Optimal. Leaf size=27 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)},x\right ) \]
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Rubi [A] time = 0.38, 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 = {} \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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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 = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} x^{4} - x^{2}\right )} \operatorname {arcosh}\left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, 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.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{x^2\,\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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