3.65 \(\int \frac{x^4}{\cosh ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=170 \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{48 a^5}+\frac{27 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5}+\frac{2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{a^3 \cosh ^{-1}(a x)}-\frac{5 x^5}{6 \cosh ^{-1}(a x)^2}-\frac{25 x^4 \sqrt{a x-1} \sqrt{a x+1}}{6 a \cosh ^{-1}(a x)}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

[Out]

-(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + (2*x^3)/(3*a^2*ArcCosh[a*x]^2) - (5*x^5)/(6*ArcCosh
[a*x]^2) + (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (25*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*
a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(48*a^5) + (27*CoshIntegral[3*ArcCosh[a*x]])/(32*a^5) + (125*Cosh
Integral[5*ArcCosh[a*x]])/(96*a^5)

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Rubi [A]  time = 0.617029, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5668, 5775, 5666, 3301} \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{48 a^5}+\frac{27 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5}+\frac{2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{a x-1} \sqrt{a x+1}}{a^3 \cosh ^{-1}(a x)}-\frac{5 x^5}{6 \cosh ^{-1}(a x)^2}-\frac{25 x^4 \sqrt{a x-1} \sqrt{a x+1}}{6 a \cosh ^{-1}(a x)}-\frac{x^4 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4/ArcCosh[a*x]^4,x]

[Out]

-(x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + (2*x^3)/(3*a^2*ArcCosh[a*x]^2) - (5*x^5)/(6*ArcCosh
[a*x]^2) + (2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (25*x^4*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(6*
a*ArcCosh[a*x]) + CoshIntegral[ArcCosh[a*x]]/(48*a^5) + (27*CoshIntegral[3*ArcCosh[a*x]])/(32*a^5) + (125*Cosh
Integral[5*ArcCosh[a*x]])/(96*a^5)

Rule 5668

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + (-Dist[(c*(m + 1))/(b*(n + 1)), Int[(x^(m + 1)*(a + b*ArcCosh
[c*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] + Dist[m/(b*c*(n + 1)), Int[(x^(m - 1)*(a + b*ArcCosh[c
*x])^(n + 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]

Rule 5775

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[((f*x)^m*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] - Dist[(f
*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(f*x)^(m - 1)*(a + b*ArcCosh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d1
, e1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5666

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(
a + b*ArcCosh[c*x])^(n + 1))/(b*c*(n + 1)), x] + Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a +
b*x)^(n + 1)*Cosh[x]^(m - 1)*(m - (m + 1)*Cosh[x]^2), x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \frac{x^4}{\cosh ^{-1}(a x)^4} \, dx &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac{4 \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3} \, dx}{3 a}+\frac{1}{3} (5 a) \int \frac{x^5}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac{25}{6} \int \frac{x^4}{\cosh ^{-1}(a x)^2} \, dx-\frac{2 \int \frac{x^2}{\cosh ^{-1}(a x)^2} \, dx}{a^2}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac{25 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{6 a \cosh ^{-1}(a x)}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 x}-\frac{3 \cosh (3 x)}{4 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{8 x}-\frac{9 \cosh (3 x)}{16 x}-\frac{5 \cosh (5 x)}{16 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{6 a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac{25 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{6 a \cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{48 a^5}+\frac{125 \operatorname{Subst}\left (\int \frac{\cosh (5 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{96 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{2 a^5}+\frac{75 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{32 a^5}\\ &=-\frac{x^4 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{2 x^3}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{5 x^5}{6 \cosh ^{-1}(a x)^2}+\frac{2 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{a^3 \cosh ^{-1}(a x)}-\frac{25 x^4 \sqrt{-1+a x} \sqrt{1+a x}}{6 a \cosh ^{-1}(a x)}+\frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{48 a^5}+\frac{27 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{32 a^5}+\frac{125 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5}\\ \end{align*}

Mathematica [A]  time = 0.608565, size = 126, normalized size = 0.74 \[ \frac{-\frac{16 a^2 x^2 \left (2 a^2 x^2 \sqrt{a x-1} \sqrt{a x+1}+a x \left (5 a^2 x^2-4\right ) \cosh ^{-1}(a x)+\sqrt{a x-1} \sqrt{a x+1} \left (25 a^2 x^2-12\right ) \cosh ^{-1}(a x)^2\right )}{\cosh ^{-1}(a x)^3}+2 \text{Chi}\left (\cosh ^{-1}(a x)\right )+81 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )+125 \text{Chi}\left (5 \cosh ^{-1}(a x)\right )}{96 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/ArcCosh[a*x]^4,x]

