Optimal. Leaf size=13 \[ -\frac {e+\log (x)}{x^2 \log (x)} \]
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Rubi [A]
time = 0.11, antiderivative size = 16, normalized size of antiderivative = 1.23, number of steps
used = 7, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6874, 2343,
2346, 2209} \begin {gather*} -\frac {1}{x^2}-\frac {e}{x^2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2343
Rule 2346
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2}{x^3}+\frac {e}{x^3 \log ^2(x)}+\frac {2 e}{x^3 \log (x)}\right ) \, dx\\ &=-\frac {1}{x^2}+e \int \frac {1}{x^3 \log ^2(x)} \, dx+(2 e) \int \frac {1}{x^3 \log (x)} \, dx\\ &=-\frac {1}{x^2}-\frac {e}{x^2 \log (x)}-(2 e) \int \frac {1}{x^3 \log (x)} \, dx+(2 e) \text {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {1}{x^2}+2 e \text {Ei}(-2 \log (x))-\frac {e}{x^2 \log (x)}-(2 e) \text {Subst}\left (\int \frac {e^{-2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {1}{x^2}-\frac {e}{x^2 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 16, normalized size = 1.23 \begin {gather*} -\frac {1}{x^2}-\frac {e}{x^2 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.04, size = 42, normalized size = 3.23
method | result | size |
risch | \(-\frac {1}{x^{2}}-\frac {{\mathrm e}}{x^{2} \ln \left (x \right )}\) | \(18\) |
norman | \(\frac {-{\mathrm e}-\ln \left (x \right )}{x^{2} \ln \left (x \right )}\) | \(20\) |
default | \(-\frac {1}{x^{2}}-2 \,{\mathrm e} \expIntegral \left (1, 2 \ln \left (x \right )\right )+{\mathrm e} \left (-\frac {1}{x^{2} \ln \left (x \right )}+2 \expIntegral \left (1, 2 \ln \left (x \right )\right )\right )\) | \(42\) |
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 = 25, normalized size = 1.92 \begin {gather*} 2 \, {\rm Ei}\left (-2 \, \log \left (x\right )\right ) e - 2 \, e \Gamma \left (-1, 2 \, \log \left (x\right )\right ) - \frac {1}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 14, normalized size = 1.08 \begin {gather*} -\frac {e + \log \left (x\right )}{x^{2} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 1.15 \begin {gather*} - \frac {1}{x^{2}} - \frac {e}{x^{2} \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 14, normalized size = 1.08 \begin {gather*} -\frac {e + \log \left (x\right )}{x^{2} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.50, size = 14, normalized size = 1.08 \begin {gather*} -\frac {\mathrm {e}+\ln \left (x\right )}{x^2\,\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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