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

Optimal. Leaf size=12 \[ \text{Unintegrable}\left (\frac{1}{x^2 \cosh ^{-1}(a x)^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0140318, 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 = {} \[ \int \frac{1}{x^2 \cosh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/(x^2*ArcCosh[a*x]^2), x]

Rubi steps

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

Mathematica [A]  time = 7.91706, size = 0, normalized size = 0. \[ \int \frac{1}{x^2 \cosh ^{-1}(a x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Integrate[1/(x^2*ArcCosh[a*x]^2), x]

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Maple [A]  time = 0.091, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  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^{4} + \sqrt{a x + 1} \sqrt{a x - 1} a^{2} x^{3} - a x^{2}\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )} - \int \frac{a^{5} x^{5} - 2 \, a^{3} x^{3} +{\left (a^{3} x^{3} - 3 \, a x\right )}{\left (a x + 1\right )}{\left (a x - 1\right )} +{\left (2 \, a^{4} x^{4} - 5 \, a^{2} x^{2} + 2\right )} \sqrt{a x + 1} \sqrt{a x - 1} + a x}{{\left (a^{5} x^{7} +{\left (a x + 1\right )}{\left (a x - 1\right )} a^{3} x^{5} - 2 \, a^{3} x^{5} + a x^{3} + 2 \,{\left (a^{4} x^{6} - a^{2} x^{4}\right )} \sqrt{a x + 1} \sqrt{a x - 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/x^2/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^4 + sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x^3 -
a*x^2)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))) - integrate((a^5*x^5 - 2*a^3*x^3 + (a^3*x^3 - 3*a*x)*(a*x + 1)
*(a*x - 1) + (2*a^4*x^4 - 5*a^2*x^2 + 2)*sqrt(a*x + 1)*sqrt(a*x - 1) + a*x)/((a^5*x^7 + (a*x + 1)*(a*x - 1)*a^
3*x^5 - 2*a^3*x^5 + a*x^3 + 2*(a^4*x^6 - a^2*x^4)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*
x - 1))), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{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(1/x^2/arccosh(a*x)^2,x, algorithm="fricas")

[Out]

integral(1/(x^2*arccosh(a*x)^2), x)

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

[Out]

Integral(1/(x**2*acosh(a*x)**2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{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(1/x^2/arccosh(a*x)^2,x, algorithm="giac")

[Out]

integrate(1/(x^2*arccosh(a*x)^2), x)