3.853 \(\int x^3 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^4(x)} \, dx\)
Optimal. Leaf size=326 \[ -\frac{3}{2} x^2 \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x^2 \cosh ^2(x) \text{PolyLog}\left (2,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x \cosh ^2(x) \text{PolyLog}\left (3,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x \cosh ^2(x) \text{PolyLog}\left (3,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{4} \cosh ^2(x) \text{PolyLog}\left (4,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{4} \cosh ^2(x) \text{PolyLog}\left (4,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^3 \sinh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)}+3 x \log \left (e^{2 x}+1\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)} \]
[Out]
(-3*x^2*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 + (x^3*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 - 2*x^3*ArcTanh[E^(2*x)]*Cosh[x]^
2*Sqrt[a*Sech[x]^4] + 3*x*Cosh[x]^2*Log[1 + E^(2*x)]*Sqrt[a*Sech[x]^4] + (3*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqr
t[a*Sech[x]^4])/2 - (3*x^2*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4])/2 + (3*x^2*Cosh[x]^2*PolyLog[2, E
^(2*x)]*Sqrt[a*Sech[x]^4])/2 + (3*x*Cosh[x]^2*PolyLog[3, -E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (3*x*Cosh[x]^2*PolyL
og[3, E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (3*Cosh[x]^2*PolyLog[4, -E^(2*x)]*Sqrt[a*Sech[x]^4])/4 + (3*Cosh[x]^2*Po
lyLog[4, E^(2*x)]*Sqrt[a*Sech[x]^4])/4 - (3*x^2*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x])/2 - (x^3*Sqrt[a*Sech[x]^4]*
Sinh[x]^2)/2
________________________________________________________________________________________
Rubi [A] time = 0.682779, antiderivative size = 326, normalized size of antiderivative = 1.,
number of steps used = 21, number of rules used = 17, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.944, Rules used
= {6720, 2620, 14, 5462, 2551, 5461, 4182, 2531, 6609, 2282, 6589, 3720, 3718, 2190, 2279, 2391,
30} \[ -\frac{3}{2} x^2 \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x^2 \cosh ^2(x) \text{PolyLog}\left (2,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x \cosh ^2(x) \text{PolyLog}\left (3,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x \cosh ^2(x) \text{PolyLog}\left (3,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{4} \cosh ^2(x) \text{PolyLog}\left (4,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{4} \cosh ^2(x) \text{PolyLog}\left (4,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^3 \sinh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)}+3 x \log \left (e^{2 x}+1\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)} \]
Antiderivative was successfully verified.
[In]
Int[x^3*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^4],x]
[Out]
(-3*x^2*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 + (x^3*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 - 2*x^3*ArcTanh[E^(2*x)]*Cosh[x]^
2*Sqrt[a*Sech[x]^4] + 3*x*Cosh[x]^2*Log[1 + E^(2*x)]*Sqrt[a*Sech[x]^4] + (3*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqr
t[a*Sech[x]^4])/2 - (3*x^2*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4])/2 + (3*x^2*Cosh[x]^2*PolyLog[2, E
^(2*x)]*Sqrt[a*Sech[x]^4])/2 + (3*x*Cosh[x]^2*PolyLog[3, -E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (3*x*Cosh[x]^2*PolyL
og[3, E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (3*Cosh[x]^2*PolyLog[4, -E^(2*x)]*Sqrt[a*Sech[x]^4])/4 + (3*Cosh[x]^2*Po
lyLog[4, E^(2*x)]*Sqrt[a*Sech[x]^4])/4 - (3*x^2*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x])/2 - (x^3*Sqrt[a*Sech[x]^4]*
