Optimal. Leaf size=21 \[ 1-\frac {1}{e \log \left (e^{-x} x^{\frac {1}{x}}\right )} \]
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Rubi [F]
time = 0.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1-x^2-\log (x)}{e x^2 \log ^2\left (e^{-x} x^{\frac {1}{x}}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {1-x^2-\log (x)}{x^2 \log ^2\left (e^{-x} x^{\frac {1}{x}}\right )} \, dx}{e}\\ &=\frac {\int \left (-\frac {1}{\log ^2\left (e^{-x} x^{\frac {1}{x}}\right )}+\frac {1}{x^2 \log ^2\left (e^{-x} x^{\frac {1}{x}}\right )}-\frac {\log (x)}{x^2 \log ^2\left (e^{-x} x^{\frac {1}{x}}\right )}\right ) \, dx}{e}\\ &=-\frac {\int \frac {1}{\log ^2\left (e^{-x} x^{\frac {1}{x}}\right )} \, dx}{e}+\frac {\int \frac {1}{x^2 \log ^2\left (e^{-x} x^{\frac {1}{x}}\right )} \, dx}{e}-\frac {\int \frac {\log (x)}{x^2 \log ^2\left (e^{-x} x^{\frac {1}{x}}\right )} \, dx}{e}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 19, normalized size = 0.90 \begin {gather*} -\frac {1}{e \log \left (e^{-x} x^{\frac {1}{x}}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs.
\(2(23)=46\).
time = 4.25, size = 54, normalized size = 2.57
method | result | size |
default | \(-\frac {{\mathrm e}^{-1} x}{-x^{2}+\left (\ln \left ({\mathrm e}^{\frac {\ln \left (x \right )}{x}} {\mathrm e}^{-x}\right )-\frac {\ln \left (x \right )}{x}+\ln \left ({\mathrm e}^{x}\right )\right ) x -\left (\ln \left ({\mathrm e}^{x}\right )-x \right ) x +\ln \left (x \right )}\) | \(54\) |
risch | \(-\frac {2 i {\mathrm e}^{-1}}{\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x^{\frac {1}{x}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} x^{\frac {1}{x}}\right )-\pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} x^{\frac {1}{x}}\right )^{2}-\pi \,\mathrm {csgn}\left (i x^{\frac {1}{x}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-x} x^{\frac {1}{x}}\right )^{2}+\pi \mathrm {csgn}\left (i {\mathrm e}^{-x} x^{\frac {1}{x}}\right )^{3}-2 i \ln \left ({\mathrm e}^{x}\right )+2 i \ln \left (x^{\frac {1}{x}}\right )}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 15, normalized size = 0.71 \begin {gather*} \frac {e^{\left (-1\right )}}{x - \log \left (x^{\left (\frac {1}{x}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 17, normalized size = 0.81 \begin {gather*} \frac {x}{x^{2} e - e \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 15, normalized size = 0.71 \begin {gather*} - \frac {x}{- e x^{2} + e \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 14, normalized size = 0.67 \begin {gather*} \frac {x e^{\left (-1\right )}}{x^{2} - \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {{\mathrm {e}}^{-1}\,\left (\ln \left (x\right )+x^2-1\right )}{x^2\,{\ln \left ({\mathrm {e}}^{-x}\,{\mathrm {e}}^{\frac {\ln \left (x\right )}{x}}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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