[Out]

((-16*a^2*x^2*(2*a^2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + a*x*(-4 + 5*a^2*x^2)*ArcCosh[a*x] + Sqrt[-1 + a*x]*Sqr
t[1 + a*x]*(-12 + 25*a^2*x^2)*ArcCosh[a*x]^2))/ArcCosh[a*x]^3 + 2*CoshIntegral[ArcCosh[a*x]] + 81*CoshIntegral
[3*ArcCosh[a*x]] + 125*CoshIntegral[5*ArcCosh[a*x]])/(96*a^5)

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Maple [A]  time = 0.043, size = 175, normalized size = 1. \begin{align*}{\frac{1}{{a}^{5}} \left ( -{\frac{1}{24\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ax}{48\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{1}{48\,{\rm arccosh} \left (ax\right )}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ) }{48}}-{\frac{\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{16\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}-{\frac{3\,\cosh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{9\,\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{32\,{\rm arccosh} \left (ax\right )}}+{\frac{27\,{\it Chi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{32}}-{\frac{\sinh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{48\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}-{\frac{5\,\cosh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{96\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{25\,\sinh \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{96\,{\rm arccosh} \left (ax\right )}}+{\frac{125\,{\it Chi} \left ( 5\,{\rm arccosh} \left (ax\right ) \right ) }{96}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/arccosh(a*x)^4,x)

[Out]