Sinh[x]^2)/2
Rule 6720
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 5462
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
Rule 2551
Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]
Rule 5461
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
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 6609
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, 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 3720
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]
Rule 3718
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && 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 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int x^3 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^4(x)} \, dx &=\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^3 \text{csch}(x) \text{sech}^3(x) \, dx\\ &=x^3 \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \left (\log (\tanh (x))-\frac{\tanh ^2(x)}{2}\right ) \, dx\\ &=x^3 \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \left (x^2 \log (\tanh (x))-\frac{1}{2} x^2 \tanh ^2(x)\right ) \, dx\\ &=x^3 \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\frac{1}{2} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \tanh ^2(x) \, dx-\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \log (\tanh (x)) \, dx\\ &=-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^3 \text{csch}(x) \text{sech}(x) \, dx+\frac{1}{2} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \, dx+\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \tanh (x) \, dx\\ &=-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^3 \text{csch}(2 x) \, dx+\left (6 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \frac{e^{2 x} x}{1+e^{2 x}} \, dx\\ &=-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+3 x \cosh ^2(x) \log \left (1+e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \log \left (1-e^{2 x}\right ) \, dx-\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \log \left (1+e^{2 x}\right ) \, dx+\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \log \left (1+e^{2 x}\right ) \, dx\\ &=-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+3 x \cosh ^2(x) \log \left (1+e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\frac{1}{2} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 x}\right )+\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \text{Li}_2\left (-e^{2 x}\right ) \, dx-\left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \text{Li}_2\left (e^{2 x}\right ) \, dx\\ &=-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+3 x \cosh ^2(x) \log \left (1+e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x \cosh ^2(x) \text{Li}_3\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x \cosh ^2(x) \text{Li}_3\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\frac{1}{2} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \text{Li}_3\left (-e^{2 x}\right ) \, dx+\frac{1}{2} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \text{Li}_3\left (e^{2 x}\right ) \, dx\\ &=-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+3 x \cosh ^2(x) \log \left (1+e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x \cosh ^2(x) \text{Li}_3\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x \cosh ^2(x) \text{Li}_3\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\frac{1}{4} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 x}\right )+\frac{1}{4} \left (3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 x}\right )\\ &=-\frac{3}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^3 