3.82 \(\int \frac{(e x)^{-1+2 n}}{(a+b \text{csch}(c+d x^n))^2} \, dx\)
Optimal. Leaf size=681 \[ -\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 d^2 e n \sqrt{a^2+b^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a^2 d^2 e n \sqrt{a^2+b^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sinh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2+b^2\right )}-\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a^2 d e n \sqrt{a^2+b^2}}+\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}+\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a^2 d e n \sqrt{a^2+b^2}}-\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \]
[Out]
(e*x)^(2*n)/(2*a^2*e*n) + (b^3*(e*x)^(2*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)^(2*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) - (b^3*(e*x)^(2*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)^(2*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) + (b^2*(
e*x)^(2*n)*Log[b + a*Sinh[c + d*x^n]])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*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)) - (2*b*(e*x)^(2*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)) - (b^3*(e*x)^(2*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)) + (2*b*(e*x)^(2*n)*PolyL
og[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)) - (b^2*(e*x)^(2*n)*Co
sh[c + d*x^n])/(a*(a^2 + b^2)*d*e*n*x^n*(b + a*Sinh[c + d*x^n]))
________________________________________________________________________________________
Rubi [A] time = 1.14658, antiderivative size = 681, normalized size of antiderivative =
1., number of steps used = 23, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.458, Rules used = {5441, 5437, 4191, 3324, 3322, 2264, 2190, 2279, 2391, 2668, 31}
\[ -\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 d^2 e n \sqrt{a^2+b^2}}+\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac{2 b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a^2 d^2 e n \sqrt{a^2+b^2}}-\frac{b^3 x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}\right )}{a^2 d^2 e n \left (a^2+b^2\right )^{3/2}}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (a \sinh \left (c+d x^n\right )+b\right )}{a^2 d^2 e n \left (a^2+b^2\right )}-\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a^2 d e n \sqrt{a^2+b^2}}+\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{b-\sqrt{a^2+b^2}}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}+\frac{2 b x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a^2 d e n \sqrt{a^2+b^2}}-\frac{b^3 x^{-n} (e x)^{2 n} \log \left (\frac{a e^{c+d x^n}}{\sqrt{a^2+b^2}+b}+1\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^n\right )+b\right )}+\frac{(e x)^{2 n}}{2 a^2 e n} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^(-1 + 2*n)/(a + b*Csch[c + d*x^n])^2,x]
[Out]
(e*x)^(2*n)/(2*a^2*e*n) + (b^3*(e*x)^(2*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)^(2*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) - (b^3*(e*x)^(2*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)^(2*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) + (b^2*(
e*x)^(2*n)*Log[b + a*Sinh[c + d*x^n]])/(a^2*(a^2 + b^2)*d^2*e*n*x^(2*n)) + (b^3*(e*x)^(2*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)) - (2*b*(e*x)^(2*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)) - (b^3*(e*x)^(2*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)) + (2*b*(e*x)^(2*n)*PolyL
og[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)) - (b^2*(e*x)^(2*n)*Co
sh[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 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]
Rule 2668
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rubi steps
\begin{align*} \int \frac{(e x)^{-1+2 n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \int \frac{x^{-1+2 n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{(a+b \text{csch}(c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \left (\frac{x}{a^2}+\frac{b^2 x}{a^2 (b+a \sinh (c+d x))^2}-\frac{2 b x}{a^2 (b+a \sinh (c+d x))}\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{(e x)^{2 n}}{2 a^2 e n}-\frac{\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 e n}+\frac{\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{(b+a \sinh (c+d x))^2} \, dx,x,x^n\right )}{a^2 e n}\\ &=\frac{(e x)^{2 n}}{2 a^2 e n}-\frac{b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{-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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{x}{b+a \sinh (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n}+\frac{\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\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)^{2 