Optimal. Leaf size=20 \[ 2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {e^x-1}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.03, 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 = {2249, 217, 206} \[ 2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {e^x-1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^{x/2}}{\sqrt {-1+e^x}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,e^{x/2}\right )\\ &=2 \operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ 2 \tanh ^{-1}\left (\frac {e^{x/2}}{\sqrt {e^x-1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{x/2}}{\sqrt {-1+e^x}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 16, normalized size = 0.80 \[ -2 \, \log \left (\sqrt {e^{x} - 1} - e^{\left (\frac {1}{2} \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.93, size = 16, normalized size = 0.80 \[ -2 \, \log \left (-\sqrt {e^{x} - 1} + e^{\left (\frac {1}{2} \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.44, size = 18, normalized size = 0.90 \[ 2 \, \log \left (2 \, \sqrt {e^{x} - 1} + 2 \, e^{\left (\frac {1}{2} \, x\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.34, size = 16, normalized size = 0.80 \[ \ln \left ({\mathrm {e}}^x+\sqrt {{\mathrm {e}}^x}\,\sqrt {{\mathrm {e}}^x-1}-\frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.74, size = 7, normalized size = 0.35 \[ 2 \operatorname {acosh}{\left (e^{\frac {x}{2}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________