3.9.50 \(\int x^3 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx\) [850]

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

[Out]

x^3*(a*sech(x)^2)^(1/2)-6*x^2*arctan(exp(x))*cosh(x)*(a*sech(x)^2)^(1/2)-2*x^3*arctanh(exp(x))*cosh(x)*(a*sech
(x)^2)^(1/2)-3*x^2*cosh(x)*polylog(2,-exp(x))*(a*sech(x)^2)^(1/2)+6*I*x*cosh(x)*polylog(2,-I*exp(x))*(a*sech(x
)^2)^(1/2)-6*I*x*cosh(x)*polylog(2,I*exp(x))*(a*sech(x)^2)^(1/2)+3*x^2*cosh(x)*polylog(2,exp(x))*(a*sech(x)^2)
^(1/2)+6*x*cosh(x)*polylog(3,-exp(x))*(a*sech(x)^2)^(1/2)-6*I*cosh(x)*polylog(3,-I*exp(x))*(a*sech(x)^2)^(1/2)
+6*I*cosh(x)*polylog(3,I*exp(x))*(a*sech(x)^2)^(1/2)-6*x*cosh(x)*polylog(3,exp(x))*(a*sech(x)^2)^(1/2)-6*cosh(
x)*polylog(4,-exp(x))*(a*sech(x)^2)^(1/2)+6*cosh(x)*polylog(4,exp(x))*(a*sech(x)^2)^(1/2)

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Rubi [A]
time = 0.44, antiderivative size = 287, normalized size of antiderivative = 1.00, 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 = {6852, 2702, 327, 213, 5570, 14, 6408, 4267, 2611, 6744, 2320, 6724, 4265} \begin {gather*} -6 x^2 \text {ArcTan}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-3 x^2 \text {Li}_2\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+3 x^2 \text {Li}_2\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 i x \text {Li}_2\left (-i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 i x \text {Li}_2\left (i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 x \text {Li}_3\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 x \text {Li}_3\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 i \text {Li}_3\left (-i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 i \text {Li}_3\left (i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-6 \text {Li}_4\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+6 \text {Li}_4\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+x^3 \sqrt {a \text {sech}^2(x)}-2 x^3 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt {a \text {sech}^2(x)} \end {gather*}

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 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 213

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

Rule 327

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2702

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 4265

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]

Rule 4267

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 5570

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 6408

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 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

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 6852

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])

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

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Mathematica [A]
time = 0.40, size = 249, normalized size = 0.87 \begin {gather*} \frac {1}{8} \left (8 x^3+\pi ^4 \cosh (x)-2 x^4 \cosh (x)+24 i x^2 \cosh (x) \log \left (1-i e^{-x}\right )-24 i x^2 \cosh (x) \log \left (1+i e^{-x}\right )-8 x^3 \cosh (x) \log \left (1+e^{-x}\right )+8 x^3 \cosh (x) \log \left (1-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 )+24 x^2 \cosh (x) \text {PolyLog}\left (2,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 x \cosh (x) \text {PolyLog}\left (3,e^x\right )+48 \cosh (x) \text {PolyLog}\left (4,-e^{-x}\right )+48 \cosh (x) \text {PolyLog}\left (4,e^x\right )\right ) \sqrt {a \text {sech}^2(x)} \end {gather*}

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 = 1.47, size = 0, normalized size = 0.00 \[\int x^{3} \mathrm {csch}\left (x \right ) \mathrm {sech}\left (x \right ) \sqrt {a \mathrm {sech}\left (x \right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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 [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (229) = 458\).
time = 0.41, size = 1202, normalized size = 4.19 \begin {gather*} \text {Too large to display} \end {gather*}

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) - I)*sinh(x)^2 - I*cosh(x)^2 + (-I*cosh(x)^2 - I)*e^(2*x) + 2*(-I*cosh(x)*e^(2*x) - I*cosh(x)
)*sinh(x) - 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) + I)*si
nh(x)^2 + I*cosh(x)^2 + (I*cosh(x)^2 + I)*e^(2*x) + 2*(I*cosh(x)*e^(2*x) + I*cosh(x))*sinh(x) + 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(co
sh(x) + sinh(x)) - 6*(I*x*cosh(x)^2 + (I*x*e^(2*x) + I*x)*sinh(x)^2 + (I*x*cosh(x)^2 + I*x)*e^(2*x) + 2*(I*x*c
osh(x)*e^(2*x) + I*x*cosh(x))*sinh(x) + I*x)*dilog(I*cosh(x) + I*sinh(x)) - 6*(-I*x*cosh(x)^2 + (-I*x*e^(2*x)
- I*x)*sinh(x)^2 + (-I*x*cosh(x)^2 - I*x)*e^(2*x) + 2*(-I*x*cosh(x)*e^(2*x) - I*x*cosh(x))*sinh(x) - I*x)*dilo
g(-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 + (-I*x^2*e^(2*x) - I*x^2)*sinh(x)^2 - I*x^2 + (-I*x^2*cosh(x)^2
- I*x^2)*e^(2*x) + 2*(-I*x^2*cosh(x)*e^(2*x) - I*x^2*cosh(x))*sinh(x))*log(I*cosh(x) + I*sinh(x) + 1) - 3*(I*x
^2*cosh(x)^2 + (I*x^2*e^(2*x) + I*x^2)*sinh(x)^2 + I*x^2 + (I*x^2*cosh(x)^2 + I*x^2)*e^(2*x) + 2*(I*x^2*cosh(x
)*e^(2*x) + I*x^2*cosh(x))*sinh(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) - s
inh(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \sqrt {a \operatorname {sech}^{2}{\left (x \right )}} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}\, dx \end {gather*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*(a/cosh(x)^2)^(1/2))/(cosh(x)*sinh(x)),x)

[Out]

int((x^3*(a/cosh(x)^2)^(1/2))/(cosh(x)*sinh(x)), x)

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