1/a^5*(-1/24/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/48*a*x/arccosh(a*x)^2-1/48/arccosh(a*x)*(a*x-1)^(1/2
)*(a*x+1)^(1/2)+1/48*Chi(arccosh(a*x))-1/16/arccosh(a*x)^3*sinh(3*arccosh(a*x))-3/32/arccosh(a*x)^2*cosh(3*arc
cosh(a*x))-9/32/arccosh(a*x)*sinh(3*arccosh(a*x))+27/32*Chi(3*arccosh(a*x))-1/48/arccosh(a*x)^3*sinh(5*arccosh
(a*x))-5/96/arccosh(a*x)^2*cosh(5*arccosh(a*x))-25/96/arccosh(a*x)*sinh(5*arccosh(a*x))+125/96*Chi(5*arccosh(a
*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^13*x^15 - 10*a^11*x^13 + 20*a^9*x^11 - 20*a^7*x^9 + 10*a^5*x^7 - 2*a^3*x^5 + 2*(a^8*x^10 - a^6*x^8)*
(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 2*(5*a^9*x^11 - 9*a^7*x^9 + 4*a^5*x^7)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x
^12 - 13*a^8*x^10 + 11*a^6*x^8 - 3*a^4*x^6)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^11*x^13 - 17*a^9*x^11 + 2
1*a^7*x^9 - 11*a^5*x^7 + 2*a^3*x^5)*(a*x + 1)*(a*x - 1) + (25*a^13*x^15 - 125*a^11*x^13 + 250*a^9*x^11 - 250*a
^7*x^9 + 125*a^5*x^7 - 25*a^3*x^5 + (25*a^8*x^10 - 49*a^6*x^8 + 27*a^4*x^6 - 3*a^2*x^4)*(a*x + 1)^(5/2)*(a*x -
1)^(5/2) + (125*a^9*x^11 - 321*a^7*x^9 + 286*a^5*x^7 - 102*a^3*x^5 + 12*a*x^3)*(a*x + 1)^2*(a*x - 1)^2 + (250
*a^10*x^12 - 794*a^8*x^10 + 946*a^6*x^8 - 519*a^4*x^6 + 129*a^2*x^4 - 12*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2)
+ 2*(125*a^11*x^13 - 473*a^9*x^11 + 696*a^7*x^9 - 497*a^5*x^7 + 173*a^3*x^5 - 24*a*x^3)*(a*x + 1)*(a*x - 1) +
(125*a^12*x^14 - 549*a^10*x^12 + 955*a^8*x^10 - 824*a^6*x^8 + 354*a^4*x^6 - 61*a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x
- 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*a^12*x^14 - 21*a^10*x^12 + 34*a^8*x^10 - 26*a^6*x^8 + 9
*a^4*x^6 - a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1) + (5*a^13*x^15 - 25*a^11*x^13 + 50*a^9*x^11 - 50*a^7*x^9 + 25*
a^5*x^7 - 5*a^3*x^5 + (5*a^8*x^10 - 8*a^6*x^8 + 3*a^4*x^6)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (25*a^9*x^11 - 57
*a^7*x^9 + 42*a^5*x^7 - 10*a^3*x^5)*(a*x + 1)^2*(a*x - 1)^2 + (50*a^10*x^12 - 148*a^8*x^10 + 158*a^6*x^8 - 71*
a^4*x^6 + 11*a^2*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(25*a^11*x^13 - 91*a^9*x^11 + 126*a^7*x^9 - 81*a^5*x
^7 + 23*a^3*x^5 - 2*a*x^3)*(a*x + 1)*(a*x - 1) + (25*a^12*x^14 - 108*a^10*x^12 + 183*a^8*x^10 - 151*a^6*x^8 +
60*a^4*x^6 - 9*a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^13*x^10 - 5*a
^11*x^8 + (a*x + 1)^(5/2)*(a*x - 1)^(5/2)*a^8*x^5 + 10*a^9*x^6 - 10*a^7*x^4 + 5*a^5*x^2 + 5*(a^9*x^6 - a^7*x^4
)*(a*x + 1)^2*(a*x - 1)^2 + 10*(a^10*x^7 - 2*a^8*x^5 + a^6*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 10*(a^11*x^8
- 3*a^9*x^6 + 3*a^7*x^4 - a^5*x^2)*(a*x + 1)*(a*x - 1) - a^3 + 5*(a^12*x^9 - 4*a^10*x^7 + 6*a^8*x^5 - 4*a^6*x
^3 + a^4*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^3) + integrate(1/6*(125*a^15*x
^16 - 750*a^13*x^14 + 1875*a^11*x^12 - 2500*a^9*x^10 + 1875*a^7*x^8 - 750*a^5*x^6 + (125*a^9*x^10 - 147*a^7*x^
8 + 27*a^5*x^6 + 3*a^3*x^4)*(a*x + 1)^3*(a*x - 1)^3 + 125*a^3*x^4 + (750*a^10*x^11 - 1485*a^8*x^9 + 901*a^6*x^
7 - 147*a^4*x^5 - 12*a^2*x^3)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (1875*a^11*x^12 - 5220*a^9*x^10 + 5209*a^7*x^8
- 2185*a^5*x^6 + 321*a^3*x^4)*(a*x + 1)^2*(a*x - 1)^2 + (2500*a^12*x^13 - 8970*a^10*x^11 + 12366*a^8*x^9 - 81
43*a^6*x^7 + 2583*a^4*x^5 - 360*a^2*x^3 + 24*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (1875*a^13*x^14 - 8235*a^11*
x^12 + 14449*a^9*x^10 - 12834*a^7*x^8 + 6030*a^5*x^6 - 1429*a^3*x^4 + 144*a*x^2)*(a*x + 1)*(a*x - 1) + (750*a^
14*x^15 - 3897*a^12*x^13 + 8293*a^10*x^11 - 9226*a^8*x^9 + 5655*a^6*x^7 - 1819*a^4*x^5 + 244*a^2*x^3)*sqrt(a*x
+ 1)*sqrt(a*x - 1))/((a^15*x^12 - 6*a^13*x^10 + (a*x + 1)^3*(a*x - 1)^3*a^9*x^6 + 15*a^11*x^8 - 20*a^9*x^6 +
15*a^7*x^4 - 6*a^5*x^2 + 6*(a^10*x^7 - a^8*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 15*(a^11*x^8 - 2*a^9*x^6 + a
^7*x^4)*(a*x + 1)^2*(a*x - 1)^2 + 20*(a^12*x^9 - 3*a^10*x^7 + 3*a^8*x^5 - a^6*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(
3/2) + 15*(a^13*x^10 - 4*a^11*x^8 + 6*a^9*x^6 - 4*a^7*x^4 + a^5*x^2)*(a*x + 1)*(a*x - 1) + a^3 + 6*(a^14*x^11
- 5*a^12*x^9 + 10*a^10*x^7 - 10*a^8*x^5 + 5*a^6*x^3 - a^4*x)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x +
1)*sqrt(a*x - 1))), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4}}{\operatorname{arcosh}\left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^4/arccosh(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{acosh}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/acosh(a*x)**4,x)

[Out]

Integral(x**4/acosh(a*x)**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4}}{\operatorname{arcosh}\left (a x\right )^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^4/arccosh(a*x)^4, x)