Optimal. Leaf size=29 \[ \frac {-\frac {\left (5-e^2\right )^2}{x^2}+x}{x^2}+(5+x) \log ^2(x) \]
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Rubi [A]
time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps
used = 9, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {14, 2388, 2338,
2332, 2333} \begin {gather*} -\frac {\left (5-e^2\right )^2}{x^4}+\frac {1}{x}+x \log ^2(x)+5 \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2332
Rule 2333
Rule 2338
Rule 2388
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {100-40 e^2+4 e^4-x^3}{x^5}+\frac {2 (5+x) \log (x)}{x}+\log ^2(x)\right ) \, dx\\ &=2 \int \frac {(5+x) \log (x)}{x} \, dx+\int \frac {100-40 e^2+4 e^4-x^3}{x^5} \, dx+\int \log ^2(x) \, dx\\ &=x \log ^2(x)+10 \int \frac {\log (x)}{x} \, dx+\int \left (\frac {4 \left (-5+e^2\right )^2}{x^5}-\frac {1}{x^2}\right ) \, dx\\ &=-\frac {\left (5-e^2\right )^2}{x^4}+\frac {1}{x}+5 \log ^2(x)+x \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 37, normalized size = 1.28 \begin {gather*} -\frac {25}{x^4}+\frac {10 e^2}{x^4}-\frac {e^4}{x^4}+\frac {1}{x}+5 \log ^2(x)+x \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 38, normalized size = 1.31
method | result | size |
risch | \(\left (5+x \right ) \ln \left (x \right )^{2}-\frac {-x^{3}+{\mathrm e}^{4}-10 \,{\mathrm e}^{2}+25}{x^{4}}\) | \(28\) |
norman | \(\frac {x^{3}+x^{5} \ln \left (x \right )^{2}+5 x^{4} \ln \left (x \right )^{2}-25-{\mathrm e}^{4}+10 \,{\mathrm e}^{2}}{x^{4}}\) | \(37\) |
default | \(x \ln \left (x \right )^{2}+5 \ln \left (x \right )^{2}+\frac {1}{x}-\frac {{\mathrm e}^{4}}{x^{4}}+\frac {10 \,{\mathrm e}^{2}}{x^{4}}-\frac {25}{x^{4}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 49, normalized size = 1.69 \begin {gather*} {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + 2 \, x \log \left (x\right ) + 5 \, \log \left (x\right )^{2} - 2 \, x + \frac {1}{x} - \frac {e^{4}}{x^{4}} + \frac {10 \, e^{2}}{x^{4}} - \frac {25}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 31, normalized size = 1.07 \begin {gather*} \frac {x^{3} + {\left (x^{5} + 5 \, x^{4}\right )} \log \left (x\right )^{2} - e^{4} + 10 \, e^{2} - 25}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 24, normalized size = 0.83 \begin {gather*} \left (x + 5\right ) \log {\left (x \right )}^{2} - \frac {- x^{3} - 10 e^{2} + 25 + e^{4}}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 34, normalized size = 1.17 \begin {gather*} \frac {x^{5} \log \left (x\right )^{2} + 5 \, x^{4} \log \left (x\right )^{2} + x^{3} - e^{4} + 10 \, e^{2} - 25}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 30, normalized size = 1.03 \begin {gather*} x\,{\ln \left (x\right )}^2+5\,{\ln \left (x\right )}^2+\frac {x^4-x\,{\left ({\mathrm {e}}^2-5\right )}^2}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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