### 3.850 $$\int x^3 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^2(x)} \, dx$$

Optimal. Leaf size=287 $-3 x^2 \cosh (x) \text{PolyLog}\left (2,-e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{PolyLog}\left (2,e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{PolyLog}\left (2,-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{PolyLog}\left (2,i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{PolyLog}\left (3,-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{PolyLog}\left (3,e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{PolyLog}\left (3,-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{PolyLog}\left (3,i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 \cosh (x) \text{PolyLog}\left (4,-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 \cosh (x) \text{PolyLog}\left (4,e^x\right ) \sqrt{a \text{sech}^2(x)}+x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt{a \text{sech}^2(x)}$

[Out]

x^3*Sqrt[a*Sech[x]^2] - 6*x^2*ArcTan[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x^3*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x
]^2] - 3*x^2*Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + (6*I)*x*Cosh[x]*PolyLog[2, (-I)*E^x]*Sqrt[a*Sech[x]^
2] - (6*I)*x*Cosh[x]*PolyLog[2, I*E^x]*Sqrt[a*Sech[x]^2] + 3*x^2*Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2] + 6
*x*Cosh[x]*PolyLog[3, -E^x]*Sqrt[a*Sech[x]^2] - (6*I)*Cosh[x]*PolyLog[3, (-I)*E^x]*Sqrt[a*Sech[x]^2] + (6*I)*C
osh[x]*PolyLog[3, I*E^x]*Sqrt[a*Sech[x]^2] - 6*x*Cosh[x]*PolyLog[3, E^x]*Sqrt[a*Sech[x]^2] - 6*Cosh[x]*PolyLog
[4, -E^x]*Sqrt[a*Sech[x]^2] + 6*Cosh[x]*PolyLog[4, E^x]*Sqrt[a*Sech[x]^2]

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Rubi [A]  time = 0.673415, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.722, Rules used = {6720, 2622, 321, 207, 5462, 14, 6273, 4182, 2531, 6609, 2282, 6589, 4180} $-3 x^2 \cosh (x) \text{PolyLog}\left (2,-e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{PolyLog}\left (2,e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{PolyLog}\left (2,-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{PolyLog}\left (2,i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{PolyLog}\left (3,-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{PolyLog}\left (3,e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{PolyLog}\left (3,-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{PolyLog}\left (3,i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 \cosh (x) \text{PolyLog}\left (4,-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 \cosh (x) \text{PolyLog}\left (4,e^x\right ) \sqrt{a \text{sech}^2(x)}+x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt{a \text{sech}^2(x)}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^2],x]

[Out]

x^3*Sqrt[a*Sech[x]^2] - 6*x^2*ArcTan[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x^3*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x
]^2] - 3*x^2*Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + (6*I)*x*Cosh[x]*PolyLog[2, (-I)*E^x]*Sqrt[a*Sech[x]^
2] - (6*I)*x*Cosh[x]*PolyLog[2, I*E^x]*Sqrt[a*Sech[x]^2] + 3*x^2*Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2] + 6
*x*Cosh[x]*PolyLog[3, -E^x]*Sqrt[a*Sech[x]^2] - (6*I)*Cosh[x]*PolyLog[3, (-I)*E^x]*Sqrt[a*Sech[x]^2] + (6*I)*C
osh[x]*PolyLog[3, I*E^x]*Sqrt[a*Sech[x]^2] - 6*x*Cosh[x]*PolyLog[3, E^x]*Sqrt[a*Sech[x]^2] - 6*Cosh[x]*PolyLog
[4, -E^x]*Sqrt[a*Sech[x]^2] + 6*Cosh[x]*PolyLog[4, E^x]*Sqrt[a*Sech[x]^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 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

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 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 6273

Int[((a_.) + ArcTanh[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcTan
h[u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/(1 - u^2), x],
x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(m
+ 1), u, x] && FalseQ[PowerVariableExpn[u, m + 1, x]]

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 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rubi steps

