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

Optimal. Leaf size=68 \[ \frac{\text{Shi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}+\frac{1}{2 a^2 \cosh ^{-1}(a x)}-\frac{x^2}{\cosh ^{-1}(a x)}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

[Out]

-(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + 1/(2*a^2*ArcCosh[a*x]) - x^2/ArcCosh[a*x] + SinhInteg
ral[2*ArcCosh[a*x]]/a^2

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Rubi [A]  time = 0.392987, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5668, 5775, 5670, 5448, 12, 3298, 5676} \[ \frac{\text{Shi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}+\frac{1}{2 a^2 \cosh ^{-1}(a x)}-\frac{x^2}{\cosh ^{-1}(a x)}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + 1/(2*a^2*ArcCosh[a*x]) - x^2/ArcCosh[a*x] + SinhInteg
ral[2*ArcCosh[a*x]]/a^2

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 5676

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

Rubi steps

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

Mathematica [A]  time = 0.0493593, size = 67, normalized size = 0.99 \[ \frac{\text{Shi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}+\frac{1-2 a^2 x^2}{2 a^2 \cosh ^{-1}(a x)}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{2 a \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(x*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(2*a*ArcCosh[a*x]^2) + (1 - 2*a^2*x^2)/(2*a^2*ArcCosh[a*x]) + SinhIntegral[2
*ArcCosh[a*x]]/a^2

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Maple [A]  time = 0.026, size = 43, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}-{\frac{\cosh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{2\,{\rm arccosh} \left (ax\right )}}+{\it Shi} \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^2*(-1/4/arccosh(a*x)^2*sinh(2*arccosh(a*x))-1/2/arccosh(a*x)*cosh(2*arccosh(a*x))+Shi(2*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/arccosh(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a^8*x^8 - 3*a^6*x^6 + 3*a^4*x^4 + (a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - a^2*x^2 + (3*a^6
*x^6 - 5*a^4*x^4 + 2*a^2*x^2)*(a*x + 1)*(a*x - 1) + (3*a^7*x^7 - 7*a^5*x^5 + 5*a^3*x^3 - a*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) + (2*a^8*x^8 - 6*a^6*x^6 + 6*a^4*x^4 + 2*(a^5*x^5 - a^3*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 2*a
^2*x^2 + (6*a^6*x^6 - 10*a^4*x^4 + 5*a^2*x^2 - 1)*(a*x + 1)*(a*x - 1) + (6*a^7*x^7 - 14*a^5*x^5 + 11*a^3*x^3 -
3*a*x)*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) + integrate(1/2*(4*a^9
*x^9 + 4*(a*x + 1)^2*(a*x - 1)^2*a^5*x^5 - 16*a^7*x^7 + 24*a^5*x^5 - 16*a^3*x^3 + (16*a^6*x^6 - 16*a^4*x^4 + 3
)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 24*(a^7*x^7 - 2*a^5*x^5 + a^3*x^3)*(a*x + 1)*(a*x - 1) + (16*a^8*x^8 - 48*
a^6*x^6 + 48*a^4*x^4 - 19*a^2*x^2 + 3)*sqrt(a*x + 1)*sqrt(a*x - 1) + 4*a*x)/((a^9*x^8 + (a*x + 1)^2*(a*x - 1)^
2*a^5*x^4 - 4*a^7*x^6 + 6*a^5*x^4 - 4*a^3*x^2 + 4*(a^6*x^5 - a^4*x^3)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 6*(a^7
*x^6 - 2*a^5*x^4 + a^3*x^2)*(a*x + 1)*(a*x - 1) + 4*(a^8*x^7 - 3*a^6*x^5 + 3*a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sq
rt(a*x - 1) + a)*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}{\operatorname{arcosh}\left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/arccosh(a*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/acosh(a*x)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arccosh(a*x)^3, x)