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^3 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+3 x \cosh ^2(x) \log \left (1+e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x^2 \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{2} x \cosh ^2(x) \text{Li}_3\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x \cosh ^2(x) \text{Li}_3\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{4} \cosh ^2(x) \text{Li}_4\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{3}{4} \cosh ^2(x) \text{Li}_4\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{3}{2} x^2 \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^3 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)\\ \end{align*}
Mathematica [A] time = 0.99359, size = 157, normalized size = 0.48 \[ \frac{1}{64} \cosh ^2(x) \sqrt{a \text{sech}^4(x)} \left (96 x^2 \text{PolyLog}\left (2,e^{2 x}\right )+96 \left (x^2-1\right ) \text{PolyLog}\left (2,-e^{-2 x}\right )+96 x \text{PolyLog}\left (3,-e^{-2 x}\right )-96 x \text{PolyLog}\left (3,e^{2 x}\right )+48 \text{PolyLog}\left (4,-e^{-2 x}\right )+48 \text{PolyLog}\left (4,e^{2 x}\right )-32 x^4+96 x^2-64 x^3 \log \left (e^{-2 x}+1\right )+64 x^3 \log \left (1-e^{2 x}\right )-96 x^2 \tanh (x)+32 x^3 \text{sech}^2(x)+192 x \log \left (e^{-2 x}+1\right )+\pi ^4\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^4],x]
[Out]
(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(Pi^4 + 96*x^2 - 32*x^4 + 192*x*Log[1 + E^(-2*x)] - 64*x^3*Log[1 + E^(-2*x)] + 64
*x^3*Log[1 - E^(2*x)] + 96*(-1 + x^2)*PolyLog[2, -E^(-2*x)] + 96*x^2*PolyLog[2, E^(2*x)] + 96*x*PolyLog[3, -E^
(-2*x)] - 96*x*PolyLog[3, E^(2*x)] + 48*PolyLog[4, -E^(-2*x)] + 48*PolyLog[4, E^(2*x)] + 32*x^3*Sech[x]^2 - 96
*x^2*Tanh[x]))/64
________________________________________________________________________________________
Maple [B] time = 0.085, size = 602, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x)
[Out]
(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*x^2*(2*x*exp(2*x)+3*exp(2*x)+3)-3*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2
)*exp(-2*x)*(exp(2*x)+1)^2*x^2+3*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*ln(exp(2*x)+1)+3
/2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(2,-exp(2*x))+(a*exp(4*x)/(exp(2*x)+1)^4)
^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x^3*ln(exp(x)+1)+3*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*
x^2*polylog(2,-exp(x))-6*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*polylog(3,-exp(x))+6*(a*
exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(4,-exp(x))+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*e
xp(-2*x)*(exp(2*x)+1)^2*x^3*ln(1-exp(x))+3*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x^2*poly
log(2,exp(x))-6*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*polylog(3,exp(x))+6*(a*exp(4*x)/(
exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(4,exp(x))-(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(e
xp(2*x)+1)^2*x^3*ln(exp(2*x)+1)-3/2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x^2*polylog(2,-
exp(2*x))+3/2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*polylog(3,-exp(2*x))-3/4*(a*exp(4*x
)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(4,-exp(2*x))
