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

Optimal. Leaf size=61 \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^4}+\frac{\text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0501213, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5666, 3301} \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{2 a^4}+\frac{\text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4}-\frac{x^3 \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.230709, size = 58, normalized size = 0.95 \[ \frac{-\frac{2 a^3 x^3 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}{\cosh ^{-1}(a x)}+\text{Chi}\left (2 \cosh ^{-1}(a x)\right )+\text{Chi}\left (4 \cosh ^{-1}(a x)\right )}{2 a^4} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 54, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{4\,{\rm arccosh} \left (ax\right )}}+{\frac{{\it Chi} \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{2}}-{\frac{\sinh \left ( 4\,{\rm arccosh} \left (ax\right ) \right ) }{8\,{\rm arccosh} \left (ax\right )}}+{\frac{{\it Chi} \left ( 4\,{\rm arccosh} \left (ax\right ) \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/arccosh(a*x)^2,x)

[Out]

1/a^4*(-1/4/arccosh(a*x)*sinh(2*arccosh(a*x))+1/2*Chi(2*arccosh(a*x))-1/8/arccosh(a*x)*sinh(4*arccosh(a*x))+1/
2*Chi(4*arccosh(a*x)))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{6} - a x^{4} +{\left (a^{2} x^{5} - x^{3}\right )} \sqrt{a x + 1} \sqrt{a x - 1}}{{\left (a^{3} x^{2} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )} + \int \frac{4 \, a^{5} x^{7} - 8 \, a^{3} x^{5} + 4 \, a x^{3} + 2 \,{\left (2 \, a^{3} x^{5} - a x^{3}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} +{\left (8 \, a^{4} x^{6} - 10 \, a^{2} x^{4} + 3 \, x^{2}\right )} \sqrt{a x + 1} \sqrt{a x - 1}}{{\left (a^{5} x^{4} +{\left (a x + 1\right )}{\left (a x - 1\right )} a^{3} x^{2} - 2 \, a^{3} x^{2} + 2 \,{\left (a^{4} x^{3} - a^{2} x\right )} \sqrt{a x + 1} \sqrt{a x - 1} + a\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*x^6 - a*x^4 + (a^2*x^5 - x^3)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^3*x^2 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x
- a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) + integrate((4*a^5*x^7 - 8*a^3*x^5 + 4*a*x^3 + 2*(2*a^3*x^5 - a*
x^3)*(a*x + 1)*(a*x - 1) + (8*a^4*x^6 - 10*a^2*x^4 + 3*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^5*x^4 + (a*x + 1)
*(a*x - 1)*a^3*x^2 - 2*a^3*x^2 + 2*(a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*
sqrt(a*x - 1))), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{3}}{\operatorname{arcosh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3/arccosh(a*x)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3/acosh(a*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3/arccosh(a*x)^2, x)