\begin{align*} \int x^3 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^2(x)} \, dx &=\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^3 \text{csch}(x) \text{sech}^2(x) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-x^3 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \left (-\tanh ^{-1}(\cosh (x))+\text{sech}(x)\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-x^3 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \left (-x^2 \tanh ^{-1}(\cosh (x))+x^2 \text{sech}(x)\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-x^3 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}+\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \tanh ^{-1}(\cosh (x)) \, dx-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \text{sech}(x) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}+\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \log \left (1-i e^x\right ) \, dx-\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \log \left (1+i e^x\right ) \, dx+\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^3 \text{csch}(x) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_2\left (-i e^x\right ) \, dx+\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_2\left (i e^x\right ) \, dx-\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \log \left (1-e^x\right ) \, dx+\left (3 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x^2 \log \left (1+e^x\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^x\right )+\left (6 i \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^x\right )+\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \text{Li}_2\left (-e^x\right ) \, dx-\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \text{Li}_2\left (e^x\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{Li}_3\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{Li}_3\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{Li}_3\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{Li}_3\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_3\left (-e^x\right ) \, dx+\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{Li}_3\left (e^x\right ) \, dx\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{Li}_3\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{Li}_3\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{Li}_3\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{Li}_3\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^x\right )+\left (6 \cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^x\right )\\ &=x^3 \sqrt{a \text{sech}^2(x)}-6 x^2 \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x^3 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-3 x^2 \cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i x \cosh (x) \text{Li}_2\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i x \cosh (x) \text{Li}_2\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}+3 x^2 \cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}+6 x \cosh (x) \text{Li}_3\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}-6 i \cosh (x) \text{Li}_3\left (-i e^x\right ) \sqrt{a \text{sech}^2(x)}+6 i \cosh (x) \text{Li}_3\left (i e^x\right ) \sqrt{a \text{sech}^2(x)}-6 x \cosh (x) \text{Li}_3\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-6 \cosh (x) \text{Li}_4\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+6 \cosh (x) \text{Li}_4\left (e^x\right ) \sqrt{a \text{sech}^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.631798, size = 249, normalized size = 0.87 $\frac{1}{8} \sqrt{a \text{sech}^2(x)} \left (24 x^2 \cosh (x) \text{PolyLog}\left (2,-e^{-x}\right )+24 x^2 \cosh (x) \text{PolyLog}\left (2,e^x\right )+48 i x \cosh (x) \text{PolyLog}\left (2,-i e^{-x}\right )-48 i x \cosh (x) \text{PolyLog}\left (2,i e^{-x}\right )+48 x \cosh (x) \text{PolyLog}\left (3,-e^{-x}\right )-48 x \cosh (x) \text{PolyLog}\left (3,e^x\right )+48 i \cosh (x) \text{PolyLog}\left (3,-i e^{-x}\right )-48 i \cosh (x) \text{PolyLog}\left (3,i e^{-x}\right )+48 \cosh (x) \text{PolyLog}\left (4,-e^{-x}\right )+48 \cosh (x) \text{PolyLog}\left (4,e^x\right )+8 x^3-2 x^4 \cosh (x)-8 x^3 \log \left (e^{-x}+1\right ) \cosh (x)+8 x^3 \log \left (1-e^x\right ) \cosh (x)+24 i x^2 \log \left (1-i e^{-x}\right ) \cosh (x)-24 i x^2 \log \left (1+i e^{-x}\right ) \cosh (x)+\pi ^4 \cosh (x)\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^2],x]

[Out]

((8*x^3 + Pi^4*Cosh[x] - 2*x^4*Cosh[x] + (24*I)*x^2*Cosh[x]*Log[1 - I/E^x] - (24*I)*x^2*Cosh[x]*Log[1 + I/E^x]
- 8*x^3*Cosh[x]*Log[1 + E^(-x)] + 8*x^3*Cosh[x]*Log[1 - E^x] + 24*x^2*Cosh[x]*PolyLog[2, -E^(-x)] + (48*I)*x*
Cosh[x]*PolyLog[2, (-I)/E^x] - (48*I)*x*Cosh[x]*PolyLog[2, I/E^x] + 24*x^2*Cosh[x]*PolyLog[2, E^x] + 48*x*Cosh
[x]*PolyLog[3, -E^(-x)] + (48*I)*Cosh[x]*PolyLog[3, (-I)/E^x] - (48*I)*Cosh[x]*PolyLog[3, I/E^x] - 48*x*Cosh[x
]*PolyLog[3, E^x] + 48*Cosh[x]*PolyLog[4, -E^(-x)] + 48*Cosh[x]*PolyLog[4, E^x])*Sqrt[a*Sech[x]^2])/8

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Maple [F]  time = 0.133, size = 0, normalized size = 0. \begin{align*} \int{x}^{3}{\rm csch} \left (x\right ){\rm sech} \left (x\right )\sqrt{a \left ({\rm sech} \left (x\right ) \right ) ^{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*csch(x)*sech(x)*(a*sech(x)^2)^(1/2),x)

[Out]

int(x^3*csch(x)*sech(x)*(a*sech(x)^2)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, \sqrt{a} x^{3} e^{x}}{e^{\left (2 \, x\right )} + 1} -{\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} - 12 \, \sqrt{a} \int \frac{x^{2} e^{x}}{2 \,{\left (e^{\left (2 \, x\right )} + 1\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)^2)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(a)*x^3*e^x/(e^(2*x) + 1) - (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) - 12
*sqrt(a)*integrate(1/2*x^2*e^x/(e^(2*x) + 1), x)

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Fricas [C]  time = 2.51009, size = 3722, normalized size = 12.97 \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)^2)^(1/2),x, algorithm="fricas")

[Out]

