3.60 \(\int \frac{x^2}{\cosh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=85 \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac{x}{a^2 \cosh ^{-1}(a x)}-\frac{3 x^3}{2 \cosh ^{-1}(a x)}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

[Out]

-(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + x/(a^2*ArcCosh[a*x]) - (3*x^3)/(2*ArcCosh[a*x]) + S
inhIntegral[ArcCosh[a*x]]/(8*a^3) + (9*SinhIntegral[3*ArcCosh[a*x]])/(8*a^3)

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Rubi [A]  time = 0.504109, antiderivative size = 85, 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, 5670, 5448, 3298, 5658} \[ \frac{\text{Shi}\left (\cosh ^{-1}(a x)\right )}{8 a^3}+\frac{9 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3}+\frac{x}{a^2 \cosh ^{-1}(a x)}-\frac{3 x^3}{2 \cosh ^{-1}(a x)}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/ArcCosh[a*x]^3,x]

[Out]

-(x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + x/(a^2*ArcCosh[a*x]) - (3*x^3)/(2*ArcCosh[a*x]) + S
inhIntegral[ArcCosh[a*x]]/(8*a^3) + (9*SinhIntegral[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 5670

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

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 3298

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

Rule 5658

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

Rubi steps

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

Mathematica [A]  time = 0.14575, size = 69, normalized size = 0.81 \[ \frac{-\frac{4 a x \left (\left (3 a^2 x^2-2\right ) \cosh ^{-1}(a x)+a x \sqrt{a x-1} \sqrt{a x+1}\right )}{\cosh ^{-1}(a x)^2}+\text{Shi}\left (\cosh ^{-1}(a x)\right )+9 \text{Shi}\left (3 \cosh ^{-1}(a x)\right )}{8 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/ArcCosh[a*x]^3,x]

[Out]

((-4*a*x*(a*x*Sqrt[-1 + a*x]*Sqrt[1 + a*x] + (-2 + 3*a^2*x^2)*ArcCosh[a*x]))/ArcCosh[a*x]^2 + SinhIntegral[Arc
Cosh[a*x]] + 9*SinhIntegral[3*ArcCosh[a*x]])/(8*a^3)

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Maple [A]  time = 0.029, size = 84, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ( -{\frac{1}{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{ax}{8\,{\rm arccosh} \left (ax\right )}}+{\frac{{\it Shi} \left ({\rm arccosh} \left (ax\right ) \right ) }{8}}-{\frac{\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{8\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{3\,\cosh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{8\,{\rm arccosh} \left (ax\right )}}+{\frac{9\,{\it Shi} \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)^3,x)

[Out]

1/a^3*(-1/8/arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1/8*a*x/arccosh(a*x)+1/8*Shi(arccosh(a*x))-1/8/arccosh(
a*x)^2*sinh(3*arccosh(a*x))-3/8/arccosh(a*x)*cosh(3*arccosh(a*x))+9/8*Shi(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)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^9 - 3*a^6*x^7 + 3*a^4*x^5 - a^2*x^3 + (a^5*x^6 - a^3*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^6
*x^7 - 5*a^4*x^5 + 2*a^2*x^3)*(a*x + 1)*(a*x - 1) + (3*a^7*x^8 - 7*a^5*x^6 + 5*a^3*x^4 - a*x^2)*sqrt(a*x + 1)*
sqrt(a*x - 1) + (3*a^8*x^9 - 9*a^6*x^7 + 9*a^4*x^5 - 3*a^2*x^3 + (3*a^5*x^6 - 4*a^3*x^4 + a*x^2)*(a*x + 1)^(3/
2)*(a*x - 1)^(3/2) + (9*a^6*x^7 - 17*a^4*x^5 + 10*a^2*x^3 - 2*x)*(a*x + 1)*(a*x - 1) + (9*a^7*x^8 - 22*a^5*x^6
+ 18*a^3*x^4 - 5*a*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/((a^8*x^6 + (a*x
+ 1)^(3/2)*(a*x - 1)^(3/2)*a^5*x^3 - 3*a^6*x^4 + 3*a^4*x^2 + 3*(a^6*x^4 - a^4*x^2)*(a*x + 1)*(a*x - 1) + 3*(a
^7*x^5 - 2*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))^2) + int
egrate(1/2*(9*a^10*x^10 - 36*a^8*x^8 + 54*a^6*x^6 - 36*a^4*x^4 + (9*a^6*x^6 - 4*a^4*x^4 - a^2*x^2)*(a*x + 1)^2
*(a*x - 1)^2 + (36*a^7*x^7 - 48*a^5*x^5 + 13*a^3*x^3 + 2*a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 9*a^2*x^2 + (5
4*a^8*x^8 - 120*a^6*x^6 + 83*a^4*x^4 - 19*a^2*x^2 + 2)*(a*x + 1)*(a*x - 1) + (36*a^9*x^9 - 112*a^7*x^7 + 123*a
^5*x^5 - 57*a^3*x^3 + 10*a*x)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^10*x^8 + (a*x + 1)^2*(a*x - 1)^2*a^6*x^4 - 4*a^
8*x^6 + 6*a^6*x^4 - 4*a^4*x^2 + 4*(a^7*x^5 - a^5*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^8*x^6 - 2*a^6*x^4
+ a^4*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^9*x^7 - 3*a^7*x^5 + 3*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 )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccosh(a*x)^3,x, algorithm="fricas")

[Out]

integral(x^2/arccosh(a*x)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{acosh}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/acosh(a*x)**3,x)

[Out]

Integral(x**2/acosh(a*x)**3, 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 )^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/arccosh(a*x)^3,x, algorithm="giac")

[Out]

integrate(x^2/arccosh(a*x)^3, x)