Optimal. Leaf size=23 \[ \frac {5 e^5}{3 x^2}+x+\frac {x^2}{4 \log ^2(x)} \]
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Rubi [A] time = 0.20, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {12, 6688, 2306, 2309, 2178} \begin {gather*} \frac {5 e^5}{3 x^2}+\frac {x^2}{4 \log ^2(x)}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2306
Rule 2309
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{6} \int \frac {-3 x^4+3 x^4 \log (x)+\left (-20 e^5+6 x^3\right ) \log ^3(x)}{x^3 \log ^3(x)} \, dx\\ &=\frac {1}{6} \int \left (6-\frac {20 e^5}{x^3}-\frac {3 x}{\log ^3(x)}+\frac {3 x}{\log ^2(x)}\right ) \, dx\\ &=\frac {5 e^5}{3 x^2}+x-\frac {1}{2} \int \frac {x}{\log ^3(x)} \, dx+\frac {1}{2} \int \frac {x}{\log ^2(x)} \, dx\\ &=\frac {5 e^5}{3 x^2}+x+\frac {x^2}{4 \log ^2(x)}-\frac {x^2}{2 \log (x)}-\frac {1}{2} \int \frac {x}{\log ^2(x)} \, dx+\int \frac {x}{\log (x)} \, dx\\ &=\frac {5 e^5}{3 x^2}+x+\frac {x^2}{4 \log ^2(x)}-\int \frac {x}{\log (x)} \, dx+\operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {5 e^5}{3 x^2}+x+\text {Ei}(2 \log (x))+\frac {x^2}{4 \log ^2(x)}-\operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {5 e^5}{3 x^2}+x+\frac {x^2}{4 \log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} \frac {5 e^5}{3 x^2}+x+\frac {x^2}{4 \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 31, normalized size = 1.35 \begin {gather*} \frac {3 \, x^{4} + 4 \, {\left (3 \, x^{3} + 5 \, e^{5}\right )} \log \relax (x)^{2}}{12 \, x^{2} \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 32, normalized size = 1.39 \begin {gather*} \frac {12 \, x^{3} \log \relax (x)^{2} + 3 \, x^{4} + 20 \, e^{5} \log \relax (x)^{2}}{12 \, x^{2} \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 19, normalized size = 0.83
method | result | size |
default | \(\frac {5 \,{\mathrm e}^{5}}{3 x^{2}}+x +\frac {x^{2}}{4 \ln \relax (x )^{2}}\) | \(19\) |
risch | \(\frac {3 x^{3}+5 \,{\mathrm e}^{5}}{3 x^{2}}+\frac {x^{2}}{4 \ln \relax (x )^{2}}\) | \(26\) |
norman | \(\frac {x^{3} \ln \relax (x )^{2}+\frac {x^{4}}{4}+\frac {5 \,{\mathrm e}^{5} \ln \relax (x )^{2}}{3}}{x^{2} \ln \relax (x )^{2}}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.47, size = 23, normalized size = 1.00 \begin {gather*} x + \frac {5 \, e^{5}}{3 \, x^{2}} + \Gamma \left (-1, -2 \, \log \relax (x)\right ) + 2 \, \Gamma \left (-2, -2 \, \log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.46, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^3+\frac {5\,{\mathrm {e}}^5}{3}}{x^2}+\frac {x^2}{4\,{\ln \relax (x)}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 20, normalized size = 0.87 \begin {gather*} \frac {x^{2}}{4 \log {\relax (x )}^{2}} + x + \frac {5 e^{5}}{3 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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