Optimal. Leaf size=19 \[ \frac {e \left (4-\frac {4}{4+\log \left (\frac {x}{15}\right )}\right )}{x^2} \]
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Rubi [A]
time = 0.38, antiderivative size = 23, normalized size of antiderivative = 1.21, number of steps
used = 9, number of rules used = 6, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6820, 12, 6874,
2343, 2346, 2209} \begin {gather*} \frac {4 e}{x^2}-\frac {4 e}{x^2 \left (\log \left (\frac {x}{15}\right )+4\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e \left (-23-14 \log \left (\frac {x}{15}\right )-2 \log ^2\left (\frac {x}{15}\right )\right )}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )^2} \, dx\\ &=(4 e) \int \frac {-23-14 \log \left (\frac {x}{15}\right )-2 \log ^2\left (\frac {x}{15}\right )}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )^2} \, dx\\ &=(4 e) \int \left (-\frac {2}{x^3}+\frac {1}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )^2}+\frac {2}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )}\right ) \, dx\\ &=\frac {4 e}{x^2}+(4 e) \int \frac {1}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )^2} \, dx+(8 e) \int \frac {1}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )} \, dx\\ &=\frac {4 e}{x^2}-\frac {4 e}{x^2 \left (4+\log \left (\frac {x}{15}\right )\right )}+\frac {1}{225} (8 e) \text {Subst}\left (\int \frac {e^{-2 x}}{4+x} \, dx,x,\log \left (\frac {x}{15}\right )\right )-(8 e) \int \frac {1}{x^3 \left (4+\log \left (\frac {x}{15}\right )\right )} \, dx\\ &=\frac {4 e}{x^2}+\frac {8}{225} e^9 \text {Ei}\left (-2 \left (4+\log \left (\frac {x}{15}\right )\right )\right )-\frac {4 e}{x^2 \left (4+\log \left (\frac {x}{15}\right )\right )}-\frac {1}{225} (8 e) \text {Subst}\left (\int \frac {e^{-2 x}}{4+x} \, dx,x,\log \left (\frac {x}{15}\right )\right )\\ &=\frac {4 e}{x^2}-\frac {4 e}{x^2 \left (4+\log \left (\frac {x}{15}\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 18, normalized size = 0.95 \begin {gather*} -\frac {4 e \left (-1+\frac {1}{4+\log \left (\frac {x}{15}\right )}\right )}{x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.35, size = 22, normalized size = 1.16
method | result | size |
derivativedivides | \(\frac {4 \,{\mathrm e} \left (\ln \left (\frac {x}{15}\right )+3\right )}{\left (4+\ln \left (\frac {x}{15}\right )\right ) x^{2}}\) | \(22\) |
default | \(\frac {4 \,{\mathrm e} \left (\ln \left (\frac {x}{15}\right )+3\right )}{\left (4+\ln \left (\frac {x}{15}\right )\right ) x^{2}}\) | \(22\) |
risch | \(\frac {4 \,{\mathrm e}}{x^{2}}-\frac {4 \,{\mathrm e}}{x^{2} \left (4+\ln \left (\frac {x}{15}\right )\right )}\) | \(24\) |
norman | \(\frac {4 \,{\mathrm e} \ln \left (\frac {x}{15}\right )+12 \,{\mathrm e}}{x^{2} \left (4+\ln \left (\frac {x}{15}\right )\right )}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 38 vs.
\(2 (17) = 34\).
time = 0.50, size = 38, normalized size = 2.00 \begin {gather*} \frac {4 \, {\left ({\left (\log \left (5\right ) + \log \left (3\right ) - 3\right )} e - e \log \left (x\right )\right )}}{x^{2} {\left (\log \left (5\right ) + \log \left (3\right ) - 4\right )} - x^{2} \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 30, normalized size = 1.58 \begin {gather*} \frac {4 \, {\left (e \log \left (\frac {1}{15} \, x\right ) + 3 \, e\right )}}{x^{2} \log \left (\frac {1}{15} \, x\right ) + 4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 26, normalized size = 1.37 \begin {gather*} - \frac {4 e}{x^{2} \log {\left (\frac {x}{15} \right )} + 4 x^{2}} + \frac {4 e}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 30, normalized size = 1.58 \begin {gather*} \frac {4 \, {\left (e \log \left (\frac {1}{15} \, x\right ) + 3 \, e\right )}}{x^{2} \log \left (\frac {1}{15} \, x\right ) + 4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.18, size = 21, normalized size = 1.11 \begin {gather*} \frac {4\,\mathrm {e}\,\left (\ln \left (\frac {x}{15}\right )+3\right )}{x^2\,\left (\ln \left (\frac {x}{15}\right )+4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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