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

Optimal. Leaf size=26 \[ \text{Unintegrable}\left (\frac{1}{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.377071, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.684712, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \sqrt{1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.141, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}{\rm arccosh} \left (ax\right )}{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} x^{4} - x^{2}\right )} \operatorname{arcosh}\left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-a^2*x^2 + 1)/((a^2*x^4 - x^2)*arccosh(a*x)), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{2} \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/x**2/acosh(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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