Optimal. Leaf size=23 \[ \frac {\log \left (\frac {1}{4 x \log \left (x^2 (x+\log (x))\right )}\right )}{x} \]
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Rubi [F]
time = 1.58, 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 {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{\left (x^3+x^2 \log (x)\right ) \log \left (x^3+x^2 \log (x)\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 {-1-3 x-2 \log (x)+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )+(-x-\log (x)) \log \left (x^3+x^2 \log (x)\right ) \log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx\\ &=\int \left (\frac {-1-3 x-2 \log (x)-x \log \left (x^2 (x+\log (x))\right )-\log (x) \log \left (x^2 (x+\log (x))\right )}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )}-\frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x^2}\right ) \, dx\\ &=\int \frac {-1-3 x-2 \log (x)-x \log \left (x^2 (x+\log (x))\right )-\log (x) \log \left (x^2 (x+\log (x))\right )}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx-\int \frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x^2} \, dx\\ &=\frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x}+\int \left (-\frac {1}{x (x+\log (x))}-\frac {\log (x)}{x^2 (x+\log (x))}+\frac {1}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )}+\frac {3}{x (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )}+\frac {2 \log (x)}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )}\right ) \, dx+\int \frac {1+3 x+x \log \left (x^2 (x+\log (x))\right )+\log (x) \left (2+\log \left (x^2 (x+\log (x))\right )\right )}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx\\ &=\frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x}+2 \int \frac {\log (x)}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx-\int \frac {1}{x (x+\log (x))} \, dx-\int \frac {\log (x)}{x^2 (x+\log (x))} \, dx+\int \left (\frac {1}{x (x+\log (x))}+\frac {\log (x)}{x^2 (x+\log (x))}+\frac {1}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )}+\frac {3}{x (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )}+\frac {2 \log (x)}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )}\right ) \, dx+\int \frac {1}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx\\ &=\frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x}+2 \int \frac {\log (x)}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+2 \int \frac {\log (x)}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+\int \frac {\log (x)}{x^2 (x+\log (x))} \, dx-\int \left (\frac {1}{x^2}-\frac {1}{x (x+\log (x))}\right ) \, dx+\int \frac {1}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+\int \frac {1}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx\\ &=\frac {1}{x}+\frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x}+2 \int \frac {\log (x)}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+2 \int \frac {\log (x)}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+\int \frac {1}{x (x+\log (x))} \, dx+\int \left (\frac {1}{x^2}-\frac {1}{x (x+\log (x))}\right ) \, dx+\int \frac {1}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+\int \frac {1}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx\\ &=\frac {\log \left (\frac {1}{4 x \log \left (x^3+x^2 \log (x)\right )}\right )}{x}+2 \int \frac {\log (x)}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+2 \int \frac {\log (x)}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+3 \int \frac {1}{x (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+\int \frac {1}{x^2 (-x-\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx+\int \frac {1}{x^2 (x+\log (x)) \log \left (x^3+x^2 \log (x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (\frac {1}{4 x \log \left (x^2 (x+\log (x))\right )}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.50, size = 2102, normalized size = 91.39
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \, \log \left (2\right ) + \log \left (x\right ) + \log \left (\log \left (x + \log \left (x\right )\right ) + 2 \, \log \left (x\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 23, normalized size = 1.00 \begin {gather*} \frac {\log \left (\frac {1}{4 \, x \log \left (x^{3} + x^{2} \log \left (x\right )\right )}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.07, size = 19, normalized size = 0.83 \begin {gather*} \frac {\log {\left (\frac {1}{4 x \log {\left (x^{3} + x^{2} \log {\left (x \right )} \right )}} \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 26, normalized size = 1.13 \begin {gather*} -\frac {\log \left (x\right )}{x} - \frac {\log \left (4 \, \log \left (x + \log \left (x\right )\right ) + 8 \, \log \left (x\right )\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.65, size = 23, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {1}{4\,x\,\ln \left (x^2\,\ln \left (x\right )+x^3\right )}\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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