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

Optimal. Leaf size=28 \[ \frac{\sqrt{a x-1} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{1-a x}} \]

[Out]

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

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Rubi [A]  time = 0.162425, antiderivative size = 41, normalized size of antiderivative = 1.46, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {5713, 5674} \[ \frac{\sqrt{a x-1} \sqrt{a x+1} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]

[Out]

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

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5674

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

Rubi steps

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

Mathematica [A]  time = 0.0591602, size = 47, normalized size = 1.68 \[ \frac{\sqrt{\frac{a x-1}{a x+1}} (a x+1) \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt{-(a x-1) (a x+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]

[Out]

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

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Maple [A]  time = 0.079, size = 48, normalized size = 1.7 \begin{align*} -{\frac{\ln \left ({\rm arccosh} \left (ax\right ) \right ) }{a \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{ax-1}\sqrt{ax+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)

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Fricas [B]  time = 2.02138, size = 117, normalized size = 4.18 \begin{align*} -\frac{\sqrt{a^{2} x^{2} - 1} \sqrt{-a^{2} x^{2} + 1} \log \left (\log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )\right )}{a^{3} x^{2} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a^2*x^2 - 1)*sqrt(-a^2*x^2 + 1)*log(log(a*x + sqrt(a^2*x^2 - 1)))/(a^3*x^2 - a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(sqrt(-(a*x - 1)*(a*x + 1))*acosh(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)), x)