3.9.49 \(\int x^2 \text {csch}(x) \text {sech}(x) \sqrt {a \text {sech}^2(x)} \, dx\) [849]

Optimal. Leaf size=187 \[ x^2 \sqrt {a \text {sech}^2(x)}-4 x \text {ArcTan}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^2 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \text {PolyLog}\left (2,-e^x\right ) \sqrt {a \text {sech}^2(x)}+2 i \cosh (x) \text {PolyLog}\left (2,-i e^x\right ) \sqrt {a \text {sech}^2(x)}-2 i \cosh (x) \text {PolyLog}\left (2,i e^x\right ) \sqrt {a \text {sech}^2(x)}+2 x \cosh (x) \text {PolyLog}\left (2,e^x\right ) \sqrt {a \text {sech}^2(x)}+2 \cosh (x) \text {PolyLog}\left (3,-e^x\right ) \sqrt {a \text {sech}^2(x)}-2 \cosh (x) \text {PolyLog}\left (3,e^x\right ) \sqrt {a \text {sech}^2(x)} \]

[Out]

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

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Rubi [A]
time = 0.36, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {6852, 2702, 327, 213, 5570, 14, 6408, 4267, 2611, 2320, 6724, 4265, 2317, 2438} \begin {gather*} -4 x \text {ArcTan}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \text {Li}_2\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+2 x \text {Li}_2\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+2 i \text {Li}_2\left (-i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 i \text {Li}_2\left (i e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+2 \text {Li}_3\left (-e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 \text {Li}_3\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+x^2 \sqrt {a \text {sech}^2(x)}-2 x^2 \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt {a \text {sech}^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^2*Sqrt[a*Sech[x]^2] - 4*x*ArcTan[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x^2*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x]^
2] - 2*x*Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + (2*I)*Cosh[x]*PolyLog[2, (-I)*E^x]*Sqrt[a*Sech[x]^2] - (
2*I)*Cosh[x]*PolyLog[2, I*E^x]*Sqrt[a*Sech[x]^2] + 2*x*Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2] + 2*Cosh[x]*P
olyLog[3, -E^x]*Sqrt[a*Sech[x]^2] - 2*Cosh[x]*PolyLog[3, 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 2317

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 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 2438

