Optimal. Leaf size=21 \[ -1+\frac {2+e^x}{x}-\frac {2 (-5+e) x}{\log ^2(x)} \]
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Rubi [A] time = 0.37, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 6, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6, 6688, 14, 2197, 2297, 2298} \begin {gather*} \frac {e^x}{x}+\frac {2}{x}+\frac {2 (5-e) x}{\log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 14
Rule 2197
Rule 2297
Rule 2298
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-20+4 e) x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx\\ &=\int \left (\frac {-2+e^x (-1+x)}{x^2}+\frac {4 (-5+e)}{\log ^3(x)}-\frac {2 (-5+e)}{\log ^2(x)}\right ) \, dx\\ &=(2 (5-e)) \int \frac {1}{\log ^2(x)} \, dx-(4 (5-e)) \int \frac {1}{\log ^3(x)} \, dx+\int \frac {-2+e^x (-1+x)}{x^2} \, dx\\ &=\frac {2 (5-e) x}{\log ^2(x)}-\frac {2 (5-e) x}{\log (x)}-(2 (5-e)) \int \frac {1}{\log ^2(x)} \, dx+(2 (5-e)) \int \frac {1}{\log (x)} \, dx+\int \left (-\frac {2}{x^2}+\frac {e^x (-1+x)}{x^2}\right ) \, dx\\ &=\frac {2}{x}+\frac {2 (5-e) x}{\log ^2(x)}+2 (5-e) \text {li}(x)-(2 (5-e)) \int \frac {1}{\log (x)} \, dx+\int \frac {e^x (-1+x)}{x^2} \, dx\\ &=\frac {2}{x}+\frac {e^x}{x}+\frac {2 (5-e) x}{\log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 23, normalized size = 1.10 \begin {gather*} \frac {2}{x}+\frac {e^x}{x}-\frac {2 (-5+e) x}{\log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 32, normalized size = 1.52 \begin {gather*} -\frac {2 \, x^{2} e - {\left (e^{x} + 2\right )} \log \relax (x)^{2} - 10 \, x^{2}}{x \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 36, normalized size = 1.71 \begin {gather*} -\frac {2 \, x^{2} e - e^{x} \log \relax (x)^{2} - 10 \, x^{2} - 2 \, \log \relax (x)^{2}}{x \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 = 21, normalized size = 1.00
method | result | size |
risch | \(\frac {{\mathrm e}^{x}+2}{x}-\frac {2 \left ({\mathrm e}-5\right ) x}{\ln \relax (x )^{2}}\) | \(21\) |
default | \(-\frac {2 \,{\mathrm e} x}{\ln \relax (x )^{2}}+\frac {10 x}{\ln \relax (x )^{2}}+\frac {{\mathrm e}^{x}}{x}+\frac {2}{x}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.44, size = 51, normalized size = 2.43 \begin {gather*} -2 \, e \Gamma \left (-1, -\log \relax (x)\right ) - 4 \, e \Gamma \left (-2, -\log \relax (x)\right ) + \frac {2}{x} + {\rm Ei}\relax (x) - \Gamma \left (-1, -x\right ) + 10 \, \Gamma \left (-1, -\log \relax (x)\right ) + 20 \, \Gamma \left (-2, -\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.29, size = 28, normalized size = 1.33 \begin {gather*} \frac {{\mathrm {e}}^x}{x}+\frac {10\,x}{{\ln \relax (x)}^2}+\frac {2}{x}-\frac {2\,x\,\mathrm {e}}{{\ln \relax (x)}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.28, size = 22, normalized size = 1.05 \begin {gather*} \frac {- 2 e x + 10 x}{\log {\relax (x )}^{2}} + \frac {e^{x}}{x} + \frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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