3.77 \(\int (e x)^{-1+3 n} (a+b \text{csch}(c+d x^n))^2 \, dx\)
Optimal. Leaf size=344 \[ \frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-e^{c+d x^n}\right )}{d^3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,e^{c+d x^n}\right )}{d^3 e n}-\frac{4 a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{b^2 x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{d e n} \]
[Out]
(a^2*(e*x)^(3*n))/(3*e*n) - (b^2*(e*x)^(3*n))/(d*e*n*x^n) - (4*a*b*(e*x)^(3*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*
x^n) - (b^2*(e*x)^(3*n)*Coth[c + d*x^n])/(d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 - E^(2*(c + d*x^n))])/(d^2*e*n
*x^(2*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[2, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (4*a*b*(e*x)^(3*n)*PolyLog[2, E
^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (b^2*(e*x)^(3*n)*PolyLog[2, E^(2*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (4*a*b*
(e*x)^(3*n)*PolyLog[3, -E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[3, E^(c + d*x^n)])/(d^3
*e*n*x^(3*n))
________________________________________________________________________________________
Rubi [A] time = 0.407818, antiderivative size = 344, normalized size of antiderivative = 1.,
number of steps used = 16, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5,
Rules used = {5441, 5437, 4190, 4182, 2531, 2282, 6589, 4184, 3716, 2190, 2279, 2391}
\[ \frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,-e^{c+d x^n}\right )}{d^3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (3,e^{c+d x^n}\right )}{d^3 e n}-\frac{4 a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-2 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{b^2 x^{-3 n} (e x)^{3 n} \text{PolyLog}\left (2,e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{d e n} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^(-1 + 3*n)*(a + b*Csch[c + d*x^n])^2,x]
[Out]
(a^2*(e*x)^(3*n))/(3*e*n) - (b^2*(e*x)^(3*n))/(d*e*n*x^n) - (4*a*b*(e*x)^(3*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*
x^n) - (b^2*(e*x)^(3*n)*Coth[c + d*x^n])/(d*e*n*x^n) + (2*b^2*(e*x)^(3*n)*Log[1 - E^(2*(c + d*x^n))])/(d^2*e*n
*x^(2*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[2, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (4*a*b*(e*x)^(3*n)*PolyLog[2, E
^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (b^2*(e*x)^(3*n)*PolyLog[2, E^(2*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + (4*a*b*
(e*x)^(3*n)*PolyLog[3, -E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) - (4*a*b*(e*x)^(3*n)*PolyLog[3, E^(c + d*x^n)])/(d^3
*e*n*x^(3*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 4190
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Rule 4182
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]
Rule 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 3716
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && 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]
Rubi steps
\begin{align*} \int (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 x^{-1+3 n} \left (a+b \text{csch}\left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 (a+b \text{csch}(c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text{csch}(c+d x)+b^2 x^2 \text{csch}^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}+\frac{\left (2 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \text{csch}(c+d x) \, dx,x,x^n\right )}{e n}+\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x^2 \text{csch}^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}-\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int x \coth (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}-\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{c+d x^n}\right )}{d^2 e n}+\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac{\left (4 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{e^{2 (c+d x)} x}{1-e^{2 (c+d x)}} \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{c+d x^n}\right )}{d^2 e n}+\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}-\frac{\left (4 a b x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}-\frac{\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-e^{c+d x^n}\right )}{d^3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (e^{c+d