2*b^3 + b^5)*d^3*cosh(3*n
- 1) + (a^4*b + 2*a^2*b^3 + b^5)*d^3*sinh(3*n - 1))*cosh(n*log(x))^2 - 2*((a^2*b^3 + b^5)*d^2*cosh(3*n - 1) +
(a^2*b^3 + b^5)*d^2*sinh(3*n - 1))*cosh(n*log(x)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c) + 6*((((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*co
sh(n*log(x)) + ((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^
2)*sinh(n*log(x)) - (a^3*b^2 + a*b^4)*cosh(3*n - 1) - (a^3*b^2 + a*b^4)*sinh(3*n - 1))*cosh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c)^2 + (((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh(3*n - 1))*sqrt((
a^2 + b^2)/a^2)*cosh(n*log(x)) + ((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh(3*n - 1))*s
qrt((a^2 + b^2)/a^2)*sinh(n*log(x)) - (a^3*b^2 + a*b^4)*cosh(3*n - 1) - (a^3*b^2 + a*b^4)*sinh(3*n - 1))*sinh(
d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - ((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh
(3*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) - ((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d
*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*sinh(n*log(x)) + 2*(((2*a^3*b^2 + a*b^4)*d*cosh(3*n - 1) + (2*a^3*b^2 +
a*b^4)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + ((2*a^3*b^2 + a*b^4)*d*cosh(3*n - 1) + (2*a^3*b
^2 + a*b^4)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*sinh(n*log(x)) - (a^2*b^3 + b^5)*cosh(3*n - 1) - (a^2*b^3 +
b^5)*sinh(3*n - 1))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^3*b^2 + a*b^4)*cosh(3*n - 1) + 2*(((2*
a^3*b^2 + a*b^4)*d*cosh(3*n - 1) + (2*a^3*b^2 + a*b^4)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) +
((2*a^3*b^2 + a*b^4)*d*cosh(3*n - 1) + (2*a^3*b^2 + a*b^4)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*sinh(n*log(
x)) + (((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*cosh(
n*log(x)) + ((2*a^4*b + a^2*b^3)*d*cosh(3*n - 1) + (2*a^4*b + a^2*b^3)*d*sinh(3*n - 1))*sqrt((a^2 + b^2)/a^2)*
sinh(n*log(x)) - (a^3*b^2 + a*b^4)*cosh(3*n - 1...

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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