3.9.44 \(\int \frac {x^3 \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx\) [844]

Optimal. Leaf size=150 \[ -\frac {2 x^3 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {3 x^2 \text {PolyLog}\left (2,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {3 x^2 \text {PolyLog}\left (2,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 x \text {PolyLog}\left (3,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 x \text {PolyLog}\left (3,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 \text {PolyLog}\left (4,-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 \text {PolyLog}\left (4,e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \]

[Out]

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

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Rubi [A]
time = 0.44, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6852, 4267, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {3 x^2 \text {Li}_2\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {3 x^2 \text {Li}_2\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 x \text {Li}_3\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 x \text {Li}_3\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {6 \text {Li}_4\left (-e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}+\frac {6 \text {Li}_4\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}}-\frac {2 x^3 \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\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]

(-2*x^3*ArcTanh[E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (3*x^2*PolyLog[2, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (3*x^2*
PolyLog[2, E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (6*x*PolyLog[3, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (6*x*PolyLog[3
, E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (6*PolyLog[4, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (6*PolyLog[4, E^x]*Sech[x
])/Sqrt[a*Sech[x]^2]

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

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Mathematica [A]
time = 0.06, size = 113, normalized size = 0.75 \begin {gather*} \frac {\left (\pi ^4-2 x^4-8 x^3 \log \left (1+e^{-x}\right )+8 x^3 \log \left (1-e^x\right )+24 x^2 \text {PolyLog}\left (2,-e^{-x}\right )+24 x^2 \text {PolyLog}\left (2,e^x\right )+48 x \text {PolyLog}\left (3,-e^{-x}\right )-48 x \text {PolyLog}\left (3,e^x\right )+48 \text {PolyLog}\left (4,-e^{-x}\right )+48 \text {PolyLog}\left (4,e^x\right )\right ) \text {sech}(x)}{8 \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]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(280\) vs. \(2(129)=258\).
time = 1.46, size = 281, normalized size = 1.87

method	result	size
risch	\(-\frac {{\mathrm e}^{x} x^{3} \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {3 \,{\mathrm e}^{x} x^{2} \polylog \left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {6 \,{\mathrm e}^{x} x \polylog \left (3, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {6 \,{\mathrm e}^{x} \polylog \left (4, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} x^{3} \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {3 \,{\mathrm e}^{x} x^{2} \polylog \left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {6 \,{\mathrm e}^{x} x \polylog \left (3, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {6 \,{\mathrm e}^{x} \polylog \left (4, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}\)	\(281\)

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,method=_RETURNVERBOSE)

[Out]

-1/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^3*ln(exp(x)+1)-3/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/
(1+exp(2*x))*exp(x)*x^2*polylog(2,-exp(x))+6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x*polylog(3
,-exp(x))-6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*polylog(4,-exp(x))+1/(a*exp(2*x)/(1+exp(2*x)
)^2)^(1/2)/(1+exp(2*x))*exp(x)*x^3*ln(1-exp(x))+3/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x^2*po
lylog(2,exp(x))-6/(a*exp(2*x)/(1+exp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*x*polylog(3,exp(x))+6/(a*exp(2*x)/(1+e
xp(2*x))^2)^(1/2)/(1+exp(2*x))*exp(x)*polylog(4,exp(x))

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Maxima [A]
time = 0.49, size = 80, normalized size = 0.53 \begin {gather*} -\frac {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})}{\sqrt {a}} + \frac {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})}{\sqrt {a}} \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]

-(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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (127) = 254\).
time = 0.38, size = 272, normalized size = 1.81 \begin {gather*} \frac {{\left (6 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (4, \cosh \left (x\right ) + \sinh \left (x\right )\right ) - 6 \, \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} {\left (e^{\left (2 \, x\right )} + 1\right )} e^{x} {\rm polylog}\left (4, -\cosh \left (x\right ) - \sinh \left (x\right )\right ) - 6 \, {\left (x e^{\left (2 \, x\right )} + x\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 ) + 6 \, {\left (x e^{\left (2 \, x\right )} + x\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 (3 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - 3 \, {\left (x^{2} e^{\left (2 \, x\right )} + x^{2}\right )} {\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) - {\left (x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + {\left (x^{3} e^{\left (2 \, x\right )} + x^{3}\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}\right )} e^{\left (-x\right )}}{a} \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*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*(e^(2*x) + 1)*e^x*polylog(4, cosh(x) + sinh(x)) - 6*sqrt(a/(e^(4*x) + 2*e
^(2*x) + 1))*(e^(2*x) + 1)*e^x*polylog(4, -cosh(x) - sinh(x)) - 6*(x*e^(2*x) + x)*sqrt(a/(e^(4*x) + 2*e^(2*x)
+ 1))*e^x*polylog(3, cosh(x) + sinh(x)) + 6*(x*e^(2*x) + x)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x*polylog(3, -
cosh(x) - sinh(x)) + (3*(x^2*e^(2*x) + x^2)*dilog(cosh(x) + sinh(x)) - 3*(x^2*e^(2*x) + x^2)*dilog(-cosh(x) -
sinh(x)) - (x^3*e^(2*x) + x^3)*log(cosh(x) + sinh(x) + 1) + (x^3*e^(2*x) + x^3)*log(-cosh(x) - sinh(x) + 1))*s
qrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x)*e^(-x)/a

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\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*csch(x)*sech(x)/sqrt(a*sech(x)**2), 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(x^3*csch(x)*sech(x)/sqrt(a*sech(x)^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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