________________________________________________________________________________________
Maxima [A] time = 1.81104, size = 279, normalized size = 0.86 \begin{align*} -3 \, \sqrt{a} x^{2} - \frac{1}{3} \,{\left (4 \, x^{3} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 6 \, x^{2}{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) - 6 \, x{\rm Li}_{3}(-e^{\left (2 \, x\right )}) + 3 \,{\rm Li}_{4}(-e^{\left (2 \, x\right )})\right )} \sqrt{a} +{\left (x^{3} \log \left (e^{x} + 1\right ) + 3 \, x^{2}{\rm Li}_2\left (-e^{x}\right ) - 6 \, x{\rm Li}_{3}(-e^{x}) + 6 \,{\rm Li}_{4}(-e^{x})\right )} \sqrt{a} +{\left (x^{3} \log \left (-e^{x} + 1\right ) + 3 \, x^{2}{\rm Li}_2\left (e^{x}\right ) - 6 \, x{\rm Li}_{3}(e^{x}) + 6 \,{\rm Li}_{4}(e^{x})\right )} \sqrt{a} + \frac{3}{2} \,{\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} \sqrt{a} + \frac{3 \, \sqrt{a} x^{2} +{\left (2 \, \sqrt{a} x^{3} + 3 \, \sqrt{a} x^{2}\right )} e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="maxima")
[Out]
-3*sqrt(a)*x^2 - 1/3*(4*x^3*log(e^(2*x) + 1) + 6*x^2*dilog(-e^(2*x)) - 6*x*polylog(3, -e^(2*x)) + 3*polylog(4,
-e^(2*x)))*sqrt(a) + (x^3*log(e^x + 1) + 3*x^2*dilog(-e^x) - 6*x*polylog(3, -e^x) + 6*polylog(4, -e^x))*sqrt(
a) + (x^3*log(-e^x + 1) + 3*x^2*dilog(e^x) - 6*x*polylog(3, e^x) + 6*polylog(4, e^x))*sqrt(a) + 3/2*(2*x*log(e
^(2*x) + 1) + dilog(-e^(2*x)))*sqrt(a) + (3*sqrt(a)*x^2 + (2*sqrt(a)*x^3 + 3*sqrt(a)*x^2)*e^(2*x))/(e^(4*x) +
2*e^(2*x) + 1)
________________________________________________________________________________________
Fricas [C] time = 3.39832, size = 13428, normalized size = 41.19 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="fricas")
[Out]
(6*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x
)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 +
(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 +
cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(
4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(4, cosh(x) + sinh(x)) - 6*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x
)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x)
+ 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(
x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*
x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(4, I*cosh
(x) + I*sinh(x)) - 6*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x)
+ cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^
2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)
^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) +
4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(4, -I*cosh(x) - I*sinh(x)) + 6*((e^(4*x) + 2*e^(2*x)
+ 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3
*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2
+ 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cos
h(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^
(2*x)*polylog(4, -cosh(x) - sinh(x)) - 6*(x*cosh(x)^4 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)
*e^(4*x) + 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)
*e^(4*x) + 2*(3*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cos
h(x)^4 + 2*x*cosh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*co
sh(x)^3 + x*cosh(x))*e^(2*x))*sinh(x) + x)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*p
olylog(3, cosh(x) + sinh(x)) + 6*(x*cosh(x)^4 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(4*x)
+ 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)*e^(4*x)
+ 2*(3*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cosh(x)^4 +
2*x*cosh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*cosh(x)^3
+ x*cosh(x))*e^(2*x))*sinh(x) + x)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3
, I*cosh(x) + I*sinh(x)) + 6*(x*cosh(x)^4 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(4*x) + 2
*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)*e^(4*x) + 2