n}}{2 a^2 e n}-\frac{b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{-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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{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 (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{1}{b+x} \, dx,x,a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}\\ &=\frac{(e x)^{2 n}}{2 a^2 e n}-\frac{2 b x^{-n} (e x)^{2 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 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{b^2 x^{-n} (e x)^{2 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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{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^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{e^{c+d x} x}{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 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \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 (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \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)^{2 n}}{2 a^2 e n}+\frac{b^3 x^{-n} (e x)^{2 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)^{2 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^3 x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{b^2 x^{-n} (e x)^{2 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 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{\left (2 b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \sqrt{a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x^n}\right )}{a^2 \sqrt{a^2+b^2} d^2 e n}-\frac{\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \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 (b^3 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \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)^{2 n}}{2 a^2 e n}+\frac{b^3 x^{-n} (e x)^{2 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)^{2 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^3 x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}-\frac{2 b x^{-2 n} (e x)^{2 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 x^{-2 n} (e x)^{2 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)^{2 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 (b^3 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b-2 \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^2 e n}+\frac{\left (b^3 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 a x}{2 b+2 \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^2 e n}\\ &=\frac{(e x)^{2 n}}{2 a^2 e n}+\frac{b^3 x^{-n} (e x)^{2 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)^{2 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^3 x^{-n} (e x)^{2 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)^{2 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^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac{b^3 x^{-2 n} (e x)^{2 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{2 b x^{-2 n} (e x)^{2 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^3 x^{-2 n} (e x)^{2 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{2 b x^{-2 n} (e x)^{2 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)^{2 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 [C] time = 33.4276, size = 3256, normalized size = 4.78 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(e*x)^(-1 + 2*n)/(a + b*Csch[c + d*x^n])^2,x]
[Out]
(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csch[c/2]*Csch[c + d*x^n]^2*Sech[c/2]*(b*Cosh[c] + a*Sinh[d*x^n])*(b + a*Sinh[
c + d*x^n]))/(2*a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + d*x^n])^2) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Coth[c]*Csch[
c + d*x^n]^2*(b + a*Sinh[c + d*x^n])^2)/(a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + d*x^n])^2) - (2*b^3*x^(1 - 2*n)*(
e*x)^(-1 + 2*n)*ArcTan[(a - b*Tanh[(c + d*x^n)/2])/Sqrt[-a^2 - b^2]]*Coth[c]*Csch[c + d*x^n]^2*(b + a*Sinh[c +
d*x^n])^2)/(a^2*Sqrt[-a^2 - b^2]*(a^2 + b^2)*d^2*n*(a + b*Csch[c + d*x^n])^2) + (2*b*x^(1 - 2*n)*(e*x)^(-1 +
2*n)*Csch[c + d*x^n]^2*((I*Pi*ArcTanh[(-a + b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (2*((-I
)*c + Pi/2 - I*d*x^n)*ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - 2*((-I)*c +
ArcCos[((-I)*b)/a])*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] + (ArcCos[((-I)*
b)/a] - (2*I)*(ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a -
b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]]))*Log[Sqrt[-a^2 - b^2]/(Sqrt[2]*Sqrt[(-I)*a]*E^((I/2)*(
(-I)*c + Pi/2 - I*d*x^n))*Sqrt[b + a*Sinh[c + d*x^n]])] + (ArcCos[((-I)*b)/a] + (2*I)*(ArcTanh[(((-I)*a + b)*C
