Optimal. Leaf size=12 \[ x-\log \left (1+3 e^x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2320, 36, 29, 31}
\begin {gather*} x-\log \left (3 e^x+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2320
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {1}{x (1+3 x)} \, dx,x,e^x\right )\\ &=-\left (3 \text {Subst}\left (\int \frac {1}{1+3 x} \, dx,x,e^x\right )\right )+\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )\\ &=x-\log \left (1+3 e^x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.25 \begin {gather*} \log \left (e^x\right )-\log \left (1+3 e^x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 14, normalized size = 1.17
method | result | size |
risch | \(x -\ln \left (\frac {1}{3}+{\mathrm e}^{x}\right )\) | \(10\) |
parallelrisch | \(x -\ln \left (\frac {1}{3}+{\mathrm e}^{x}\right )\) | \(10\) |
norman | \(x -\ln \left (1+3 \,{\mathrm e}^{x}\right )\) | \(12\) |
derivativedivides | \(-\ln \left (1+3 \,{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(14\) |
default | \(-\ln \left (1+3 \,{\mathrm e}^{x}\right )+\ln \left ({\mathrm e}^{x}\right )\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 11, normalized size = 0.92 \begin {gather*} x - \log \left (3 \, e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.58, size = 11, normalized size = 0.92 \begin {gather*} x - \log \left (3 \, e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} x - \log {\left (e^{x} + \frac {1}{3} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 11, normalized size = 0.92 \begin {gather*} x - \log \left (3 \, e^{x} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 11, normalized size = 0.92 \begin {gather*} x-\ln \left (3\,{\mathrm {e}}^x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 1.58 \begin {gather*} \frac {\ln \left (1+3 \,{\mathrm e}^{x}\right )}{3}-\frac {\ln \left (-2+3 \,{\mathrm e}^{x}\right )}{3} \end {gather*}
Warning: Unable to verify antiderivative.
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