*(3*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cosh(x)^4 + 2*x
*cosh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*cosh(x)^3 + x*
cosh(x))*e^(2*x))*sinh(x) + x)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, -I
*cosh(x) - I*sinh(x)) - 6*(x*cosh(x)^4 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(4*x) + 2*x*
cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)*e^(4*x) + 2*(3
*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cosh(x)^4 + 2*x*co
sh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*cosh(x)^3 + x*cos
h(x))*e^(2*x))*sinh(x) + x)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, -cosh
(x) - sinh(x)) - (3*x^2*cosh(x)^4 + 3*(x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sinh(x)^4 + 12*(x^2*cosh(x)*e^(4*x)
+ 2*x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x)^3 - (2*x^3 - 3*x^2)*cosh(x)^2 + (18*x^2*cosh(x)^2 - 2*x^3 + 3*x
^2 + (18*x^2*cosh(x)^2 - 2*x^3 + 3*x^2)*e^(4*x) + 2*(18*x^2*cosh(x)^2 - 2*x^3 + 3*x^2)*e^(2*x))*sinh(x)^2 - 3*
(x^2*cosh(x)^4 + (x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sinh(x)^4 + 2*x^2*cosh(x)^2 + 4*(x^2*cosh(x)*e^(4*x) + 2*
x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x)^3 + 2*(3*x^2*cosh(x)^2 + x^2 + (3*x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*
(3*x^2*cosh(x)^2 + x^2)*e^(2*x))*sinh(x)^2 + x^2 + (x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*(x^2*co
sh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(2*x) + 4*(x^2*cosh(x)^3 + x^2*cosh(x) + (x^2*cosh(x)^3 + x^2*cosh(x))*e^(4
*x) + 2*(x^2*cosh(x)^3 + x^2*cosh(x))*e^(2*x))*sinh(x))*dilog(cosh(x) + sinh(x)) + 3*((x^2 - 1)*cosh(x)^4 + (x
^2 + (x^2 - 1)*e^(4*x) + 2*(x^2 - 1)*e^(2*x) - 1)*sinh(x)^4 + 4*((x^2 - 1)*cosh(x)*e^(4*x) + 2*(x^2 - 1)*cosh(
x)*e^(2*x) + (x^2 - 1)*cosh(x))*sinh(x)^3 + 2*(x^2 - 1)*cosh(x)^2 + 2*(3*(x^2 - 1)*cosh(x)^2 + x^2 + (3*(x^2 -
1)*cosh(x)^2 + x^2 - 1)*e^(4*x) + 2*(3*(x^2 - 1)*cosh(x)^2 + x^2 - 1)*e^(2*x) - 1)*sinh(x)^2 + x^2 + ((x^2 -
1)*cosh(x)^4 + 2*(x^2 - 1)*cosh(x)^2 + x^2 - 1)*e^(4*x) + 2*((x^2 - 1)*cosh(x)^4 + 2*(x^2 - 1)*cosh(x)^2 + x^2
- 1)*e^(2*x) + 4*((x^2 - 1)*cosh(x)^3 + (x^2 - 1)*cosh(x) + ((x^2 - 1)*cosh(x)^3 + (x^2 - 1)*cosh(x))*e^(4*x)
+ 2*((x^2 - 1)*cosh(x)^3 + (x^2 - 1)*cosh(x))*e^(2*x))*sinh(x) - 1)*dilog(I*cosh(x) + I*sinh(x)) + 3*((x^2 -
1)*cosh(x)^4 + (x^2 + (x^2 - 1)*e^(4*x) + 2*(x^2 - 1)*e^(2*x) - 1)*sinh(x)^4 + 4*((x^2 - 1)*cosh(x)*e^(4*x) +
2*(x^2 - 1)*cosh(x)*e^(2*x) + (x^2 - 1)*cosh(x))*sinh(x)^3 + 2*(x^2 - 1)*cosh(x)^2 + 2*(3*(x^2 - 1)*cosh(x)^2
+ x^2 + (3*(x^2 - 1)*cosh(x)^2 + x^2 - 1)*e^(4*x) + 2*(3*(x^2 - 1)*cosh(x)^2 + x^2 - 1)*e^(2*x) - 1)*sinh(x)^2
+ x^2 + ((x^2 - 1)*cosh(x)^4 + 2*(x^2 - 1)*cosh(x)^2 + x^2 - 1)*e^(4*x) + 2*((x^2 - 1)*cosh(x)^4 + 2*(x^2 - 1
)*cosh(x)^2 + x^2 - 1)*e^(2*x) + 4*((x^2 - 1)*cosh(x)^3 + (x^2 - 1)*cosh(x) + ((x^2 - 1)*cosh(x)^3 + (x^2 - 1)
*cosh(x))*e^(4*x) + 2*((x^2 - 1)*cosh(x)^3 + (x^2 - 1)*cosh(x))*e^(2*x))*sinh(x) - 1)*dilog(-I*cosh(x) - I*sin
h(x)) - 3*(x^2*cosh(x)^4 + (x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sinh(x)^4 + 2*x^2*cosh(x)^2 + 4*(x^2*cosh(x)*e^
(4*x) + 2*x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x)^3 + 2*(3*x^2*cosh(x)^2 + x^2 + (3*x^2*cosh(x)^2 + x^2)*e^
(4*x) + 2*(3*x^2*cosh(x)^2 + x^2)*e^(2*x))*sinh(x)^2 + x^2 + (x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(4*x) +
2*(x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(2*x) + 4*(x^2*cosh(x)^3 + x^2*cosh(x) + (x^2*cosh(x)^3 + x^2*cos
h(x))*e^(4*x) + 2*(x^2*cosh(x)^3 + x^2*cosh(x))*e^(2*x))*sinh(x))*dilog(-cosh(x) - sinh(x)) + (3*x^2*cosh(x)^4