x^n}\right )}{d^3 e n}-\frac{\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}\\ &=\frac{a^2 (e x)^{3 n}}{3 e n}-\frac{b^2 x^{-n} (e x)^{3 n}}{d e n}-\frac{4 a b x^{-n} (e x)^{3 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{3 n} \coth \left (c+d x^n\right )}{d e n}+\frac{2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1-e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (-e^{c+d x^n}\right )}{d^2 e n}+\frac{4 a b x^{-2 n} (e x)^{3 n} \text{Li}_2\left (e^{c+d x^n}\right )}{d^2 e n}+\frac{b^2 x^{-3 n} (e x)^{3 n} \text{Li}_2\left (e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (-e^{c+d x^n}\right )}{d^3 e n}-\frac{4 a b x^{-3 n} (e x)^{3 n} \text{Li}_3\left (e^{c+d x^n}\right )}{d^3 e n}\\ \end{align*}
Mathematica [F] time = 101.672, size = 0, normalized size = 0. \[ \int (e x)^{-1+3 n} \left (a+b \text{csch}\left (c+d x^n\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(e*x)^(-1 + 3*n)*(a + b*Csch[c + d*x^n])^2,x]
[Out]
Integrate[(e*x)^(-1 + 3*n)*(a + b*Csch[c + d*x^n])^2, x]
________________________________________________________________________________________
Maple [F] time = 0.307, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+3\,n} \left ( a+b{\rm csch} \left (c+d{x}^{n}\right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^(-1+3*n)*(a+b*csch(c+d*x^n))^2,x)
[Out]
int((e*x)^(-1+3*n)*(a+b*csch(c+d*x^n))^2,x)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+3*n)*(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [C] time = 3.12592, size = 15032, normalized size = 43.7 \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+3*n)*(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")
[Out]
-1/3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 + 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*n
- 1)*log(e)) + a^2*d^3*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^3 - (a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x
))^3 - 6*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 + 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3
*n - 1)*log(e)) + a^2*d^3*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^3 + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*
log(x)) - 2*b^2*d^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh(n*log(x)) - 2*b^2*d^2)*sinh((3*n - 1)*log(e)))*sinh
(n*log(x))^2 + (a^2*d^3*cosh(n*log(x))^3 - 6*b^2*d^2*cosh(n*log(x))^2 + 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*
(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 4*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2*d^3*
cosh(n*log(x))^2 - 4*b^2*d^2*cosh(n*log(x)))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d
*sinh(n*log(x)) + c)^2 - 2*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 - 6*b^2*d^2*cosh((3*n - 1)*log(e))
*cosh(n*log(x))^2 + 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*n - 1)*log(e)) + a^2*d^3*sinh((3*n - 1
)*log(e)))*sinh(n*log(x))^3 + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 2*b^2*d^2*cosh((3*n - 1)*log(
e)) + (a^2*d^3*cosh(n*log(x)) - 2*b^2*d^2)*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^3*cosh(n*log(x))^
3 - 6*b^2*d^2*cosh(n*log(x))^2 + 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*
log(x))^2 - 4*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2*d^3*cosh(n*log(x))^2 - 4*b^2*d^2*cosh(n*log
(x)))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(
x)) + d*sinh(n*log(x)) + c) - (a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 - 6*b^2*d^2*cosh((3*n - 1)*log(
e))*cosh(n*log(x))^2 + 6*b^2*c^2*cosh((3*n - 1)*log(e)) + (a^2*d^3*cosh((3*n - 1)*log(e)) + a^2*d^3*sinh((3*n
- 1)*log(e)))*sinh(n*log(x))^3 + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - 2*b^2*d^2*cosh((3*n - 1)*l
og(e)) + (a^2*d^3*cosh(n*log(x)) - 2*b^2*d^2)*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^3*cosh(n*log(x
))^3 - 6*b^2*d^2*cosh(n*log(x))^2 + 6*b^2*c^2)*sinh((3*n - 1)*log(e)) + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh
(n*log(x))^2 - 4*b^2*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a^2*d^3*cosh(n*log(x))^2 - 4*b^2*d^2*cosh(n*
log(x)))*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 3*(a^2*d^3*