ot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/S
qrt[-a^2 - b^2]]))*Log[(Sqrt[-a^2 - b^2]*E^((I/2)*((-I)*c + Pi/2 - I*d*x^n)))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a
*Sinh[c + d*x^n]])] - (ArcCos[((-I)*b)/a] + (2*I)*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt
[-a^2 - b^2]])*Log[1 - (I*(b - I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n
)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] + (-ArcCos[((-I)*b)/a] + (2*I)*Ar
cTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]])*Log[1 - (I*(b + I*Sqrt[-a^2 - b^2])*(
(-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c
+ Pi/2 - I*d*x^n)/2]))] + I*(PolyLog[2, (I*(b - I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*
c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] - PolyLog[2, (I
*(b + I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + S
qrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))]))/Sqrt[-a^2 - b^2])*(b + a*Sinh[c + d*x^n])^2)/((a^2 + b^2
)*d^2*n*(a + b*Csch[c + d*x^n])^2) + (b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csch[c + d*x^n]^2*((I*Pi*ArcTanh[(-a +
b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (2*((-I)*c + Pi/2 - I*d*x^n)*ArcTanh[(((-I)*a + b)*
Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - 2*((-I)*c + ArcCos[((-I)*b)/a])*ArcTanh[(((-I)*a - b)*Ta
n[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] + (ArcCos[((-I)*b)/a] - (2*I)*(ArcTanh[(((-I)*a + b)*Cot[((-
I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a
^2 - b^2]]))*Log[Sqrt[-a^2 - b^2]/(Sqrt[2]*Sqrt[(-I)*a]*E^((I/2)*((-I)*c + Pi/2 - I*d*x^n))*Sqrt[b + a*Sinh[c
+ d*x^n]])] + (ArcCos[((-I)*b)/a] + (2*I)*(ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 -
b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]]))*Log[(Sqrt[-a^2 - b^2]*E^(
(I/2)*((-I)*c + Pi/2 - I*d*x^n)))/(Sqrt[2]*Sqrt[(-I)*a]*Sqrt[b + a*Sinh[c + d*x^n]])] - (ArcCos[((-I)*b)/a] +
(2*I)*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]])*Log[1 - (I*(b - I*Sqrt[-a^2 -
b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan
[((-I)*c + Pi/2 - I*d*x^n)/2]))] + (-ArcCos[((-I)*b)/a] + (2*I)*ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d
*x^n)/2])/Sqrt[-a^2 - b^2]])*Log[1 - (I*(b + I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c +
Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] + I*(PolyLog[2, (I*
(b - I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sq
rt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))] - PolyLog[2, (I*(b + I*Sqrt[-a^2 - b^2])*((-I)*a + b - Sqrt
[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n)/2]))/(a*((-I)*a + b + Sqrt[-a^2 - b^2]*Tan[((-I)*c + Pi/2 - I*d*x^n
)/2]))]))/Sqrt[-a^2 - b^2])*(b + a*Sinh[c + d*x^n])^2)/(a^2*(a^2 + b^2)*d^2*n*(a + b*Csch[c + d*x^n])^2) + (x^
(1 - n)*(e*x)^(-1 + 2*n)*Csch[c/2]*Csch[c + d*x^n]^2*Sech[c/2]*(-2*b^2*Cosh[c] + a^2*d*x^n*Sinh[c] + b^2*d*x^n
*Sinh[c])*(b + a*Sinh[c + d*x^n])^2)/(4*a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + d*x^n])^2) - (b^2*x^(1 - 2*n)*(e*x
)^(-1 + 2*n)*Csch[c]*Csch[c + d*x^n]^2*(-(a*d*x^n*Cosh[c]) + a*Log[b + a*Cosh[d*x^n]*Sinh[c] + a*Cosh[c]*Sinh[
d*x^n]]*Sinh[c] + (2*a*b*ArcTan[(a*Cosh[c] + (-b + a*Sinh[c])*Tanh[(d*x^n)/2])/Sqrt[-b^2 - a^2*Cosh[c]^2 + a^2
*Sinh[c]^2]]*Cosh[c])/Sqrt[-b^2 - a^2*Cosh[c]^2 + a^2*Sinh[c]^2])*(b + a*Sinh[c + d*x^n])^2)/(a*(a^2 + b^2)*d^
2*n*(a + b*Csch[c + d*x^n])^2*(-(a^2*Cosh[c]^2) + a^2*Sinh[c]^2))
________________________________________________________________________________________
Maple [F] time = 0.583, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{-1+2\,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+2*n)/(a+b*csch(c+d*x^n))^2,x)
[Out]
int((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{4 \, a b^{2} e^{2 \, n} x^{n} +{\left (a^{3} d e^{2 \, n} + a b^{2} d e^{2 \, n}\right )} x^{2 \, n} -{\left (a^{3} d e^{2 \, n} e^{\left (2 \, c\right )} + a b^{2} d e^{2 \, n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + 2 \, n \log \left (x\right )\right )} - 2 \,{\left (2 \, b^{3} e^{2 \, n} e^{\left (n \log \left (x\right ) + c\right )} +{\left (a^{2} b d e^{2 \, n} e^{c} + b^{3} d e^{2 \, n} e^{c}\right )} x^{2 \, n}\right )} e^{\left (d x^{n}\right )}}{2 \,{\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 (a b^{2} e^{2 \, n} x^{n} -{\left (b^{3} e^{2 \, n} e^{\left (n \log \left (x\right ) + c\right )} +{\left (2 \, a^{2} b d e^{2 \, n} e^{c} + b^{3} d e^{2 \, n} e^{c}\right )} x^{2 \, 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+2*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")
[Out]
1/2*(4*a*b^2*e^(2*n)*x^n + (a^3*d*e^(2*n) + a*b^2*d*e^(2*n))*x^(2*n) - (a^3*d*e^(2*n)*e^(2*c) + a*b^2*d*e^(2*n
