3.66 \(\int \frac{x^3}{\cosh ^{-1}(a x)^4} \, dx\)

Optimal. Leaf size=155 \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac{x^2}{2 a^2 \cosh ^{-1}(a x)^2}+\frac{x \sqrt{a x-1} \sqrt{a x+1}}{a^3 \cosh ^{-1}(a x)}-\frac{2 x^4}{3 \cosh ^{-1}(a x)^2}-\frac{8 x^3 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

[Out]

-(x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + x^2/(2*a^2*ArcCosh[a*x]^2) - (2*x^4)/(3*ArcCosh[a*x
]^2) + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (8*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh
[a*x]) + CoshIntegral[2*ArcCosh[a*x]]/(3*a^4) + (4*CoshIntegral[4*ArcCosh[a*x]])/(3*a^4)

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Rubi [A]  time = 0.587555, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5668, 5775, 5666, 3301} \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac{4 \text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{3 a^4}+\frac{x^2}{2 a^2 \cosh ^{-1}(a x)^2}+\frac{x \sqrt{a x-1} \sqrt{a x+1}}{a^3 \cosh ^{-1}(a x)}-\frac{2 x^4}{3 \cosh ^{-1}(a x)^2}-\frac{8 x^3 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{3 a \cosh ^{-1}(a x)^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcCosh[a*x]^4,x]

[Out]

-(x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh[a*x]^3) + x^2/(2*a^2*ArcCosh[a*x]^2) - (2*x^4)/(3*ArcCosh[a*x
]^2) + (x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a^3*ArcCosh[a*x]) - (8*x^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(3*a*ArcCosh
[a*x]) + CoshIntegral[2*ArcCosh[a*x]]/(3*a^4) + (4*CoshIntegral[4*ArcCosh[a*x]])/(3*a^4)

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

Mathematica [A]  time = 0.373256, size = 188, normalized size = 1.21 \[ \frac{\sqrt{a x-1} \left (a x \sqrt{\frac{a x-1}{a x+1}} \left (-2 a^4 x^4+2 a^2 x^2-a x \sqrt{a x-1} \sqrt{a x+1} \left (4 a^2 x^2-3\right ) \cosh ^{-1}(a x)-2 \left (8 a^4 x^4-11 a^2 x^2+3\right ) \cosh ^{-1}(a x)^2\right )+2 (a x-1) \cosh ^{-1}(a x)^3 \text{Chi}\left (2 \cosh ^{-1}(a x)\right )+8 (a x-1) \cosh ^{-1}(a x)^3 \text{Chi}\left (4 \cosh ^{-1}(a x)\right )\right )}{6 a^4 \left (\frac{a x-1}{a x+1}\right )^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3/ArcCosh[a*x]^4,x]

[Out]

(Sqrt[-1 + a*x]*(a*x*Sqrt[(-1 + a*x)/(1 + a*x)]*(2*a^2*x^2 - 2*a^4*x^4 - a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-3
+ 4*a^2*x^2)*ArcCosh[a*x] - 2*(3 - 11*a^2*x^2 + 8*a^4*x^4)*ArcCosh[a*x]^2) + 2*(-1 + a*x)*ArcCosh[a*x]^3*CoshI
ntegral[2*ArcCosh[a*x]] + 8*(-1 + a*x)*ArcCosh[a*x]^3*CoshIntegral[4*ArcCosh[a*x]]))/(6*a^4*((-1 + a*x)/(1 + a
*x))^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^3)

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Maple [A]  time = 0.043, size = 114, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}-{\frac{\cosh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{6\,{\rm arccosh} \left (ax\right )}}+{\frac{{\it Chi} \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{3}}-{\frac{\sinh \left ( 4\,{\rm arccosh} \left (ax\right ) \right ) }{24\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}}-{\frac{\cosh \left ( 4\,{\rm arccosh} \left (ax\right ) \right ) }{12\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{\sinh \left ( 4\,{\rm arccosh} \left (ax\right ) \right ) }{3\,{\rm arccosh} \left (ax\right )}}+{\frac{4\,{\it Chi} \left ( 4\,{\rm arccosh} \left (ax\right ) \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arccosh(a*x)^4,x)

[Out]