(6*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)
*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(4, cosh(x) + sinh(x)) - 6*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2
+ (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*p
olylog(4, -cosh(x) - sinh(x)) - 6*(x*cosh(x)^2 + (x*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^2 + x)*e^(2*x) + 2*(x*
cosh(x)*e^(2*x) + x*cosh(x))*sinh(x) + x)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, cosh(x) + sinh(x))
+ ((6*I*e^(2*x) + 6*I)*sinh(x)^2 + 6*I*cosh(x)^2 + (6*I*cosh(x)^2 + 6*I)*e^(2*x) + (12*I*cosh(x)*e^(2*x) + 12*
I*cosh(x))*sinh(x) + 6*I)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, I*cosh(x) + I*sinh(x)) + ((-6*I*e^(
2*x) - 6*I)*sinh(x)^2 - 6*I*cosh(x)^2 + (-6*I*cosh(x)^2 - 6*I)*e^(2*x) + (-12*I*cosh(x)*e^(2*x) - 12*I*cosh(x)
)*sinh(x) - 6*I)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, -I*cosh(x) - I*sinh(x)) + 6*(x*cosh(x)^2 + (
x*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^2 + x)*e^(2*x) + 2*(x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x) + x)*sqrt(a/(
e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, -cosh(x) - sinh(x)) + (2*x^3*cosh(x)*e^(2*x) + 2*x^3*cosh(x) + 3*(x^2
*cosh(x)^2 + (x^2*e^(2*x) + x^2)*sinh(x)^2 + x^2 + (x^2*cosh(x)^2 + x^2)*e^(2*x) + 2*(x^2*cosh(x)*e^(2*x) + x^
2*cosh(x))*sinh(x))*dilog(cosh(x) + sinh(x)) + (-6*I*x*cosh(x)^2 + (-6*I*x*e^(2*x) - 6*I*x)*sinh(x)^2 + (-6*I*
x*cosh(x)^2 - 6*I*x)*e^(2*x) + (-12*I*x*cosh(x)*e^(2*x) - 12*I*x*cosh(x))*sinh(x) - 6*I*x)*dilog(I*cosh(x) + I
*sinh(x)) + (6*I*x*cosh(x)^2 + (6*I*x*e^(2*x) + 6*I*x)*sinh(x)^2 + (6*I*x*cosh(x)^2 + 6*I*x)*e^(2*x) + (12*I*x
*cosh(x)*e^(2*x) + 12*I*x*cosh(x))*sinh(x) + 6*I*x)*dilog(-I*cosh(x) - I*sinh(x)) - 3*(x^2*cosh(x)^2 + (x^2*e^
(2*x) + x^2)*sinh(x)^2 + x^2 + (x^2*cosh(x)^2 + x^2)*e^(2*x) + 2*(x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x))*
dilog(-cosh(x) - sinh(x)) - (x^3*cosh(x)^2 + x^3 + (x^3*e^(2*x) + x^3)*sinh(x)^2 + (x^3*cosh(x)^2 + x^3)*e^(2*
x) + 2*(x^3*cosh(x)*e^(2*x) + x^3*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + (3*I*x^2*cosh(x)^2 + (3*I*x^2
*e^(2*x) + 3*I*x^2)*sinh(x)^2 + 3*I*x^2 + (3*I*x^2*cosh(x)^2 + 3*I*x^2)*e^(2*x) + (6*I*x^2*cosh(x)*e^(2*x) + 6
*I*x^2*cosh(x))*sinh(x))*log(I*cosh(x) + I*sinh(x) + 1) + (-3*I*x^2*cosh(x)^2 + (-3*I*x^2*e^(2*x) - 3*I*x^2)*s
inh(x)^2 - 3*I*x^2 + (-3*I*x^2*cosh(x)^2 - 3*I*x^2)*e^(2*x) + (-6*I*x^2*cosh(x)*e^(2*x) - 6*I*x^2*cosh(x))*sin
h(x))*log(-I*cosh(x) - I*sinh(x) + 1) + (x^3*cosh(x)^2 + x^3 + (x^3*e^(2*x) + x^3)*sinh(x)^2 + (x^3*cosh(x)^2
+ x^3)*e^(2*x) + 2*(x^3*cosh(x)*e^(2*x) + x^3*cosh(x))*sinh(x))*log(-cosh(x) - sinh(x) + 1) + 2*(x^3*e^(2*x) +
x^3)*sinh(x))*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x)/(2*cosh(x)*e^x*sinh(x) + e^x*sinh(x)^2 + (cosh(x)^2 + 1)
*e^x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \sqrt{a \operatorname{sech}^{2}{\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)**2)**(1/2),x)

[Out]

Integral(x**3*sqrt(a*sech(x)**2)*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 )^{2}} 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)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sech(x)^2)*x^3*csch(x)*sech(x), x)