)*e^(2*c))*e^(2*d*x^n + 2*n*log(x)) - 2*(2*b^3*e^(2*n)*e^(n*log(x) + c) + (a^2*b*d*e^(2*n)*e^c + b^3*d*e^(2*n)
*e^c)*x^(2*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*(a*b^2*e^(2*n)*x^n - (b^3*e^(2*n)*e^(n*lo
g(x) + c) + (2*a^2*b*d*e^(2*n)*e^c + b^3*d*e^(2*n)*e^c)*x^(2*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 [B] time = 3.75371, size = 21689, normalized size = 31.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")
[Out]
-1/2*((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 - ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*c
osh((2*n - 1)*log(e))*cosh(n*log(x))^2 - 4*(a^3*b^2 + a*b^4)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*(a^3*
b^2 + a*b^4)*c*cosh((2*n - 1)*log(e)) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 + 2*a^3*b
^2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))^2 - 4
*(a^3*b^2 + a*b^4)*d*cosh(n*log(x)) - 4*(a^3*b^2 + a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 + 2*a^3*b^2 + a*
b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 2*(a^3*b^2 + a*b^4)*d*cosh((2*n - 1)*log(e)) + ((a^5 + 2*a^3*
b^2 + a*b^4)*d^2*cosh(n*log(x)) - 2*(a^3*b^2 + a*b^4)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n
*log(x)) + d*sinh(n*log(x)) + c)^2 - 4*(a^3*b^2 + a*b^4)*c*cosh((2*n - 1)*log(e)) - ((a^5 + 2*a^3*b^2 + a*b^4)
*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 - 4*(a^3*b^2 + a*b^4)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4
*(a^3*b^2 + a*b^4)*c*cosh((2*n - 1)*log(e)) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 + 2
*a^3*b^2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))
^2 - 4*(a^3*b^2 + a*b^4)*d*cosh(n*log(x)) - 4*(a^3*b^2 + a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 + 2*a^3*b^
2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 2*(a^3*b^2 + a*b^4)*d*cosh((2*n - 1)*log(e)) + ((a^5 +
2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x)) - 2*(a^3*b^2 + a*b^4)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*
cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 + 2*a^
3*b^2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 - 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*
log(e))*cosh(n*log(x))^2 - 2*(a^2*b^3 + b^5)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*(a^2*b^3 + b^5)*c*cos
h((2*n - 1)*log(e)) + ((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e)) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*si
nh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(n*log(x))^2 - 2*(a^2*b^3 + b^5)*d
*cosh(n*log(x)) - 4*(a^2*b^3 + b^5)*c)*sinh((2*n - 1)*log(e)) + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1
)*log(e))*cosh(n*log(x)) - (a^2*b^3 + b^5)*d*cosh((2*n - 1)*log(e)) + ((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(n*lo
g(x)) - (a^2*b^3 + b^5)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c) + 2*(((2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^
2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^4*b + a^2*b^3)*sqrt((a
^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh
(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))
- (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)) + 2*((2*a^3*b^2 + a*b^4)*sqrt((a^2 + b^2)/
a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 + a*b^4)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*
log(x)) + d*sinh(n*log(x)) + c) + 2*((2*a^3*b^2 + a*b^4)*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3
*b^2 + a*b^4)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*cosh((
2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d
*sinh(n*log(x)) + c))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*dilog(((a*sqrt((a^2 + b^2)/a^2) + b)*cosh
(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt((a^2 + b^2)/a^2) + b)*sinh(d*cosh(n*log(x)) + d*sinh(n*log
(x)) + c) - a)/a + 1) - 2*(((2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*
b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^4*b
+ a^2*b^3)*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n
- 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*cos
h((2*n - 1)*log(e)) - (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)) + 2*((2*a^3*b^2 + a*b^4
)*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 + a*b^4)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(
e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*((2*a^3*b^2 + a*b^4)*sqrt((a^2 + b^2)/a^2)*cosh((2*n -
1)*log(e)) + (2*a^3*b^2 + a*b^4)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*sqrt((a^2
+ b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*cosh(d
*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*dilog(-((a*sqrt((a^2 +