cosh((3*n - 1)*log(e))*cosh(n*log(x)) + a^2*d^3*cosh(n*log(x))*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + 6*(2
*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - (2*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*cosh((3*n
- 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + 2*(a*b*d*cosh((3*n - 1)*log(e)) + a*b*d
*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + b^2*cosh((3*n - 1)*
log(e)) - 2*(2*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*cosh((3*n - 1)*log(e)) + (2*a*b*d*cosh(n*log(
x)) + b^2)*sinh((3*n - 1)*log(e)) + 2*(a*b*d*cosh((3*n - 1)*log(e)) + a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log
(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (2*a*b*d*c
osh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*cosh((3*n - 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n -
1)*log(e)) + 2*(a*b*d*cosh((3*n - 1)*log(e)) + a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c)^2 + (2*a*b*d*cosh(n*log(x)) + b^2)*sinh((3*n - 1)*log(e)) + 2*(a*b*d*cosh((3*n -
1)*log(e)) + a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*dilog(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c
) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) - 6*(2*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - (2*a*b
*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - b^2*cosh((3*n - 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) - b^2)*sinh((3
*n - 1)*log(e)) + 2*(a*b*d*cosh((3*n - 1)*log(e)) + a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(
n*log(x)) + d*sinh(n*log(x)) + c)^2 - b^2*cosh((3*n - 1)*log(e)) - 2*(2*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*lo
g(x)) - b^2*cosh((3*n - 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) - b^2)*sinh((3*n - 1)*log(e)) + 2*(a*b*d*cosh((3*
n - 1)*log(e)) + a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*s
inh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (2*a*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - b^2*cosh((3*n
- 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) - b^2)*sinh((3*n - 1)*log(e)) + 2*(a*b*d*cosh((3*n - 1)*log(e)) + a*b*d
*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2*a*b*d*cosh(n*log
(x)) - b^2)*sinh((3*n - 1)*log(e)) + 2*(a*b*d*cosh((3*n - 1)*log(e)) + a*b*d*sinh((3*n - 1)*log(e)))*sinh(n*lo
g(x)))*dilog(-cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) -
6*(a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - (a*b*d^2*c
osh((3*n - 1)*log(e))*cosh(n*log(x))^2 - b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a*b*d^2*cosh((3*n - 1)
*log(e)) + a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a*b*d^2*cosh(n*log(x))^2 - b^2*d*cosh(n*log(x))
)*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - b^2*d*cosh((3*n - 1)*log(e)) + (
2*a*b*d^2*cosh(n*log(x)) - b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log
(x)) + c)^2 - 2*(a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x))
+ (a*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a*b*d^2*cosh(n*log(x)
)^2 - b^2*d*cosh(n*log(x)))*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - b^2*d*
cosh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh(n*log(x)) - b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a*b*d^2*cosh((3*n - 1)*lo
g(e))*cosh(n*log(x))^2 - b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d
^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a*b*d^2*cosh(n*log(x))^2 - b^2*d*cosh(n*log(x)))*sinh((3*n - 1)
*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - b^2*d*cosh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh(n
*log(x)) - b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (a
*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (a*b*d^2*cosh(n*log(x))^2 -
b^2*d*cosh(n*log(x)))*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - b^2*d*cosh(
(3*n - 1)*log(e)) + (2*a*b*d^2*cosh(n*log(x)) - b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*log(cosh(d*cosh
(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) - 6*(((a*b*c^2 - b^2*c
)*cosh((3*n - 1)*log(e)) + (a*b*c^2 - b^2*c)*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c)^2 + 2*((a*b*c^2 - b^2*c)*cosh((3*n - 1)*log(e)) + (a*b*c^2 - b^2*c)*sinh((3*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) + ((a*b*c^2 - b^2*c)*cosh((3*n
