Optimal. Leaf size=34 \[ e^{3/x}-x+e^{\frac {4+x}{\log (5)}} \left (4-e^3-x+\log (x)\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(34)=68\).
time = 0.50, antiderivative size = 95, normalized size of antiderivative = 2.79, number of steps
used = 12, number of rules used = 8, integrand size = 158, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {6820, 2240,
2230, 2207, 2225, 2209, 2634, 12} \begin {gather*} -x+e^{3/x}+x \left (-e^{\frac {x}{\log (5)}+\frac {4}{\log (5)}}\right )+e^{\frac {x}{\log (5)}+\frac {4}{\log (5)}} \log (x)+\log (5) e^{\frac {x}{\log (5)}+\frac {4}{\log (5)}}+\left (4-125^{\frac {1}{\log (5)}}-\log (5)\right ) e^{\frac {x}{\log (5)}+\frac {4}{\log (5)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2240
Rule 2634
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1-\frac {3 e^{3/x}}{x^2}-\frac {e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \left (x^2-x \left (4-125^{\frac {1}{\log (5)}}-\log (5)\right )-\log (5)\right )}{x \log (5)}+\frac {e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (x)}{\log (5)}\right ) \, dx\\ &=-x-3 \int \frac {e^{3/x}}{x^2} \, dx-\frac {\int \frac {e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \left (x^2-x \left (4-125^{\frac {1}{\log (5)}}-\log (5)\right )-\log (5)\right )}{x} \, dx}{\log (5)}+\frac {\int e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (x) \, dx}{\log (5)}\\ &=e^{3/x}-x+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (x)-\frac {\int \frac {e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (5)}{x} \, dx}{\log (5)}-\frac {\int \left (e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} x-\frac {e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (5)}{x}+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \left (-4+125^{\frac {1}{\log (5)}}+\log (5)\right )\right ) \, dx}{\log (5)}\\ &=e^{3/x}-x+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (x)-\frac {\int e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} x \, dx}{\log (5)}+\frac {\left (4-125^{\frac {1}{\log (5)}}-\log (5)\right ) \int e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \, dx}{\log (5)}\\ &=e^{3/x}-x-e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} x+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \left (4-125^{\frac {1}{\log (5)}}-\log (5)\right )+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (x)+\int e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \, dx\\ &=e^{3/x}-x-e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} x+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \left (4-125^{\frac {1}{\log (5)}}-\log (5)\right )+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (5)+e^{\frac {4}{\log (5)}+\frac {x}{\log (5)}} \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.29, size = 45, normalized size = 1.32 \begin {gather*} e^{3/x}-x-e^{\frac {4+x}{\log (5)}} \left (-4+125^{\frac {1}{\log (5)}}+x\right )+e^{\frac {4+x}{\log (5)}} \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.79, size = 32, normalized size = 0.94
method | result | size |
risch | \({\mathrm e}^{\frac {3}{x}}-x +\left (\ln \left (x \right )+4-{\mathrm e}^{3}-x \right ) {\mathrm e}^{\frac {4+x}{\ln \left (5\right )}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 58, normalized size = 1.71 \begin {gather*} -{\left (x e^{\frac {4}{\log \left (5\right )}} - e^{\frac {4}{\log \left (5\right )}} \log \left (x\right ) - 4 \, e^{\frac {4}{\log \left (5\right )}} + e^{\left (\frac {4}{\log \left (5\right )} + 3\right )}\right )} e^{\frac {x}{\log \left (5\right )}} - x + e^{\frac {3}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 34, normalized size = 1.00 \begin {gather*} -x + e^{\left (\frac {\log \left (5\right ) \log \left (-x - e^{3} + \log \left (x\right ) + 4\right ) + x + 4}{\log \left (5\right )}\right )} + e^{\frac {3}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.84, size = 29, normalized size = 0.85 \begin {gather*} - x + e^{\frac {3}{x}} + e^{\frac {x + \log {\left (5 \right )} \log {\left (- x + \log {\left (x \right )} - e^{3} + 4 \right )} + 4}{\log {\left (5 \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs.
\(2 (32) = 64\).
time = 0.47, size = 77, normalized size = 2.26 \begin {gather*} -x e^{\left (\frac {x}{\log \left (5\right )} + \frac {4}{\log \left (5\right )}\right )} + e^{\left (\frac {x}{\log \left (5\right )} + \frac {4}{\log \left (5\right )}\right )} \log \left (x\right ) - x - e^{\left (\frac {x}{\log \left (5\right )} + \frac {4}{\log \left (5\right )} + 3\right )} + 4 \, e^{\left (\frac {x}{\log \left (5\right )} + \frac {4}{\log \left (5\right )}\right )} + e^{\frac {3}{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.60, size = 77, normalized size = 2.26 \begin {gather*} 4\,{\mathrm {e}}^{\frac {x}{\ln \left (5\right )}+\frac {4}{\ln \left (5\right )}}-x-{\mathrm {e}}^{\frac {x}{\ln \left (5\right )}+\frac {4}{\ln \left (5\right )}+3}+{\mathrm {e}}^{3/x}-x\,{\mathrm {e}}^{\frac {x}{\ln \left (5\right )}+\frac {4}{\ln \left (5\right )}}+{\mathrm {e}}^{\frac {x}{\ln \left (5\right )}+\frac {4}{\ln \left (5\right )}}\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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