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

Optimal. Leaf size=59 \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0472113, antiderivative size = 59, 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} \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3}-\frac{x^2 \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.226049, size = 58, normalized size = 0.98 \[ \frac{-\frac{4 a^2 x^2 \sqrt{\frac{a x-1}{a x+1}} (a x+1)}{\cosh ^{-1}(a x)}+\text{Chi}\left (\cosh ^{-1}(a x)\right )+3 \text{Chi}\left (3 \cosh ^{-1}(a x)\right )}{4 a^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.03, size = 59, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ( -{\frac{1}{4\,{\rm arccosh} \left (ax\right )}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ) }{4}}-{\frac{\sinh \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{4\,{\rm arccosh} \left (ax\right )}}+{\frac{3\,{\it Chi} \left ( 3\,{\rm arccosh} \left (ax\right ) \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(-1/4/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/4*Chi(arccosh(a*x))-1/4/arccosh(a*x)*sinh(3*arccosh(a*x
))+3/4*Chi(3*arccosh(a*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{3} x^{5} - a x^{3} +{\left (a^{2} x^{4} - x^{2}\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{3 \, a^{5} x^{6} - 6 \, a^{3} x^{4} +{\left (3 \, a^{3} x^{4} - a x^{2}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} + 3 \, a x^{2} +{\left (6 \, a^{4} x^{5} - 7 \, a^{2} x^{3} + 2 \, x\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^2/arccosh(a*x)^2,x, algorithm="maxima")

[Out]

-(a^3*x^5 - a*x^3 + (a^2*x^4 - x^2)*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((3*a^5*x^6 - 6*a^3*x^4 + (3*a^3*x^4 - a*x^2)*(a*x +
1)*(a*x - 1) + 3*a*x^2 + (6*a^4*x^5 - 7*a^2*x^3 + 2*x)*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)

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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 )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arccosh(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^2, x)