Optimal. Leaf size=12 \[ \frac {1}{2} e^x \log \left (\log \left (x^2\right )\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {12, 6820, 2326}
\begin {gather*} \frac {1}{2} e^x \log \left (\log \left (x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2 e^x+e^x x \log \left (x^2\right ) \log \left (\log \left (x^2\right )\right )}{x \log \left (x^2\right )} \, dx\\ &=\frac {1}{2} \int e^x \left (\frac {2}{x \log \left (x^2\right )}+\log \left (\log \left (x^2\right )\right )\right ) \, dx\\ &=\frac {1}{2} e^x \log \left (\log \left (x^2\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{2} e^x \log \left (\log \left (x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.04, size = 39, normalized size = 3.25
method | result | size |
risch | \(\frac {{\mathrm e}^{x} \ln \left (2 \ln \left (x \right )-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \left (-\mathrm {csgn}\left (i x^{2}\right )+\mathrm {csgn}\left (i x \right )\right )^{2}}{2}\right )}{2}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 14, normalized size = 1.17 \begin {gather*} \frac {1}{2} \, e^{x} \log \left (2\right ) + \frac {1}{2} \, e^{x} \log \left (\log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 9, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, e^{x} \log \left (\log \left (x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.37, size = 10, normalized size = 0.83 \begin {gather*} \frac {e^{x} \log {\left (\log {\left (x^{2} \right )} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 9, normalized size = 0.75 \begin {gather*} \frac {1}{2} \, e^{x} \log \left (\log \left (x^{2}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.90, size = 9, normalized size = 0.75 \begin {gather*} \frac {{\mathrm {e}}^x\,\ln \left (\ln \left (x^2\right )\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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