Optimal. Leaf size=23 \[ 7+e^3+e^x+\frac {x}{2}-\frac {1}{x \log (\log (3))} \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps
used = 6, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {12, 14, 2225}
\begin {gather*} \frac {x}{2}+e^x-\frac {1}{x \log (\log (3))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2225
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {2+\left (x^2+2 e^x x^2\right ) \log (\log (3))}{x^2} \, dx}{2 \log (\log (3))}\\ &=\frac {\int \left (2 e^x \log (\log (3))+\frac {2+x^2 \log (\log (3))}{x^2}\right ) \, dx}{2 \log (\log (3))}\\ &=\frac {\int \frac {2+x^2 \log (\log (3))}{x^2} \, dx}{2 \log (\log (3))}+\int e^x \, dx\\ &=e^x+\frac {\int \left (\frac {2}{x^2}+\log (\log (3))\right ) \, dx}{2 \log (\log (3))}\\ &=e^x+\frac {x}{2}-\frac {1}{x \log (\log (3))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 0.83 \begin {gather*} e^x+\frac {x}{2}-\frac {1}{x \log (\log (3))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 26, normalized size = 1.13
method | result | size |
risch | \(-\frac {1}{x \ln \left (\ln \left (3\right )\right )}+{\mathrm e}^{x}+\frac {x}{2}\) | \(17\) |
norman | \(\frac {{\mathrm e}^{x} x +\frac {x^{2}}{2}-\frac {1}{\ln \left (\ln \left (3\right )\right )}}{x}\) | \(22\) |
default | \(\frac {-\frac {2}{x}+2 \,{\mathrm e}^{x} \ln \left (\ln \left (3\right )\right )+\ln \left (\ln \left (3\right )\right ) x}{2 \ln \left (\ln \left (3\right )\right )}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 1.09 \begin {gather*} \frac {x \log \left (\log \left (3\right )\right ) + 2 \, e^{x} \log \left (\log \left (3\right )\right ) - \frac {2}{x}}{2 \, \log \left (\log \left (3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 25, normalized size = 1.09 \begin {gather*} \frac {{\left (x^{2} + 2 \, x e^{x}\right )} \log \left (\log \left (3\right )\right ) - 2}{2 \, x \log \left (\log \left (3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 19, normalized size = 0.83 \begin {gather*} \frac {x \log {\left (\log {\left (3 \right )} \right )} - \frac {2}{x}}{2 \log {\left (\log {\left (3 \right )} \right )}} + e^{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 27, normalized size = 1.17 \begin {gather*} \frac {x^{2} \log \left (\log \left (3\right )\right ) + 2 \, x e^{x} \log \left (\log \left (3\right )\right ) - 2}{2 \, x \log \left (\log \left (3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 16, normalized size = 0.70 \begin {gather*} \frac {x}{2}+{\mathrm {e}}^x-\frac {1}{x\,\ln \left (\ln \left (3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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