Optimal. Leaf size=27 \[ \frac{1}{2} e^x \sqrt{e^{2 x}+1}+\frac{1}{2} \sinh ^{-1}\left (e^x\right ) \]
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Rubi [A] time = 0.022784, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2249, 195, 215} \[ \frac{1}{2} e^x \sqrt{e^{2 x}+1}+\frac{1}{2} \sinh ^{-1}\left (e^x\right ) \]
Antiderivative was successfully verified.
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Rule 2249
Rule 195
Rule 215
Rubi steps
\begin{align*} \int e^x \sqrt{1+e^{2 x}} \, dx &=\operatorname{Subst}\left (\int \sqrt{1+x^2} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{1+e^{2 x}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,e^x\right )\\ &=\frac{1}{2} e^x \sqrt{1+e^{2 x}}+\frac{1}{2} \sinh ^{-1}\left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.0093585, size = 24, normalized size = 0.89 \[ \frac{1}{2} \left (e^x \sqrt{e^{2 x}+1}+\sinh ^{-1}\left (e^x\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 19, normalized size = 0.7 \begin{align*}{\frac{{{\rm e}^{x}}}{2}\sqrt{1+ \left ({{\rm e}^{x}} \right ) ^{2}}}+{\frac{{\it Arcsinh} \left ({{\rm e}^{x}} \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48817, size = 24, normalized size = 0.89 \begin{align*} \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} + \frac{1}{2} \, \operatorname{arsinh}\left (e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.657948, size = 84, normalized size = 3.11 \begin{align*} \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} - \frac{1}{2} \, \log \left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e^{2 x} + 1} e^{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22607, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{2} \, \sqrt{e^{\left (2 \, x\right )} + 1} e^{x} - \frac{1}{2} \, \log \left (\sqrt{e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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