3.3.98 \(\int \frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx\) [298]

Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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*}

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Mathematica [A]
time = 0.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \end {gather*}

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 = 2.99, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \mathrm {arccosh}\left (a x \right ) \sqrt {-a^{2} x^{2}+1}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{x^2\,\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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