Optimal. Leaf size=28 \[ \frac {\sqrt {a x-1} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt {1-a x}} \]
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Rubi [A] time = 0.16, 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.
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Rule 5674
Rule 5713
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*}
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Mathematica [A] time = 0.06, 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.
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fricas [B] time = 0.56, size = 55, normalized size = 1.96 \[ -\frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )\right )}{a^{3} x^{2} - a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 48, normalized size = 1.71 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\mathrm {arccosh}\left (a x \right )\right )}{a \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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