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

Optimal. Leaf size=39 \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{a}-\frac{\sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.184479, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5656, 5781, 3301} \[ \frac{\text{Chi}\left (\cosh ^{-1}(a x)\right )}{a}-\frac{\sqrt{a x-1} \sqrt{a x+1}}{a \cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^(-2),x]

[Out]

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

Rule 5656

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

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

Mathematica [A]  time = 0.100184, size = 60, normalized size = 1.54 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x) \text{Chi}\left (\cosh ^{-1}(a x)\right )-a x+1}{a \sqrt{\frac{a x-1}{a x+1}} \cosh ^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^(-2),x]

[Out]

(1 - a*x + Sqrt[(-1 + a*x)/(1 + a*x)]*ArcCosh[a*x]*CoshIntegral[ArcCosh[a*x]])/(a*Sqrt[(-1 + a*x)/(1 + a*x)]*A
rcCosh[a*x])

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Maple [A]  time = 0.026, size = 33, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{{\rm arccosh} \left (ax\right )}\sqrt{ax-1}\sqrt{ax+1}}+{\it Chi} \left ({\rm arccosh} \left (ax\right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^2,x)

[Out]

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

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^(-2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{acosh}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**2,x)

[Out]

Integral(acosh(a*x)**(-2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{arcosh}\left (a x\right )^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^(-2), x)