b^2)/a^2) - b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt((a^2 + b^2)/a^2) - b)*sinh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c) + a)/a + 1) - 2*(((a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)
)*cosh((2*n - 1)*log(e)) + (a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(
e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b
^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n -
1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((a^2*b^3 + b^5 - (2*a^3*b^2 + a*b^4)*c*sqrt(
(a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^2*b^3 + b^5 - (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2))*sinh(
(2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sq
rt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + 2*(((a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^
2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*lo
g(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^2*b^3 + b^5 - (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)
/a^2))*cosh((2*n - 1)*log(e)) + (a^2*b^3 + b^5 - (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*l
og(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^3*b^2 + a*b^4 - (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b
^2)/a^2))*sinh((2*n - 1)*log(e)))*log(2*a*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*a*sinh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c) + 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) - 2*(((a^3*b^2 + a*b^4 + (2*a^4*b + a^2*b^3)*
c*sqrt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 + a*b^4 + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^
2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^3*b^2 + a*b^4 + (2*a^4*b + a
^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 + a*b^4 + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 +
b^2)/a^2))*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((a^2*b^3 + b^5 + (2*a
^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^2*b^3 + b^5 + (2*a^3*b^2 + a*b^4)*c*sqrt(
(a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^3*b^2 + a*b^4 + (
2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + 2*(((a^3*b^2 + a*b^4 + (2*a^4*b + a^2*b^3
)*c*sqrt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^3*b^2 + a*b^4 + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/
a^2))*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^2*b^3 + b^5 + (2*a^3*b^2 + a*
b^4)*c*sqrt((a^2 + b^2)/a^2))*cosh((2*n - 1)*log(e)) + (a^2*b^3 + b^5 + (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)
/a^2))*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^3*b^2 + a*b^4 + (2*a^4*b + a
^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)))*log(2*a*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)
+ 2*a*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 2*a*sqrt((a^2 + b^2)/a^2) + 2*b) - 2*((2*a^4*b + a^2*b^
3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)
*cosh((2*n - 1)*log(e)) - ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) +
(2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/
a^2)*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^
3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)
*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 +
b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*
log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)
/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b
+ a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*l
og(x)) + c)^2 - 2*((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^3*
b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cos
h(n*log(x)) + (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 + a*b^4)*d*sqr
t((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e))
)*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 2*((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^
2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))
+ ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*s
qrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2
*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^
2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x))
)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x))
+ (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2
)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log
(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(
x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 +
b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n
*log(x)))*log(-((a*sqrt((a^2 + b^2)/a^2) + b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt((a^2 + b
