Optimal. Leaf size=12 \[ -2 \tanh ^{-1}\left (\sqrt {1+e^x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2320, 65, 213}
\begin {gather*} -2 \tanh ^{-1}\left (\sqrt {e^x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 213
Rule 2320
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+e^x}} \, dx &=\text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,e^x\right )\\ &=2 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+e^x}\right )\\ &=-2 \tanh ^{-1}\left (\sqrt {1+e^x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 12, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\sqrt {1+e^x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(12)=24\).
time = 2.91, size = 23, normalized size = 1.92 \begin {gather*} \text {Log}\left [-1+\frac {1}{\sqrt {1+E^x}}\right ]-\text {Log}\left [1+\frac {1}{\sqrt {1+E^x}}\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 10, normalized size = 0.83
method | result | size |
derivativedivides | \(-2 \arctanh \left (\sqrt {1+{\mathrm e}^{x}}\right )\) | \(10\) |
default | \(-2 \arctanh \left (\sqrt {1+{\mathrm e}^{x}}\right )\) | \(10\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (9) = 18\).
time = 0.25, size = 21, normalized size = 1.75 \begin {gather*} -\log \left (\sqrt {e^{x} + 1} + 1\right ) + \log \left (\sqrt {e^{x} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 21 vs.
\(2 (9) = 18\).
time = 0.34, size = 21, normalized size = 1.75 \begin {gather*} -\log \left (\sqrt {e^{x} + 1} + 1\right ) + \log \left (\sqrt {e^{x} + 1} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs.
\(2 (12) = 24\)
time = 0.68, size = 26, normalized size = 2.17 \begin {gather*} \log {\left (-1 + \frac {1}{\sqrt {e^{x} + 1}} \right )} - \log {\left (1 + \frac {1}{\sqrt {e^{x} + 1}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 23 vs.
\(2 (9) = 18\).
time = 0.00, size = 32, normalized size = 2.67 \begin {gather*} 2 \left (\frac {\ln \left (\sqrt {\mathrm {e}^{x}+1}-1\right )}{2}-\frac {\ln \left (\sqrt {\mathrm {e}^{x}+1}+1\right )}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 9, normalized size = 0.75 \begin {gather*} -2\,\mathrm {atanh}\left (\sqrt {{\mathrm {e}}^x+1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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