1/a^4*(-1/12/arccosh(a*x)^3*sinh(2*arccosh(a*x))-1/12/arccosh(a*x)^2*cosh(2*arccosh(a*x))-1/6/arccosh(a*x)*sin
h(2*arccosh(a*x))+1/3*Chi(2*arccosh(a*x))-1/24/arccosh(a*x)^3*sinh(4*arccosh(a*x))-1/12/arccosh(a*x)^2*cosh(4*
arccosh(a*x))-1/3/arccosh(a*x)*sinh(4*arccosh(a*x))+4/3*Chi(4*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^3/arccosh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^13*x^14 - 10*a^11*x^12 + 20*a^9*x^10 - 20*a^7*x^8 + 10*a^5*x^6 - 2*a^3*x^4 + 2*(a^8*x^9 - a^6*x^7)*(
a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 2*(5*a^9*x^10 - 9*a^7*x^8 + 4*a^5*x^6)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x^
11 - 13*a^8*x^9 + 11*a^6*x^7 - 3*a^4*x^5)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^11*x^12 - 17*a^9*x^10 + 21*
a^7*x^8 - 11*a^5*x^6 + 2*a^3*x^4)*(a*x + 1)*(a*x - 1) + (16*a^13*x^14 - 80*a^11*x^12 + 160*a^9*x^10 - 160*a^7*
x^8 + 80*a^5*x^6 - 16*a^3*x^4 + 4*(4*a^8*x^9 - 7*a^6*x^7 + 3*a^4*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (80*a^
9*x^10 - 192*a^7*x^8 + 154*a^5*x^6 - 45*a^3*x^4 + 3*a*x^2)*(a*x + 1)^2*(a*x - 1)^2 + (160*a^10*x^11 - 488*a^8*
x^9 + 550*a^6*x^7 - 279*a^4*x^5 + 63*a^2*x^3 - 6*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (160*a^11*x^12 - 592*a^9
*x^10 + 846*a^7*x^8 - 583*a^5*x^6 + 196*a^3*x^4 - 27*a*x^2)*(a*x + 1)*(a*x - 1) + (80*a^12*x^13 - 348*a^10*x^1
1 + 598*a^8*x^9 - 509*a^6*x^7 + 216*a^4*x^5 - 37*a^2*x^3)*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^13 - 21*a^10*x^11 + 34*a^8*x^9 - 26*a^6*x^7 + 9*a^4*x^5 - a^2*x^3)*sqrt(a*x +
1)*sqrt(a*x - 1) + (4*a^13*x^14 - 20*a^11*x^12 + 40*a^9*x^10 - 40*a^7*x^8 + 20*a^5*x^6 - 4*a^3*x^4 + 2*(2*a^8*
x^9 - 3*a^6*x^7 + a^4*x^5)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + (20*a^9*x^10 - 44*a^7*x^8 + 31*a^5*x^6 - 7*a^3*x^
4)*(a*x + 1)^2*(a*x - 1)^2 + (40*a^10*x^11 - 116*a^8*x^9 + 121*a^6*x^7 - 53*a^4*x^5 + 8*a^2*x^3)*(a*x + 1)^(3/
2)*(a*x - 1)^(3/2) + (40*a^11*x^12 - 144*a^9*x^10 + 197*a^7*x^8 - 125*a^5*x^6 + 35*a^3*x^4 - 3*a*x^2)*(a*x + 1
)*(a*x - 1) + (20*a^12*x^13 - 86*a^10*x^11 + 145*a^8*x^9 - 119*a^6*x^7 + 47*a^4*x^5 - 7*a^2*x^3)*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*(64*a^15*x^15 - 384*a^13*x^13 + 960*a^11*x^11 -
1280*a^9*x^9 + 960*a^7*x^7 - 384*a^5*x^5 + 8*(8*a^9*x^9 - 7*a^7*x^7)*(a*x + 1)^3*(a*x - 1)^3 + (384*a^10*x^10
- 664*a^8*x^8 + 308*a^6*x^6 - 12*a^4*x^4 - 9*a^2*x^2)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 64*a^3*x^3 + 2*(480*a^
11*x^11 - 1240*a^9*x^9 + 1096*a^7*x^7 - 360*a^5*x^5 + 15*a^3*x^3 + 9*a*x)*(a*x + 1)^2*(a*x - 1)^2 + 2*(640*a^1
2*x^12 - 2200*a^10*x^10 + 2844*a^8*x^8 - 1684*a^6*x^6 + 433*a^4*x^4 - 36*a^2*x^2 + 3)*(a*x + 1)^(3/2)*(a*x - 1
)^(3/2) + 2*(480*a^13*x^13 - 2060*a^11*x^11 + 3496*a^9*x^9 - 2952*a^7*x^7 + 1283*a^5*x^5 - 274*a^3*x^3 + 27*a*
x)*(a*x + 1)*(a*x - 1) + (384*a^14*x^14 - 1976*a^12*x^12 + 4148*a^10*x^10 - 4524*a^8*x^8 + 2699*a^6*x^6 - 842*
a^4*x^4 + 111*a^2*x^2)*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^{3}}{\operatorname{arcosh}\left (a x\right )^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(x^3/arccosh(a*x)^4, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{acosh}^{4}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/acosh(a*x)**4,x)

[Out]

Integral(x**3/acosh(a*x)**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{arcosh}\left (a x\right )^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(x^3/arccosh(a*x)^4, x)