3.67 \(\int \frac{x^2}{\cosh ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=153 \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{24 a^3}+\frac{9 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac{x}{3 a^2 \cosh ^{-1}(a x)^2}+\frac{\sqrt{a x-1} \sqrt{a x+1}}{3 a^3 \cosh ^{-1}(a x)}-\frac{x^3}{2 \cosh ^{-1}(a x)^2}-\frac{3 x^2 \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

[Out]

-(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + x/(3*a^2*ArcCosh[a*x]^2) - x^3/(2*ArcCosh[a*x]^2) +
(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^3*ArcCosh[a*x]) - (3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x])
+ CoshIntegral[ArcCosh[a*x]]/(24*a^3) + (9*CoshIntegral[3*ArcCosh[a*x]])/(8*a^3)

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Rubi [A]  time = 0.675705, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5668, 5775, 5666, 3301, 5656, 5781} \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{24 a^3}+\frac{9 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac{x}{3 a^2 \cosh ^{-1}(a x)^2}+\frac{\sqrt{a x-1} \sqrt{a x+1}}{3 a^3 \cosh ^{-1}(a x)}-\frac{x^3}{2 \cosh ^{-1}(a x)^2}-\frac{3 x^2 \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcCosh[a*x]^4,x]

[Out]

-(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + x/(3*a^2*ArcCosh[a*x]^2) - x^3/(2*ArcCosh[a*x]^2) +
(Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a^3*ArcCosh[a*x]) - (3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x])
+ CoshIntegral[ArcCosh[a*x]]/(24*a^3) + (9*CoshIntegral[3*ArcCosh[a*x]])/(8*a^3)

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]

Rule 5656

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rubi steps

\begin{align*} \int \frac{x^2}{\cosh ^{-1}(a x)^4} \, dx &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}-\frac{2 \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3} \, dx}{3 a}+a \int \frac{x^3}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3} \, dx\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{x^3}{2 \cosh ^{-1}(a x)^2}+\frac{3}{2} \int \frac{x^2}{\cosh ^{-1}(a x)^2} \, dx-\frac{\int \frac{1}{\cosh ^{-1}(a x)^2} \, dx}{3 a^2}\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{x^3}{2 \cosh ^{-1}(a x)^2}+\frac{\sqrt{-1+a x} \sqrt{1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)}-\frac{3 \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 )}{2 a^3}-\frac{\int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)} \, dx}{3 a}\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{x^3}{2 \cosh ^{-1}(a x)^2}+\frac{\sqrt{-1+a x} \sqrt{1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{3 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{8 a^3}\\ &=-\frac{x^2 \sqrt{-1+a x} \sqrt{1+a x}}{3 a \cosh ^{-1}(a x)^3}+\frac{x}{3 a^2 \cosh ^{-1}(a x)^2}-\frac{x^3}{2 \cosh ^{-1}(a x)^2}+\frac{\sqrt{-1+a x} \sqrt{1+a x}}{3 a^3 \cosh ^{-1}(a x)}-\frac{3 x^2 \sqrt{-1+a x} \sqrt{1+a x}}{2 a \cosh ^{-1}(a x)}+\frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{24 a^3}+\frac{9 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}\\ \end{align*}

Mathematica [A]  time = 0.364899, size = 183, normalized size = 1.2 \[ \frac{\sqrt{a x-1} \left (-4 \sqrt{\frac{a x-1}{a x+1}} \left (2 a^2 x^2 \left (a^2 x^2-1\right )+a x \sqrt{a x-1} \sqrt{a x+1} \left (3 a^2 x^2-2\right ) \cosh ^{-1}(a x)+\left (9 a^4 x^4-11 a^2 x^2+2\right ) \cosh ^{-1}(a x)^2\right )+(a x-1) \cosh ^{-1}(a x)^3 \text{Chi}\left (\cosh ^{-1}(a x)\right )+27 (a x-1) \cosh ^{-1}(a x)^3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )\right )}{24 a^3 \left (\frac{a x-1}{a x+1}\right )^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^2/ArcCosh[a*x]^4,x]

[Out]

(Sqrt[-1 + a*x]*(-4*Sqrt[(-1 + a*x)/(1 + a*x)]*(2*a^2*x^2*(-1 + a^2*x^2) + a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-
2 + 3*a^2*x^2)*ArcCosh[a*x] + (2 - 11*a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x]^2) + (-1 + a*x)*ArcCosh[a*x]^3*CoshInt
egral[ArcCosh[a*x]] + 27*(-1 + a*x)*ArcCosh[a*x]^3*CoshIntegral[3*ArcCosh[a*x]]))/(24*a^3*((-1 + a*x)/(1 + a*x
))^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^3)

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Maple [A]  time = 0.035, size = 121, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ( -{\frac{1}{12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ax}{24\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{1}{24\,{\rm arccosh} \left (ax\right )}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ) }{24}}-{\frac{\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}-{\frac{\cosh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{3\,\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{8\,{\rm arccosh} \left (ax\right )}}+{\frac{9\,{\it Chi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/arccosh(a*x)^4,x)

[Out]

