Optimal. Leaf size=17 \[ \frac{\tan ^{-1}\left (\sqrt{a^{2 x}-1}\right )}{\log (a)} \]
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Rubi [A] time = 0.0129283, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2282, 63, 203} \[ \frac{\tan ^{-1}\left (\sqrt{a^{2 x}-1}\right )}{\log (a)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 63
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1+a^{2 x}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+x} x} \, dx,x,a^{2 x}\right )}{2 \log (a)}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{-1+a^{2 x}}\right )}{\log (a)}\\ &=\frac{\tan ^{-1}\left (\sqrt{-1+a^{2 x}}\right )}{\log (a)}\\ \end{align*}
Mathematica [A] time = 0.0044183, size = 17, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{a^{2 x}-1}\right )}{\log (a)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 16, normalized size = 0.9 \begin{align*}{\frac{1}{\ln \left ( a \right ) }\arctan \left ( \sqrt{-1+{a}^{2\,x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.445, size = 20, normalized size = 1.18 \begin{align*} \frac{\arctan \left (\sqrt{a^{2 \, x} - 1}\right )}{\log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09896, size = 46, normalized size = 2.71 \begin{align*} \frac{\arctan \left (\sqrt{a^{2 \, x} - 1}\right )}{\log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a^{2 x} - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08796, size = 20, normalized size = 1.18 \begin{align*} \frac{\arctan \left (\sqrt{a^{2 \, x} - 1}\right )}{\log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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