Optimal. Leaf size=30 \[ x \cosh ^{-1}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5654, 74} \[ x \cosh ^{-1}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a} \]
Antiderivative was successfully verified.
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Rule 74
Rule 5654
Rubi steps
\begin {align*} \int \cosh ^{-1}(a x) \, dx &=x \cosh ^{-1}(a x)-a \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{a}+x \cosh ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 1.00 \[ x \cosh ^{-1}(a x)-\frac {\sqrt {a x-1} \sqrt {a x+1}}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 37, normalized size = 1.23 \[ \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \sqrt {a^{2} x^{2} - 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 35, normalized size = 1.17 \[ x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.97 \[ \frac {a x \,\mathrm {arccosh}\left (a x \right )-\sqrt {a x -1}\, \sqrt {a x +1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 25, normalized size = 0.83 \[ \frac {a x \operatorname {arcosh}\left (a x\right ) - \sqrt {a^{2} x^{2} - 1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 26, normalized size = 0.87 \[ x\,\mathrm {acosh}\left (a\,x\right )-\frac {\sqrt {a\,x-1}\,\sqrt {a\,x+1}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 26, normalized size = 0.87 \[ \begin {cases} x \operatorname {acosh}{\left (a x \right )} - \frac {\sqrt {a^{2} x^{2} - 1}}{a} & \text {for}\: a \neq 0 \\\frac {i \pi x}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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