1/a^3*(-1/12/arccosh(a*x)^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/24*a*x/arccosh(a*x)^2-1/24/arccosh(a*x)*(a*x-1)^(1/2
)*(a*x+1)^(1/2)+1/24*Chi(arccosh(a*x))-1/12/arccosh(a*x)^3*sinh(3*arccosh(a*x))-1/8/arccosh(a*x)^2*cosh(3*arcc
osh(a*x))-3/8/arccosh(a*x)*sinh(3*arccosh(a*x))+9/8*Chi(3*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^2/arccosh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^13*x^13 - 10*a^11*x^11 + 20*a^9*x^9 - 20*a^7*x^7 + 10*a^5*x^5 + 2*(a^8*x^8 - a^6*x^6)*(a*x + 1)^(5/2
)*(a*x - 1)^(5/2) - 2*a^3*x^3 + 2*(5*a^9*x^9 - 9*a^7*x^7 + 4*a^5*x^5)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x^10
- 13*a^8*x^8 + 11*a^6*x^6 - 3*a^4*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^11*x^11 - 17*a^9*x^9 + 21*a^7
*x^7 - 11*a^5*x^5 + 2*a^3*x^3)*(a*x + 1)*(a*x - 1) + (9*a^13*x^13 - 45*a^11*x^11 + 90*a^9*x^9 - 90*a^7*x^7 + 4
5*a^5*x^5 + (9*a^8*x^8 - 13*a^6*x^6 + 3*a^4*x^4 + a^2*x^2)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) - 9*a^3*x^3 + (45*a
^9*x^9 - 97*a^7*x^7 + 64*a^5*x^5 - 10*a^3*x^3 - 2*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (90*a^10*x^10 - 258*a^8*x^8 +
264*a^6*x^6 - 113*a^4*x^4 + 19*a^2*x^2 - 2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(45*a^11*x^11 - 161*a^9*x^9 +
219*a^7*x^7 - 141*a^5*x^5 + 44*a^3*x^3 - 6*a*x)*(a*x + 1)*(a*x - 1) + (45*a^12*x^12 - 193*a^10*x^10 + 325*a^8
*x^8 - 270*a^6*x^6 + 112*a^4*x^4 - 19*a^2*x^2)*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^12 - 21*a^10*x^10 + 34*a^8*x^8 - 26*a^6*x^6 + 9*a^4*x^4 - a^2*x^2)*sqrt(a*x + 1)*sqrt(a*x
- 1) + (3*a^13*x^13 - 15*a^11*x^11 + 30*a^9*x^9 - 30*a^7*x^7 + 15*a^5*x^5 + (3*a^8*x^8 - 4*a^6*x^6 + a^4*x^4)
*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) - 3*a^3*x^3 + (15*a^9*x^9 - 31*a^7*x^7 + 20*a^5*x^5 - 4*a^3*x^3)*(a*x + 1)^2*
(a*x - 1)^2 + (30*a^10*x^10 - 84*a^8*x^8 + 84*a^6*x^6 - 35*a^4*x^4 + 5*a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2
) + 2*(15*a^11*x^11 - 53*a^9*x^9 + 71*a^7*x^7 - 44*a^5*x^5 + 12*a^3*x^3 - a*x)*(a*x + 1)*(a*x - 1) + (15*a^12*
x^12 - 64*a^10*x^10 + 107*a^8*x^8 - 87*a^6*x^6 + 34*a^4*x^4 - 5*a^2*x^2)*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*(27*a^14*x^14 - 162*a^12*x^12 + 405*a^10*x^10 - 540*a^8*x^8 + 405*a^6*x^
6 - 162*a^4*x^4 + (27*a^8*x^8 - 13*a^6*x^6 - 3*a^4*x^4 - 3*a^2*x^2)*(a*x + 1)^3*(a*x - 1)^3 + (162*a^9*x^9 - 2
27*a^7*x^7 + 63*a^5*x^5 + 3*a^3*x^3 + 6*a*x)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (405*a^10*x^10 - 940*a^8*x^8 +
687*a^6*x^6 - 143*a^4*x^4 - 21*a^2*x^2 + 12)*(a*x + 1)^2*(a*x - 1)^2 + (540*a^11*x^11 - 1750*a^9*x^9 + 2058*a^
7*x^7 - 1017*a^5*x^5 + 145*a^3*x^3 + 24*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 27*a^2*x^2 + (405*a^12*x^12 - 1
685*a^10*x^10 + 2727*a^8*x^8 - 2118*a^6*x^6 + 782*a^4*x^4 - 123*a^2*x^2 + 12)*(a*x + 1)*(a*x - 1) + (162*a^13*
x^13 - 823*a^11*x^11 + 1695*a^9*x^9 - 1790*a^7*x^7 + 1015*a^5*x^5 - 297*a^3*x^3 + 38*a*x)*sqrt(a*x + 1)*sqrt(a
*x - 1))/((a^14*x^12 - 6*a^12*x^10 + (a*x + 1)^3*(a*x - 1)^3*a^8*x^6 + 15*a^10*x^8 - 20*a^8*x^6 + 15*a^6*x^4 +
6*(a^9*x^7 - a^7*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) - 6*a^4*x^2 + 15*(a^10*x^8 - 2*a^8*x^6 + a^6*x^4)*(a*x
+ 1)^2*(a*x - 1)^2 + 20*(a^11*x^9 - 3*a^9*x^7 + 3*a^7*x^5 - a^5*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 15*(a^1
2*x^10 - 4*a^10*x^8 + 6*a^8*x^6 - 4*a^6*x^4 + a^4*x^2)*(a*x + 1)*(a*x - 1) + 6*(a^13*x^11 - 5*a^11*x^9 + 10*a^
9*x^7 - 10*a^7*x^5 + 5*a^5*x^3 - a^3*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a^2)*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^{2}}{\operatorname{arcosh}\left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^2/arccosh(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{acosh}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/acosh(a*x)**4,x)

[Out]

Integral(x**2/acosh(a*x)**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{arcosh}\left (a x\right )^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^2/arccosh(a*x)^4, x)