Optimal. Leaf size=4 \[ \sinh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2249, 215} \[ \sinh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 2249
Rubi steps
\begin {align*} \int \frac {e^x}{\sqrt {1+e^{2 x}}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,e^x\right )\\ &=\sinh ^{-1}\left (e^x\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 4, normalized size = 1.00 \[ \sinh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 16, normalized size = 4.00 \[ -\log \left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 16, normalized size = 4.00 \[ -\log \left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 4, normalized size = 1.00 \[ \arcsinh \left ({\mathrm e}^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.65, size = 3, normalized size = 0.75 \[ \operatorname {arsinh}\left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 3, normalized size = 0.75 \[ \mathrm {asinh}\left ({\mathrm {e}}^x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 3, normalized size = 0.75 \[ \operatorname {asinh}{\left (e^{x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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