Optimal. Leaf size=29 \[ \log \left (e^{x/4}+\frac {x}{\log \left (\frac {5}{x^2}\right )}+\left (-1+\log \left (\frac {x}{3}\right )\right )^2\right ) \]
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Rubi [F]
time = 11.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\left (-8+e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )}{4 x^2 \log \left (\frac {5}{x^2}\right )+\left (4 x+4 e^{x/4} x\right ) \log ^2\left (\frac {5}{x^2}\right )-8 x \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )+4 x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\log ^2\left (\frac {5}{x^2}\right ) \left (-8+e^{x/4} x+8 \log \left (\frac {x}{3}\right )\right )}{4 x \log \left (\frac {5}{x^2}\right ) \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )} \, dx\\ &=\frac {1}{4} \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )+\log ^2\left (\frac {5}{x^2}\right ) \left (-8+e^{x/4} x+8 \log \left (\frac {x}{3}\right )\right )}{x \log \left (\frac {5}{x^2}\right ) \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )} \, dx\\ &=\frac {1}{4} \int \left (1+\frac {8 x+4 x \log \left (\frac {5}{x^2}\right )-x^2 \log \left (\frac {5}{x^2}\right )-8 \log ^2\left (\frac {5}{x^2}\right )-x (1+\log (9)) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )-x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+2 x \log ^2\left (\frac {5}{x^2}\right ) \log (x)}{x \log \left (\frac {5}{x^2}\right ) \left (x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )+\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )}\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{4} \int \frac {8 x+4 x \log \left (\frac {5}{x^2}\right )-x^2 \log \left (\frac {5}{x^2}\right )-8 \log ^2\left (\frac {5}{x^2}\right )-x (1+\log (9)) \log ^2\left (\frac {5}{x^2}\right )+8 \log ^2\left (\frac {5}{x^2}\right ) \log \left (\frac {x}{3}\right )-x \log ^2\left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )+2 x \log ^2\left (\frac {5}{x^2}\right ) \log (x)}{x \log \left (\frac {5}{x^2}\right ) \left (x+e^{x/4} \log \left (\frac {5}{x^2}\right )+(1+\log (9)) \log \left (\frac {5}{x^2}\right )+\log \left (\frac {5}{x^2}\right ) \log ^2\left (\frac {x}{3}\right )-2 \log \left (\frac {5}{x^2}\right ) \log (x)\right )} \, dx\\ &=\frac {x}{4}+\frac {1}{4} \int \frac {8 x-(-4+x) x \log \left (\frac {5}{x^2}\right )-\log ^2\left (\frac {5}{x^2}\right ) \left (8+x+x \log (9)-8 \log \left (\frac {x}{3}\right )+x \log ^2\left (\frac {x}{3}\right )-2 x \log (x)\right )}{x \log \left (\frac {5}{x^2}\right ) \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.33, size = 45, normalized size = 1.55 \begin {gather*} -\log \left (\log \left (\frac {5}{x^2}\right )\right )+\log \left (4 \left (x+\log \left (\frac {5}{x^2}\right ) \left (1+e^{x/4}+\log (9)+\log ^2\left (\frac {x}{3}\right )-2 \log (x)\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {8 \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )+\left (x \,{\mathrm e}^{\frac {x}{4}}-8\right ) \ln \left (\frac {5}{x^{2}}\right )^{2}+4 x \ln \left (\frac {5}{x^{2}}\right )+8 x}{4 x \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )^{2}-8 x \ln \left (\frac {5}{x^{2}}\right )^{2} \ln \left (\frac {x}{3}\right )+\left (4 x \,{\mathrm e}^{\frac {x}{4}}+4 x \right ) \ln \left (\frac {5}{x^{2}}\right )^{2}+4 x^{2} \ln \left (\frac {5}{x^{2}}\right )}\, dx\]
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. 78 vs.
