Optimal. Leaf size=16 \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {e^{2 x}-3}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2249, 217, 206} \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {e^{2 x}-3}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{\sqrt {-3+e^{2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {-3+x^2}} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {e^x}{\sqrt {-3+e^{2 x}}}\right )\\ &=\tanh ^{-1}\left (\frac {e^x}{\sqrt {-3+e^{2 x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 16, normalized size = 1.00 \[ \tanh ^{-1}\left (\frac {e^x}{\sqrt {e^{2 x}-3}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 16, normalized size = 1.00 \[ -\log \left (\sqrt {e^{\left (2 \, x\right )} - 3} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 16, normalized size = 1.00 \[ -\log \left (-\sqrt {e^{\left (2 \, x\right )} - 3} + e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 13, normalized size = 0.81 \[ \ln \left ({\mathrm e}^{x}+\sqrt {{\mathrm e}^{2 x}-3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.26, size = 16, normalized size = 1.00 \[ \log \left (2 \, \sqrt {e^{\left (2 \, x\right )} - 3} + 2 \, e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.77, size = 12, normalized size = 0.75 \[ \ln \left ({\mathrm {e}}^x+\sqrt {{\mathrm {e}}^{2\,x}-3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.67, size = 10, normalized size = 0.62 \[ \operatorname {acosh}{\left (\frac {\sqrt {3} e^{x}}{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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