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

Optimal. Leaf size=42 \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0251702, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5666, 3301} \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}-\frac{x \sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.240567, size = 44, normalized size = 1.05 \[ \frac{\text{Chi}\left (2 \cosh ^{-1}(a x)\right )-\frac{a x \sqrt{\frac{a x-1}{a x+1}} (a x+1)}{\cosh ^{-1}(a x)}}{a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 28, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{2}} \left ( -{\frac{\sinh \left ( 2\,{\rm arccosh} \left (ax\right ) \right ) }{2\,{\rm arccosh} \left (ax\right )}}+{\it Chi} \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)^2,x)

[Out]

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

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

[Out]

-(a^3*x^4 - a*x^2 + (a^2*x^3 - x)*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((2*a^5*x^5 + 2*(a*x + 1)*(a*x - 1)*a^3*x^3 - 4*a^3*x^3
+ (4*a^4*x^4 - 4*a^2*x^2 + 1)*sqrt(a*x + 1)*sqrt(a*x - 1) + 2*a*x)/((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}{\operatorname{arcosh}\left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/arccosh(a*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x/arccosh(a*x)^2, x)