Optimal. Leaf size=28 \[ x \left (-1+\frac {x}{e}\right )+\left (-x+\frac {1}{81} \left (x+\frac {\log (x)}{x}\right )\right )^2 \]
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Rubi [A]
time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.29, number of steps
used = 8, number of rules used = 4, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 14, 2341,
2342} \begin {gather*} \frac {(6561+6400 e) x^2}{6561 e}+\frac {\log ^2(x)}{6561 x^2}-x-\frac {160 \log (x)}{6561} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2341
Rule 2342
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {13122 x^4+e \left (-160 x^2-6561 x^3+12800 x^4\right )+2 e \log (x)-2 e \log ^2(x)}{x^3} \, dx}{6561 e}\\ &=\frac {\int \left (\frac {-160 e-6561 e x+2 (6561+6400 e) x^2}{x}+\frac {2 e \log (x)}{x^3}-\frac {2 e \log ^2(x)}{x^3}\right ) \, dx}{6561 e}\\ &=\frac {2 \int \frac {\log (x)}{x^3} \, dx}{6561}-\frac {2 \int \frac {\log ^2(x)}{x^3} \, dx}{6561}+\frac {\int \frac {-160 e-6561 e x+2 (6561+6400 e) x^2}{x} \, dx}{6561 e}\\ &=-\frac {1}{13122 x^2}-\frac {\log (x)}{6561 x^2}+\frac {\log ^2(x)}{6561 x^2}-\frac {2 \int \frac {\log (x)}{x^3} \, dx}{6561}+\frac {\int \left (-6561 e-\frac {160 e}{x}+2 (6561+6400 e) x\right ) \, dx}{6561 e}\\ &=-x+\frac {(6561+6400 e) x^2}{6561 e}-\frac {160 \log (x)}{6561}+\frac {\log ^2(x)}{6561 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.03, size = 35, normalized size = 1.25 \begin {gather*} \frac {-6561 e x+(6561+6400 e) x^2-160 e \log (x)+\frac {e \log ^2(x)}{x^2}}{6561 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs.
\(2(24)=48\).
time = 0.05, size = 74, normalized size = 2.64
method | result | size |
risch | \(\frac {\ln \left (x \right )^{2}}{6561 x^{2}}+\frac {6400 x^{2}}{6561}-x +{\mathrm e}^{-1} x^{2}-\frac {160 \ln \left (x \right )}{6561}\) | \(29\) |
norman | \(\frac {-\frac {160 x^{2} \ln \left (x \right )}{6561}-x^{3}+\frac {\ln \left (x \right )^{2}}{6561}+\frac {\left (6400 \,{\mathrm e}+6561\right ) {\mathrm e}^{-1} x^{4}}{6561}}{x^{2}}\) | \(39\) |
default | \(\frac {{\mathrm e}^{-1} \left (6400 x^{2} {\mathrm e}-6561 x \,{\mathrm e}+6561 x^{2}-2 \,{\mathrm e} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-160 \,{\mathrm e} \ln \left (x \right )+2 \,{\mathrm e} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )\right )}{6561}\) | \(74\) |
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. 61 vs.
\(2 (24) = 48\).
time = 0.26, size = 61, normalized size = 2.18 \begin {gather*} \frac {1}{13122} \, {\left (12800 \, x^{2} e + 13122 \, x^{2} - 13122 \, x e - {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} e - 320 \, e \log \left (x\right ) + \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} e}{x^{2}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 43, normalized size = 1.54 \begin {gather*} \frac {{\left (6561 \, x^{4} - 160 \, x^{2} e \log \left (x\right ) + e \log \left (x\right )^{2} + {\left (6400 \, x^{4} - 6561 \, x^{3}\right )} e\right )} e^{\left (-1\right )}}{6561 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 39, normalized size = 1.39 \begin {gather*} \frac {x^{2} \cdot \left (6561 + 6400 e\right ) - 6561 e x - 160 e \log {\left (x \right )}}{6561 e} + \frac {\log {\left (x \right )}^{2}}{6561 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 43, normalized size = 1.54 \begin {gather*} \frac {{\left (6400 \, x^{4} e + 6561 \, x^{4} - 6561 \, x^{3} e - 160 \, x^{2} e \log \left (x\right ) + e \log \left (x\right )^{2}\right )} e^{\left (-1\right )}}{6561 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.45, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\ln \left (x\right )}^2}{6561\,x^2}-\frac {160\,\ln \left (x\right )}{6561}-x+x^2\,\left ({\mathrm {e}}^{-1}+\frac {6400}{6561}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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