\(2 (24) = 48\).
time = 0.56, size = 78, normalized size = 2.69 \begin {gather*} \log \left (\frac {\log \left (5\right ) \log \left (3\right )^{2} + {\left (\log \left (5\right ) + 4 \, \log \left (3\right ) + 4\right )} \log \left (x\right )^{2} - 2 \, \log \left (x\right )^{3} + {\left (\log \left (5\right ) - 2 \, \log \left (x\right )\right )} e^{\left (\frac {1}{4} \, x\right )} + 2 \, \log \left (5\right ) \log \left (3\right ) - 2 \, {\left ({\left (\log \left (5\right ) + 2\right )} \log \left (3\right ) + \log \left (3\right )^{2} + \log \left (5\right ) + 1\right )} \log \left (x\right ) + x + \log \left (5\right )}{\log \left (5\right ) - 2 \, \log \left (x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (24) = 48\).
time = 0.32, size = 60, normalized size = 2.07 \begin {gather*} \log \left (-2 \, {\left (\log \left (\frac {5}{9}\right ) - 2\right )} \log \left (\frac {5}{x^{2}}\right )^{2} + \log \left (\frac {5}{x^{2}}\right )^{3} + {\left (\log \left (\frac {5}{9}\right )^{2} + 4 \, e^{\left (\frac {1}{4} \, x\right )} - 4 \, \log \left (\frac {5}{9}\right ) + 4\right )} \log \left (\frac {5}{x^{2}}\right ) + 4 \, x\right ) - \log \left (\log \left (\frac {5}{x^{2}}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 97 vs.
\(2 (22) = 44\).
time = 3.07, size = 97, normalized size = 3.34 \begin {gather*} \log {\left (e^{\frac {x}{4}} + \frac {- x + 2 \log {\left (\frac {x}{3} \right )}^{3} - 4 \log {\left (\frac {x}{3} \right )}^{2} - \log {\left (5 \right )} \log {\left (\frac {x}{3} \right )}^{2} + 2 \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )}^{2} - 4 \log {\left (3 \right )} \log {\left (\frac {x}{3} \right )} + 2 \log {\left (\frac {x}{3} \right )} + 2 \log {\left (5 \right )} \log {\left (\frac {x}{3} \right )} - \log {\left (5 \right )} + 2 \log {\left (3 \right )}}{2 \log {\left (\frac {x}{3} \right )} - \log {\left (5 \right )} + 2 \log {\left (3 \right )}} \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. 103 vs.
\(2 (24) = 48\).
time = 0.55, size = 103, normalized size = 3.55 \begin {gather*} \log \left (\log \left (5\right ) \log \left (3\right )^{2} - 2 \, \log \left (5\right ) \log \left (3\right ) \log \left (x\right ) - 2 \, \log \left (3\right )^{2} \log \left (x\right ) + \log \left (5\right ) \log \left (x\right )^{2} + 4 \, \log \left (3\right ) \log \left (x\right )^{2} - 2 \, \log \left (x\right )^{3} + e^{\left (\frac {1}{4} \, x\right )} \log \left (5\right ) + 2 \, \log \left (5\right ) \log \left (3\right ) - 2 \, e^{\left (\frac {1}{4} \, x\right )} \log \left (x\right ) - 2 \, \log \left (5\right ) \log \left (x\right ) - 4 \, \log \left (3\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2} + x + \log \left (5\right ) - 2 \, \log \left (x\right )\right ) - \log \left (\log \left (5\right ) - 2 \, \log \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {8\,x+4\,x\,\ln \left (\frac {5}{x^2}\right )+8\,\ln \left (\frac {x}{3}\right )\,{\ln \left (\frac {5}{x^2}\right )}^2+{\ln \left (\frac {5}{x^2}\right )}^2\,\left (x\,{\mathrm {e}}^{x/4}-8\right )}{{\ln \left (\frac {5}{x^2}\right )}^2\,\left (4\,x+4\,x\,{\mathrm {e}}^{x/4}\right )+4\,x^2\,\ln \left (\frac {5}{x^2}\right )-8\,x\,\ln \left (\frac {x}{3}\right )\,{\ln \left (\frac {5}{x^2}\right )}^2+4\,x\,{\ln \left (\frac {x}{3}\right )}^2\,{\ln \left (\frac {5}{x^2}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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