Optimal. Leaf size=28 \[ \frac {\sqrt {a x-1} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {1-a x}} \]
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Rubi [A] time = 0.30, antiderivative size = 41, normalized size of antiderivative = 1.46, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5798, 5781, 3301} \[ \frac {\sqrt {a x-1} \sqrt {a x+1} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 3301
Rule 5781
Rule 5798
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^2 \sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 50, normalized size = 1.79 \[ -\frac {\sqrt {-((a x-1) (a x+1))} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} \operatorname {arcosh}\left (a x\right )}, x\right ) \]
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 \[ \int \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 100, normalized size = 3.57 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \Ei \left (1, \mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \Ei \left (1, -\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )} \]
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 \[ \int \frac {x}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}\,{d x} \]
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 \[ \int \frac {x}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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