Optimal. Leaf size=17 \[ \frac {\tan ^{-1}\left (\sqrt {-1+a^{2 x}}\right )}{\log (a)} \]
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Rubi [A]
time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 = {2320, 65, 209}
\begin {gather*} \frac {\text {ArcTan}\left (\sqrt {a^{2 x}-1}\right )}{\log (a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+a^{2 x}}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,a^{2 x}\right )}{2 \log (a)}\\ &=\frac {\text {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*}
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Mathematica [A]
time = 0.02, size = 17, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\sqrt {-1+a^{2 x}}\right )}{\log (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 16, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\arctan \left (\sqrt {-1+a^{2 x}}\right )}{\ln \left (a \right )}\) | \(16\) |
default | \(\frac {\arctan \left (\sqrt {-1+a^{2 x}}\right )}{\ln \left (a \right )}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.48, size = 15, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (\sqrt {a^{2 \, x} - 1}\right )}{\log \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 15, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (\sqrt {a^{2 \, x} - 1}\right )}{\log \left (a\right )} \end {gather*}
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 \begin {gather*} \int \frac {1}{\sqrt {a^{2 x} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 15, normalized size = 0.88 \begin {gather*} \frac {\arctan \left (\sqrt {a^{2 \, x} - 1}\right )}{\log \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 37, normalized size = 2.18 \begin {gather*} -\frac {a^x\,\mathrm {asin}\left (\frac {1}{a^x}\right )\,\sqrt {1-\frac {1}{a^{2\,x}}}}{\ln \left (a\right )\,\sqrt {a^{2\,x}-1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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