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 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 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^2 \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^2 \text {csch}(x) \text {sech}^2(x) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-x^2 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x \left (-\tanh ^{-1}(\cosh (x))+\text {sech}(x)\right ) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-x^2 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}-\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \left (-x \tanh ^{-1}(\cosh (x))+x \text {sech}(x)\right ) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-x^2 \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt {a \text {sech}^2(x)}+\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x \tanh ^{-1}(\cosh (x)) \, dx-\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x \text {sech}(x) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-4 x \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+\left (2 i \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \log \left (1-i e^x\right ) \, dx-\left (2 i \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \log \left (1+i e^x\right ) \, dx+\left (\cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x^2 \text {csch}(x) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-4 x \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^2 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}+\left (2 i \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^x\right )-\left (2 i \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^x\right )-\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x \log \left (1-e^x\right ) \, dx+\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int x \log \left (1+e^x\right ) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-4 x \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^2 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \text {Li}_2\left (-e^x\right ) \sqrt {a \text {sech}^2(x)}+2 i \cosh (x) \text {Li}_2\left (-i e^x\right ) \sqrt {a \text {sech}^2(x)}-2 i \cosh (x) \text {Li}_2\left (i e^x\right ) \sqrt {a \text {sech}^2(x)}+2 x \cosh (x) \text {Li}_2\left (e^x\right ) \sqrt {a \text {sech}^2(x)}+\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \text {Li}_2\left (-e^x\right ) \, dx-\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \int \text {Li}_2\left (e^x\right ) \, dx\\ &=x^2 \sqrt {a \text {sech}^2(x)}-4 x \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^2 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \text {Li}_2\left (-e^x\right ) \sqrt {a \text {sech}^2(x)}+2 i \cosh (x) \text {Li}_2\left (-i e^x\right ) \sqrt {a \text {sech}^2(x)}-2 i \cosh (x) \text {Li}_2\left (i e^x\right ) \sqrt {a \text {sech}^2(x)}+2 x \cosh (x) \text {Li}_2\left (e^x\right ) \sqrt {a \text {sech}^2(x)}+\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^x\right )-\left (2 \cosh (x) \sqrt {a \text {sech}^2(x)}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^x\right )\\ &=x^2 \sqrt {a \text {sech}^2(x)}-4 x \tan ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x^2 \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt {a \text {sech}^2(x)}-2 x \cosh (x) \text {Li}_2\left (-e^x\right ) \sqrt {a \text {sech}^2(x)}+2 i \cosh (x) \text {Li}_2\left (-i e^x\right ) \sqrt {a \text {sech}^2(x)}-2 i \cosh (x) \text {Li}_2\left (i e^x\right ) \sqrt {a \text {sech}^2(x)}+2 x \cosh (x) \text {Li}_2\left (e^x\right ) \sqrt {a \text {sech}^2(x)}+2 \cosh (x) \text {Li}_3\left (-e^x\right ) \sqrt {a \text {sech}^2(x)}-2 \cosh (x) \text {Li}_3\left (e^x\right ) \sqrt {a \text {sech}^2(x)}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 154, normalized size = 0.82 \begin {gather*} \left (x^2+2 i x \cosh (x) \left (\log \left (1-i e^{-x}\right )-\log \left (1+i e^{-x}\right )\right )+x^2 \cosh (x) \left (\log \left (1-e^{-x}\right )-\log \left (1+e^{-x}\right )\right )+2 i \cosh (x) \left (\text {PolyLog}\left (2,-i e^{-x}\right )-\text {PolyLog}\left (2,i e^{-x}\right )\right )+2 x \cosh (x) \left (\text {PolyLog}\left (2,-e^{-x}\right )-\text {PolyLog}\left (2,e^{-x}\right )\right )+2 \cosh (x) \left (\text {PolyLog}\left (3,-e^{-x}\right )-\text {PolyLog}\left (3,e^{-x}\right )\right )\right ) \sqrt {a \text {sech}^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 1.58, size = 0, normalized size = 0.00 \[\int x^{2} \mathrm {csch}\left (x \right ) \mathrm {sech}\left (x \right ) \sqrt {a \mathrm {sech}\left (x \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*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^2*csch(x)*sech(x)*(a*sech(x)^2)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(a)*x^2*e^x/(e^(2*x) + 1) - (x^2*log(e^x + 1) + 2*x*dilog(-e^x) - 2*polylog(3, -e^x))*sqrt(a) + (x^2*log
(-e^x + 1) + 2*x*dilog(e^x) - 2*polylog(3, e^x))*sqrt(a) - 4*sqrt(a)*integrate(x*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. 786 vs. \(2 (149) = 298\).
