Optimal. Leaf size=31 \[ \frac {1}{2} \log \left (x \left (-x+x (x-\log (x-\log (x (x+\log (x)))))^2\right )\right ) \]
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Rubi [F]
time = 69.33, 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 {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x^3-x^5+\left (x^2-x^4\right ) \log (x)+\left (-x^2+x^4+\left (-x+x^3\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (2 x^4+2 x^3 \log (x)+\left (-2 x^3-2 x^2 \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^3-x^2 \log (x)+\left (x^2+x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\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 {-x-x^2+x^3-2 x^4+\left (x^2-2 x^3\right ) \log (x)+\left (-x+2 x^3+\left (-1+2 x^2\right ) \log (x)\right ) \log \left (x^2+x \log (x)\right )+\left (1+2 x-x^2+3 x^3+\left (1-x+3 x^2\right ) \log (x)+\left (-3 x^2-3 x \log (x)\right ) \log \left (x^2+x \log (x)\right )\right ) \log \left (x-\log \left (x^2+x \log (x)\right )\right )+\left (-x^2-x \log (x)+(x+\log (x)) \log \left (x^2+x \log (x)\right )\right ) \log ^2\left (x-\log \left (x^2+x \log (x)\right )\right )}{x (x+\log (x)) (x-\log (x (x+\log (x)))) \left (1-x^2+2 x \log (x-\log (x (x+\log (x))))-\log ^2(x-\log (x (x+\log (x))))\right )} \, dx\\ &=\int \left (\frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {2 x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {x (-1+2 x) \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}-\frac {\left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right ) \log (x-\log (x (x+\log (x))))}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {\log ^2(x-\log (x (x+\log (x))))}{x (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x))))) (1-x+\log (x-\log (x (x+\log (x)))))}\right ) \, dx\\ &=2 \int \frac {x^3}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {1}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {x}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\int \frac {x^2}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {x (-1+2 x) \log (x)}{(x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx-\int \frac {\left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right ) \log (x-\log (x (x+\log (x))))}{x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {\log ^2(x-\log (x (x+\log (x))))}{x (-1+x-\log (x-\log (x (x+\log (x))))) (1+x-\log (x-\log (x (x+\log (x)))))} \, dx+\int \frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x))))) (1-x+\log (x-\log (x (x+\log (x)))))} \, dx\\ &=2 \int \left (\frac {x^3}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x^3}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {1}{x}+\frac {(-1+x)^2}{2 x (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {(1+x)^2}{2 x (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {1}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {1}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {x}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx-\int \left (\frac {x^2}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x^2}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {x (-1+2 x) \log (x)}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {x (-1+2 x) \log (x)}{2 (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx-\int \left (\frac {(-1+x) \left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right )}{2 x (x+\log (x)) (x-\log (x (x+\log (x)))) (-1+x-\log (x-\log (x (x+\log (x)))))}-\frac {(1+x) \left (1+2 x-x^2+3 x^3+\log (x)-x \log (x)+3 x^2 \log (x)-3 x^2 \log (x (x+\log (x)))-3 x \log (x) \log (x (x+\log (x)))\right )}{2 x (x+\log (x)) (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}\right ) \, dx+\int \left (\frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{2 x (x-\log (x (x+\log (x)))) (1+x-\log (x-\log (x (x+\log (x)))))}+\frac {\left (-1+2 x^2\right ) \log \left (x^2+x \log (x)\right )}{2 x (x-\log (x (x+\log (x)))) (1-x+\log (x-\log (x (x+\log (x)))))}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.26, size = 46, normalized size = 1.48 \begin {gather*} \log (x)+\frac {1}{2} \log \left (1-x^2+2 x \log (x-\log (x (x+\log (x))))-\log ^2(x-\log (x (x+\log (x))))\right ) \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 = 4.04, size = 145, normalized size = 4.68
method | result | size |
default | \(\ln \left (x \right )+\frac {\ln \left (x^{2}-2 x \ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )+\ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )^{2}-1\right )}{2}\) | \(145\) |
risch | \(\ln \left (x \right )+\frac {\ln \left (x^{2}-2 x \ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )+\ln \left (-\ln \left (x \right )-\ln \left (x +\ln \left (x \right )\right )+\frac {i \pi \,\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i x \left (x +\ln \left (x \right )\right )\right )+\mathrm {csgn}\left (i \left (x +\ln \left (x \right )\right )\right )\right )}{2}+x \right )^{2}-1\right )}{2}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 47, normalized size = 1.52 \begin {gather*} \log \left (x\right ) + \frac {1}{2} \, \log \left (-x + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) + 1\right ) + \frac {1}{2} \, \log \left (-x + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 44, normalized size = 1.42 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 2 \, x \log \left (x - \log \left (x^{2} + x \log \left (x\right )\right )\right ) + \log \left (x - \log \left (x^{2} + x \log \left (x\right )\right )\right )^{2} - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 5.76, size = 42, normalized size = 1.35 \begin {gather*} \log {\left (x \right )} + \frac {\log {\left (x^{2} - 2 x \log {\left (x - \log {\left (x^{2} + x \log {\left (x \right )} \right )} \right )} + \log {\left (x - \log {\left (x^{2} + x \log {\left (x \right )} \right )} \right )}^{2} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 44, normalized size = 1.42 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 2 \, x \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right ) + \log \left (x - \log \left (x + \log \left (x\right )\right ) - \log \left (x\right )\right )^{2} - 1\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.04, size = 40, normalized size = 1.29 \begin {gather*} \frac {\ln \left (x^2-2\,x\,\ln \left (x-\ln \left (x\,\left (x+\ln \left (x\right )\right )\right )\right )+{\ln \left (x-\ln \left (x\,\left (x+\ln \left (x\right )\right )\right )\right )}^2-1\right )}{2}+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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