- (2*x^3 - 3*x^2)*cosh(x)^2)*e^(4*x) + 2*(3*x^2*cosh(x)^4 - (2*x^3 - 3*x^2)*cosh(x)^2)*e^(2*x) - (x^3*cosh(x)
^4 + 2*x^3*cosh(x)^2 + (x^3*e^(4*x) + 2*x^3*e^(2*x) + x^3)*sinh(x)^4 + 4*(x^3*cosh(x)*e^(4*x) + 2*x^3*cosh(x)*
e^(2*x) + x^3*cosh(x))*sinh(x)^3 + x^3 + 2*(3*x^3*cosh(x)^2 + x^3 + (3*x^3*cosh(x)^2 + x^3)*e^(4*x) + 2*(3*x^3
*cosh(x)^2 + x^3)*e^(2*x))*sinh(x)^2 + (x^3*cosh(x)^4 + 2*x^3*cosh(x)^2 + x^3)*e^(4*x) + 2*(x^3*cosh(x)^4 + 2*
x^3*cosh(x)^2 + x^3)*e^(2*x) + 4*(x^3*cosh(x)^3 + x^3*cosh(x) + (x^3*cosh(x)^3 + x^3*cosh(x))*e^(4*x) + 2*(x^3
*cosh(x)^3 + x^3*cosh(x))*e^(2*x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + ((x^3 - 3*x)*cosh(x)^4 + (x^3 + (x^3
- 3*x)*e^(4*x) + 2*(x^3 - 3*x)*e^(2*x) - 3*x)*sinh(x)^4 + 4*((x^3 - 3*x)*cosh(x)*e^(4*x) + 2*(x^3 - 3*x)*cosh(
x)*e^(2*x) + (x^3 - 3*x)*cosh(x))*sinh(x)^3 + x^3 + 2*(x^3 - 3*x)*cosh(x)^2 + 2*(x^3 + 3*(x^3 - 3*x)*cosh(x)^2
+ (x^3 + 3*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(4*x) + 2*(x^3 + 3*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(2*x) - 3*x)*sinh
(x)^2 + ((x^3 - 3*x)*cosh(x)^4 + x^3 + 2*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(4*x) + 2*((x^3 - 3*x)*cosh(x)^4 + x^3
+ 2*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(2*x) + 4*((x^3 - 3*x)*cosh(x)^3 + (x^3 - 3*x)*cosh(x) + ((x^3 - 3*x)*cosh
(x)^3 + (x^3 - 3*x)*cosh(x))*e^(4*x) + 2*((x^3 - 3*x)*cosh(x)^3 + (x^3 - 3*x)*cosh(x))*e^(2*x))*sinh(x) - 3*x)
*log(I*cosh(x) + I*sinh(x) + 1) + ((x^3 - 3*x)*cosh(x)^4 + (x^3 + (x^3 - 3*x)*e^(4*x) + 2*(x^3 - 3*x)*e^(2*x)
- 3*x)*sinh(x)^4 + 4*((x^3 - 3*x)*cosh(x)*e^(4*x) + 2*(x^3 - 3*x)*cosh(x)*e^(2*x) + (x^3 - 3*x)*cosh(x))*sinh(
x)^3 + x^3 + 2*(x^3 - 3*x)*cosh(x)^2 + 2*(x^3 + 3*(x^3 - 3*x)*cosh(x)^2 + (x^3 + 3*(x^3 - 3*x)*cosh(x)^2 - 3*x
)*e^(4*x) + 2*(x^3 + 3*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(2*x) - 3*x)*sinh(x)^2 + ((x^3 - 3*x)*cosh(x)^4 + x^3 +
2*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(4*x) + 2*((x^3 - 3*x)*cosh(x)^4 + x^3 + 2*(x^3 - 3*x)*cosh(x)^2 - 3*x)*e^(2*
x) + 4*((x^3 - 3*x)*cosh(x)^3 + (x^3 - 3*x)*cosh(x) + ((x^3 - 3*x)*cosh(x)^3 + (x^3 - 3*x)*cosh(x))*e^(4*x) +
2*((x^3 - 3*x)*cosh(x)^3 + (x^3 - 3*x)*cosh(x))*e^(2*x))*sinh(x) - 3*x)*log(-I*cosh(x) - I*sinh(x) + 1) - (x^3
*cosh(x)^4 + 2*x^3*cosh(x)^2 + (x^3*e^(4*x) + 2*x^3*e^(2*x) + x^3)*sinh(x)^4 + 4*(x^3*cosh(x)*e^(4*x) + 2*x^3*
cosh(x)*e^(2*x) + x^3*cosh(x))*sinh(x)^3 + x^3 + 2*(3*x^3*cosh(x)^2 + x^3 + (3*x^3*cosh(x)^2 + x^3)*e^(4*x) +
2*(3*x^3*cosh(x)^2 + x^3)*e^(2*x))*sinh(x)^2 + (x^3*cosh(x)^4 + 2*x^3*cosh(x)^2 + x^3)*e^(4*x) + 2*(x^3*cosh(x
)^4 + 2*x^3*cosh(x)^2 + x^3)*e^(2*x) + 4*(x^3*cosh(x)^3 + x^3*cosh(x) + (x^3*cosh(x)^3 + x^3*cosh(x))*e^(4*x)
+ 2*(x^3*cosh(x)^3 + x^3*cosh(x))*e^(2*x))*sinh(x))*log(-cosh(x) - sinh(x) + 1) + 2*(6*x^2*cosh(x)^3 - (2*x^3
- 3*x^2)*cosh(x) + (6*x^2*cosh(x)^3 - (2*x^3 - 3*x^2)*cosh(x))*e^(4*x) + 2*(6*x^2*cosh(x)^3 - (2*x^3 - 3*x^2)*
cosh(x))*e^(2*x))*sinh(x))*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x))/(4*cosh(x)*e^(2*
x)*sinh(x)^3 + e^(2*x)*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^(2*x)*sin
h(x) + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{a \operatorname{sech}^{4}{\left (x \right )}} \operatorname{csch}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*csch(x)*sech(x)*(a*sech(x)**4)**(1/2),x)
[Out]
Integral(x**3*sqrt(a*sech(x)**4)*csch(x)*sech(x), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \operatorname{sech}\left (x\right )^{4}} x^{3} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(a*sech(x)^4)*x^3*csch(x)*sech(x), x)