time = 0.41, size = 786, normalized size = 4.20 \begin {gather*} -\frac {2 \, {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} {\rm polylog}\left (3, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x} {\rm polylog}\left (3, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (2 \, x^{2} \cosh \left (x\right ) e^{\left (2 \, x\right )} + 2 \, x^{2} \cosh \left (x\right ) + 2 \, {\left (x \cosh \left (x\right )^{2} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{2} + x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) + x\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 2 \, {\left ({\left (i \, e^{\left (2 \, x\right )} + i\right )} \sinh \left (x\right )^{2} + i \, \cosh \left (x\right )^{2} + {\left (i \, \cosh \left (x\right )^{2} + i\right )} e^{\left (2 \, x\right )} + 2 \, {\left (i \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + i \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + i\right )} {\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) - 2 \, {\left ({\left (-i \, e^{\left (2 \, x\right )} - i\right )} \sinh \left (x\right )^{2} - i \, \cosh \left (x\right )^{2} + {\left (-i \, \cosh \left (x\right )^{2} - i\right )} e^{\left (2 \, x\right )} + 2 \, {\left (-i \, \cosh \left (x\right ) e^{\left (2 \, x\right )} - i \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - i\right )} {\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) - 2 \, {\left (x \cosh \left (x\right )^{2} + {\left (x e^{\left (2 \, x\right )} + x\right )} \sinh \left (x\right )^{2} + {\left (x \cosh \left (x\right )^{2} + x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x \cosh \left (x\right ) e^{\left (2 \, x\right )} + x \cosh \left (x\right )\right )} \sinh \left (x\right ) + x\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x^{2} \cosh \left (x\right )^{2} + {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} \sinh \left (x\right )^{2} + x^{2} + {\left (x^{2} \cosh \left (x\right )^{2} + x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} \cosh \left (x\right ) e^{\left (2 \, x\right )} + x^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) - 2 \, {\left (-i \, x \cosh \left (x\right )^{2} + {\left (-i \, x e^{\left (2 \, x\right )} - i \, x\right )} \sinh \left (x\right )^{2} + {\left (-i \, x \cosh \left (x\right )^{2} - i \, x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (-i \, x \cosh \left (x\right ) e^{\left (2 \, x\right )} - i \, x \cosh \left (x\right )\right )} \sinh \left (x\right ) - i \, x\right )} \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) - 2 \, {\left (i \, x \cosh \left (x\right )^{2} + {\left (i \, x e^{\left (2 \, x\right )} + i \, x\right )} \sinh \left (x\right )^{2} + {\left (i \, x \cosh \left (x\right )^{2} + i \, x\right )} e^{\left (2 \, x\right )} + 2 \, {\left (i \, x \cosh \left (x\right ) e^{\left (2 \, x\right )} + i \, x \cosh \left (x\right )\right )} \sinh \left (x\right ) + i \, x\right )} \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) + {\left (x^{2} \cosh \left (x\right )^{2} + {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} \sinh \left (x\right )^{2} + x^{2} + {\left (x^{2} \cosh \left (x\right )^{2} + x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{2} \cosh \left (x\right ) e^{\left (2 \, x\right )} + x^{2} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right ) + 2 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right ) + e^{x} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} + 1\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*csch(x)*sech(x)*(a*sech(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-(2*((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(3, cosh(x) + sinh(x)) - 2*((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(3, -cosh(x) - sinh(x)) - (2*x^2*cosh(x)*e^(2*x) + 2*x^2*cosh(x) + 2*(x*cosh(x)^2 + (x*e^(2*x) + x)*sin
h(x)^2 + (x*cosh(x)^2 + x)*e^(2*x) + 2*(x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x) + x)*dilog(cosh(x) + sinh(x)) -
2*((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))*si
nh(x) + I)*dilog(I*cosh(x) + I*sinh(x)) - 2*((-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)*dilog(-I*cosh(x) - I*sinh(x)) - 2*(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)*dilog(-cosh(x
) - sinh(x)) - (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*c
osh(x)*e^(2*x) + x^2*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - 2*(-I*x*cosh(x)^2 + (-I*x*e^(2*x) - I*x)*s
inh(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)*log(I*cosh(x
) + I*sinh(x) + 1) - 2*(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)*log(-I*cosh(x) - I*sinh(x) + 1) + (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))*log(-
cosh(x) - sinh(x) + 1) + 2*(x^2*e^(2*x) + x^2)*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^{2} \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**2*csch(x)*sech(x)*(a*sech(x)**2)**(1/2),x)

[Out]

Integral(x**2*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^2*csch(x)*sech(x)*(a*sech(x)^2)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\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^2*(a/cosh(x)^2)^(1/2))/(cosh(x)*sinh(x)),x)

[Out]

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

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