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 = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 195
Rule 215
Rule 2249
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.41, size = 29, normalized size = 1.07 \[ \frac {1}{2} \, \sqrt {e^{\left (2 \, x\right )} + 1} e^{x} - \frac {1}{2} \, \log \left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 29, normalized size = 1.07 \[ \frac {1}{2} \, \sqrt {e^{\left (2 \, x\right )} + 1} e^{x} - \frac {1}{2} \, \log \left (\sqrt {e^{\left (2 \, x\right )} + 1} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 0.70 \[ \frac {\arcsinh \left ({\mathrm e}^{x}\right )}{2}+\frac {\sqrt {{\mathrm e}^{2 x}+1}\, {\mathrm e}^{x}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.69, size = 18, normalized size = 0.67 \[ \frac {1}{2} \, \sqrt {e^{\left (2 \, x\right )} + 1} e^{x} + \frac {1}{2} \, \operatorname {arsinh}\left (e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 18, normalized size = 0.67 \[ \frac {\mathrm {asinh}\left ({\mathrm {e}}^x\right )}{2}+\frac {{\mathrm {e}}^x\,\sqrt {{\mathrm {e}}^{2\,x}+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e^{2 x} + 1} e^{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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