Optimal. Leaf size=26 \[ \frac {x^2+4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)} \]
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Rubi [A]
time = 0.67, antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps
used = 25, number of rules used = 11, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.234, Rules used = {12, 6874,
6820, 2334, 2335, 2395, 2346, 2209, 2343, 2413, 6617} \begin {gather*} \frac {4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)}+\frac {x}{e^5 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2334
Rule 2335
Rule 2343
Rule 2346
Rule 2395
Rule 2413
Rule 6617
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-x^2+\log (x) \left (8+x^2-4 \log \left (\frac {x^2}{\log ^2(5)}\right )\right )-4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log ^2(x)} \, dx}{e^5}\\ &=\frac {\int \left (\frac {-x^2+8 \log (x)+x^2 \log (x)}{x^2 \log ^2(x)}-\frac {4 (1+\log (x)) \log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log ^2(x)}\right ) \, dx}{e^5}\\ &=\frac {\int \frac {-x^2+8 \log (x)+x^2 \log (x)}{x^2 \log ^2(x)} \, dx}{e^5}-\frac {4 \int \frac {(1+\log (x)) \log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log ^2(x)} \, dx}{e^5}\\ &=\frac {\int \frac {-1+\log (x)+\frac {8 \log (x)}{x^2}}{\log ^2(x)} \, dx}{e^5}-\frac {4 \int \left (\frac {\log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log ^2(x)}+\frac {\log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log (x)}\right ) \, dx}{e^5}\\ &=\frac {\int \left (-\frac {1}{\log ^2(x)}+\frac {8+x^2}{x^2 \log (x)}\right ) \, dx}{e^5}-\frac {4 \int \frac {\log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log ^2(x)} \, dx}{e^5}-\frac {4 \int \frac {\log \left (\frac {x^2}{\log ^2(5)}\right )}{x^2 \log (x)} \, dx}{e^5}\\ &=\frac {4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)}-\frac {\int \frac {1}{\log ^2(x)} \, dx}{e^5}+\frac {\int \frac {8+x^2}{x^2 \log (x)} \, dx}{e^5}+\frac {8 \int \frac {\text {Ei}(-\log (x))}{x} \, dx}{e^5}+\frac {8 \int \frac {-1-x \text {Ei}(-\log (x)) \log (x)}{x^2 \log (x)} \, dx}{e^5}\\ &=\frac {x}{e^5 \log (x)}+\frac {4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)}+\frac {\int \left (\frac {1}{\log (x)}+\frac {8}{x^2 \log (x)}\right ) \, dx}{e^5}-\frac {\int \frac {1}{\log (x)} \, dx}{e^5}+\frac {8 \int \left (-\frac {\text {Ei}(-\log (x))}{x}-\frac {1}{x^2 \log (x)}\right ) \, dx}{e^5}+\frac {8 \text {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x))}{e^5}\\ &=\frac {8}{e^5 x}+\frac {x}{e^5 \log (x)}+\frac {8 \text {Ei}(-\log (x)) \log (x)}{e^5}+\frac {4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)}-\frac {\text {li}(x)}{e^5}+\frac {\int \frac {1}{\log (x)} \, dx}{e^5}-\frac {8 \int \frac {\text {Ei}(-\log (x))}{x} \, dx}{e^5}\\ &=\frac {8}{e^5 x}+\frac {x}{e^5 \log (x)}+\frac {8 \text {Ei}(-\log (x)) \log (x)}{e^5}+\frac {4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)}-\frac {8 \text {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x))}{e^5}\\ &=\frac {x}{e^5 \log (x)}+\frac {4 \log \left (\frac {x^2}{\log ^2(5)}\right )}{e^5 x \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 26, normalized size = 1.00 \begin {gather*} \frac {x^2+4 \log \left (x^2\right )-8 \log (\log (5))}{e^5 x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 48, normalized size = 1.85
method | result | size |
default | \({\mathrm e}^{-5} \left (\frac {x}{\ln \left (x \right )}+\frac {4 \ln \left (x^{2}\right )-8 \ln \left (x \right )}{x \ln \left (x \right )}-\frac {8 \ln \left (\ln \left (5\right )\right )}{x \ln \left (x \right )}+\frac {8}{x}\right )\) | \(48\) |
risch | \(\frac {8 \,{\mathrm e}^{-5}}{x}-\frac {{\mathrm e}^{-5} \left (2 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-x^{2}+8 \ln \left (\ln \left (5\right )\right )\right )}{x \ln \left (x \right )}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.29, size = 68, normalized size = 2.62 \begin {gather*} {\left (8 \, {\rm Ei}\left (-\log \left (x\right )\right ) \log \left (x\right ) - 8 \, \Gamma \left (-1, \log \left (x\right )\right ) \log \left (x\right ) - 4 \, {\rm Ei}\left (-\log \left (x\right )\right ) \log \left (\frac {x^{2}}{\log \left (5\right )^{2}}\right ) + 4 \, \Gamma \left (-1, \log \left (x\right )\right ) \log \left (\frac {x^{2}}{\log \left (5\right )^{2}}\right ) + \frac {8}{x} + {\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right )\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 42, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (x^{2} + 4 \, \log \left (\frac {x^{2}}{\log \left (5\right )^{2}}\right )\right )}}{x e^{5} \log \left (\log \left (5\right )^{2}\right ) + x e^{5} \log \left (\frac {x^{2}}{\log \left (5\right )^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 24, normalized size = 0.92 \begin {gather*} \frac {x^{2} - 8 \log {\left (\log {\left (5 \right )} \right )}}{x e^{5} \log {\left (x \right )}} + \frac {8}{x e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 26, normalized size = 1.00 \begin {gather*} {\left (\frac {8}{x} + \frac {x^{2} - 8 \, \log \left (\log \left (5\right )\right )}{x \log \left (x\right )}\right )} e^{\left (-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.15, size = 25, normalized size = 0.96 \begin {gather*} \frac {{\mathrm {e}}^{-5}\,\left (4\,\ln \left (x^2\right )-8\,\ln \left (\ln \left (5\right )\right )+x^2\right )}{x\,\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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