Optimal. Leaf size=26 \[ \sqrt {e^{2 x}-1}-\tan ^{-1}\left (\sqrt {e^{2 x}-1}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2282, 50, 63, 203} \[ \sqrt {e^{2 x}-1}-\tan ^{-1}\left (\sqrt {e^{2 x}-1}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 2282
Rubi steps
\begin {align*} \int \sqrt {-1+e^{2 x}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,e^{2 x}\right )\\ &=\sqrt {-1+e^{2 x}}-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,e^{2 x}\right )\\ &=\sqrt {-1+e^{2 x}}-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+e^{2 x}}\right )\\ &=\sqrt {-1+e^{2 x}}-\tan ^{-1}\left (\sqrt {-1+e^{2 x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \sqrt {e^{2 x}-1}-\tan ^{-1}\left (\sqrt {e^{2 x}-1}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 20, normalized size = 0.77 \[ \sqrt {e^{\left (2 \, x\right )} - 1} - \arctan \left (\sqrt {e^{\left (2 \, x\right )} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 20, normalized size = 0.77 \[ \sqrt {e^{\left (2 \, x\right )} - 1} - \arctan \left (\sqrt {e^{\left (2 \, x\right )} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.81 \[ -\arctan \left (\sqrt {{\mathrm e}^{2 x}-1}\right )+\sqrt {{\mathrm e}^{2 x}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.96, size = 20, normalized size = 0.77 \[ \sqrt {e^{\left (2 \, x\right )} - 1} - \arctan \left (\sqrt {e^{\left (2 \, x\right )} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 31, normalized size = 1.19 \[ \sqrt {{\mathrm {e}}^{2\,x}-1}\,\left (\frac {{\mathrm {e}}^{-x}\,\mathrm {asin}\left ({\mathrm {e}}^{-x}\right )}{\sqrt {1-{\mathrm {e}}^{-2\,x}}}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.39, size = 19, normalized size = 0.73 \[ \begin {cases} \sqrt {e^{2 x} - 1} - \operatorname {acos}{\left (e^{- x} \right )} & \text {for}\: e^{x} < 0 \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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