- 1)*log(e)) + (a*b*c^2 - b^2*c)*sinh((3*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - (a*
b*c^2 - b^2*c)*cosh((3*n - 1)*log(e)) - (a*b*c^2 - b^2*c)*sinh((3*n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 1) + 6*(a*b*d^2*cosh((3*n - 1)*log(e))
*cosh(n*log(x))^2 + b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - (a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x
))^2 + b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d^2*sinh((3*n - 1)*
log(e)))*sinh(n*log(x))^2 - (a*b*c^2 - b^2*c)*cosh((3*n - 1)*log(e)) + (a*b*d^2*cosh(n*log(x))^2 - a*b*c^2 + b
^2*d*cosh(n*log(x)) + b^2*c)*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*d
*cosh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh(n*log(x)) + b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*co
sh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 2*(a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 + b^2*d*cosh((3*n
- 1)*log(e))*cosh(n*log(x)) + (a*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*log(x))
^2 - (a*b*c^2 - b^2*c)*cosh((3*n - 1)*log(e)) + (a*b*d^2*cosh(n*log(x))^2 - a*b*c^2 + b^2*d*cosh(n*log(x)) + b
^2*c)*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*d*cosh((3*n - 1)*log(e))
+ (2*a*b*d^2*cosh(n*log(x)) + b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n
*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2
+ b^2*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + (a*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d^2*sinh((3*n - 1)*log(
e)))*sinh(n*log(x))^2 - (a*b*c^2 - b^2*c)*cosh((3*n - 1)*log(e)) + (a*b*d^2*cosh(n*log(x))^2 - a*b*c^2 + b^2*d
*cosh(n*log(x)) + b^2*c)*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*d*cos
h((3*n - 1)*log(e)) + (2*a*b*d^2*cosh(n*log(x)) + b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n
*log(x)) + d*sinh(n*log(x)) + c)^2 + (a*b*d^2*cosh((3*n - 1)*log(e)) + a*b*d^2*sinh((3*n - 1)*log(e)))*sinh(n*
log(x))^2 - (a*b*c^2 - b^2*c)*cosh((3*n - 1)*log(e)) + (a*b*d^2*cosh(n*log(x))^2 - a*b*c^2 + b^2*d*cosh(n*log(
x)) + b^2*c)*sinh((3*n - 1)*log(e)) + (2*a*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x)) + b^2*d*cosh((3*n - 1)*
log(e)) + (2*a*b*d^2*cosh(n*log(x)) + b^2*d)*sinh((3*n - 1)*log(e)))*sinh(n*log(x)))*log(-cosh(d*cosh(n*log(x)
) + d*sinh(n*log(x)) + c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + 12*((a*b*cosh((3*n - 1)*log(e
)) + a*b*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*cosh((3*n - 1)*log(e))
+ 2*(a*b*cosh((3*n - 1)*log(e)) + a*b*sinh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*si
nh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a*b*cosh((3*n - 1)*log(e)) + a*b*sinh((3*n - 1)*log(e)))*sinh(d
*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*sinh((3*n - 1)*log(e)))*polylog(3, cosh(d*cosh(n*log(x)) + d*s
inh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) - 12*((a*b*cosh((3*n - 1)*log(e)) + a*b*si
nh((3*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*cosh((3*n - 1)*log(e)) + 2*(a*b*co
sh((3*n - 1)*log(e)) + a*b*sinh((3*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) + (a*b*cosh((3*n - 1)*log(e)) + a*b*sinh((3*n - 1)*log(e)))*sinh(d*cosh(n*log
(x)) + d*sinh(n*log(x)) + c)^2 - a*b*sinh((3*n - 1)*log(e)))*polylog(3, -cosh(d*cosh(n*log(x)) + d*sinh(n*log(
x)) + c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) + (a^2*d^3*cosh(n*log(x))^3 + 6*b^2*c^2)*sinh((3*n -
1)*log(e)) + 3*(a^2*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 + a^2*d^3*cosh(n*log(x))^2*sinh((3*n - 1)*log
(e)))*sinh(n*log(x)))/(d^3*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*d^3*n*cosh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + d^3*n*sinh(d*cosh(n*log(x)) + d*sinh(n*
log(x)) + c)^2 - d^3*n)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{3 n - 1} \left (a + b \operatorname{csch}{\left (c + d x^{n} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**(-1+3*n)*(a+b*csch(c+d*x**n))**2,x)
[Out]
Integral((e*x)**(3*n - 1)*(a + b*csch(c + d*x**n))**2, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{csch}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+3*n)*(a+b*csch(c+d*x^n))^2,x, algorithm="giac")
[Out]
integrate((b*csch(d*x^n + c) + a)^2*(e*x)^(3*n - 1), x)