^2)/a^2) + b)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - a)/a) + 2*((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2
)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(
e)) - ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)
*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x))
+ (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2
)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log
(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n
- 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b
+ a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1
)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((
a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 2*((
2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^3*b^2 + a*b^4)*c*sqrt(
(a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2*a^3
*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*c
osh((2*n - 1)*log(e)) + (2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*co
sh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 2*((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*lo
g(e))*cosh(n*log(x)) + (2*a^3*b^2 + a*b^4)*c*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^
3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*sqrt((a^2 + b^2)/a^2)
*cosh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b + a^2*b^3)*c*
sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1)*lo
g(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(
x)) + d*sinh(n*log(x)) + c) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2*a^3*b^2 + a*b^4
)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n - 1
)*log(e)) + (2*a^3*b^2 + a*b^4)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*
log(x)) + d*sinh(n*log(x)) + c) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh(n*log(x)) + (2*a^4*b + a^2
*b^3)*c*sqrt((a^2 + b^2)/a^2))*sinh((2*n - 1)*log(e)) + ((2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*cosh((2*n
- 1)*log(e)) + (2*a^4*b + a^2*b^3)*d*sqrt((a^2 + b^2)/a^2)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*log(((a*sq
rt((a^2 + b^2)/a^2) - b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*sqrt((a^2 + b^2)/a^2) - b)*sinh(d*
cosh(n*log(x)) + d*sinh(n*log(x)) + c) + a)/a) - 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e))*cosh(
n*log(x))^2 - 2*(a^2*b^3 + b^5)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*(a^2*b^3 + b^5)*c*cosh((2*n - 1)*l
og(e)) + ((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e)) + (a^4*b + 2*a^2*b^3 + b^5)*d^2*sinh((2*n - 1)*
log(e)))*sinh(n*log(x))^2 + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 - 4*(a^3*b^
2 + a*b^4)*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*(a^3*b^2 + a*b^4)*c*cosh((2*n - 1)*log(e)) + ((a^5 + 2*
a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e)) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log
(x))^2 + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))^2 - 4*(a^3*b^2 + a*b^4)*d*cosh(n*log(x)) - 4*(a^3*b^2 +
a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 2
*(a^3*b^2 + a*b^4)*d*cosh((2*n - 1)*log(e)) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x)) - 2*(a^3*b^2 + a*b
^4)*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((a^4*b + 2*a^2
*b^3 + b^5)*d^2*cosh(n*log(x))^2 - 2*(a^2*b^3 + b^5)*d*cosh(n*log(x)) - 4*(a^2*b^3 + b^5)*c)*sinh((2*n - 1)*lo
g(e)) + 2*((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - (a^2*b^3 + b^5)*d*cosh((2*n -
1)*log(e)) + ((a^4*b + 2*a^2*b^3 + b^5)*d^2*cosh(n*log(x)) - (a^2*b^3 + b^5)*d)*sinh((2*n - 1)*log(e)))*sinh(
n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))^2 - 4
*(a^3*b^2 + a*b^4)*c)*sinh((2*n - 1)*log(e)) + 2*((a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh((2*n - 1)*log(e))*cosh(n*
log(x)) + (a^5 + 2*a^3*b^2 + a*b^4)*d^2*cosh(n*log(x))*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))/((a^7 + 2*a^5*b
^2 + a^3*b^4)*d^2*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (a^7 + 2*a^5*b^2 + a^3*b^4)*d^2*n*sinh(d
*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2*b^5)*d^2*n*cosh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c) - (a^7 + 2*a^5*b^2 + a^3*b^4)*d^2*n + 2*((a^7 + 2*a^5*b^2 + a^3*b^4)*d^2*n*cosh(d*cosh(n*log
(x)) + d*sinh(n*log(x)) + c) + (a^6*b + 2*a^4*b^3 + a^2*b^5)*d^2*n)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{2 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+2*n)/(a+b*csch(c+d*x**n))**2,x)
[Out]
Integral((e*x)**(2*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 )^{2 \, 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+2*n)/(a+b*csch(c+d*x^n))^2,x, algorithm="giac")
[Out]
integrate((e*x)^(2*n - 1)/(b*csch(d*x^n + c) + a)^2, x)