Optimal. Leaf size=20 \[ 2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {-1+e^x}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2281, 223, 212}
\begin {gather*} 2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {e^x-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 2281
Rubi steps
\begin {align*} \int \frac {e^{x/2}}{\sqrt {-1+e^x}} \, dx &=2 \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,e^{x/2}\right )\\ &=2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {e^{x/2}}{\sqrt {-1+e^x}}\right )\\ &=2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {-1+e^x}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} 2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {-1+e^x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.88, size = 8, normalized size = 0.40 \begin {gather*} 2 \text {ArcCosh}\left [E^{\frac {x}{2}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{\frac {x}{2}}}{\sqrt {-1+{\mathrm e}^{x}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 0.90 \begin {gather*} 2 \, \log \left (2 \, \sqrt {e^{x} - 1} + 2 \, e^{\left (\frac {1}{2} \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 16, normalized size = 0.80 \begin {gather*} -2 \, \log \left (\sqrt {e^{x} - 1} - e^{\left (\frac {1}{2} \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 7, normalized size = 0.35 \begin {gather*} 2 \operatorname {acosh}{\left (e^{\frac {x}{2}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 24, normalized size = 1.20 \begin {gather*} -2 \ln \left (-\sqrt {\left (\mathrm {e}^{\frac {x}{2}}\right )^{2}-1}+\mathrm {e}^{\frac {x}{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 16, normalized size = 0.80 \begin {gather*} \ln \left ({\mathrm {e}}^x+\sqrt {{\mathrm {e}}^x}\,\sqrt {{\mathrm {e}}^x-1}-\frac {1}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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