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 = 0.006411, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 5654
Rule 74
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*}
Mathematica [A] time = 0.016369, size = 30, normalized size = 1. \[ x \cosh ^{-1}(a x)-\frac{\sqrt{a x-1} \sqrt{a x+1}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 29, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax{\rm arccosh} \left (ax\right )-\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.15911, size = 34, normalized size = 1.13 \begin{align*} \frac{a x \operatorname{arcosh}\left (a x\right ) - \sqrt{a^{2} x^{2} - 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36269, size = 78, normalized size = 2.6 \begin{align*} \frac{a x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \sqrt{a^{2} x^{2} - 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.208176, size = 26, normalized size = 0.87 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37339, size = 47, normalized size = 1.57 \begin{align*} x \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{\sqrt{a^{2} x^{2} - 1}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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