Optimal. Leaf size=23 \[ \frac {2 \left (\frac {13}{3}-x+(5+x) \log (x \log (x))\right )^2}{x} \]
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Rubi [F] time = 1.51, 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 {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{9 x^2 \log (x)} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {780-24 x-36 x^2+\left (442-24 x-18 x^2\right ) \log (x)+\left (900+360 x+36 x^2+(120+360 x) \log (x)\right ) \log (x \log (x))+\left (-450+18 x^2\right ) \log (x) \log ^2(x \log (x))}{x^2 \log (x)} \, dx\\ &=\frac {1}{9} \int \frac {2 (13-3 x+3 (5+x) \log (x \log (x))) (6 (5+x)+\log (x) (17+3 x+3 (-5+x) \log (x \log (x))))}{x^2 \log (x)} \, dx\\ &=\frac {2}{9} \int \frac {(13-3 x+3 (5+x) \log (x \log (x))) (6 (5+x)+\log (x) (17+3 x+3 (-5+x) \log (x \log (x))))}{x^2 \log (x)} \, dx\\ &=\frac {2}{9} \int \left (-\frac {(-13+3 x) (30+6 x+17 \log (x)+3 x \log (x))}{x^2 \log (x)}+\frac {6 \left (75+30 x+3 x^2+10 \log (x)+30 x \log (x)\right ) \log (x \log (x))}{x^2 \log (x)}+\frac {9 (-5+x) (5+x) \log ^2(x \log (x))}{x^2}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {(-13+3 x) (30+6 x+17 \log (x)+3 x \log (x))}{x^2 \log (x)} \, dx\right )+\frac {4}{3} \int \frac {\left (75+30 x+3 x^2+10 \log (x)+30 x \log (x)\right ) \log (x \log (x))}{x^2 \log (x)} \, dx+2 \int \frac {(-5+x) (5+x) \log ^2(x \log (x))}{x^2} \, dx\\ &=-\left (\frac {2}{9} \int \left (\frac {(-13+3 x) (17+3 x)}{x^2}+\frac {6 (5+x) (-13+3 x)}{x^2 \log (x)}\right ) \, dx\right )+\frac {4}{3} \int \left (\frac {10 \log (x \log (x))}{x^2}+\frac {30 \log (x \log (x))}{x}+\frac {3 \log (x \log (x))}{\log (x)}+\frac {75 \log (x \log (x))}{x^2 \log (x)}+\frac {30 \log (x \log (x))}{x \log (x)}\right ) \, dx+2 \int \left (\log ^2(x \log (x))-\frac {25 \log ^2(x \log (x))}{x^2}\right ) \, dx\\ &=-\left (\frac {2}{9} \int \frac {(-13+3 x) (17+3 x)}{x^2} \, dx\right )-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx+4 \int \frac {\log (x \log (x))}{\log (x)} \, dx+\frac {40}{3} \int \frac {\log (x \log (x))}{x^2} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {40 \log (x \log (x))}{3 x}+4 \log (x \log (x)) \text {li}(x)-\frac {2}{9} \int \left (9-\frac {221}{x^2}+\frac {12}{x}\right ) \, dx-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \frac {(1+\log (x)) \text {li}(x)}{x \log (x)} \, dx+\frac {40}{3} \int \frac {1+\log (x)}{x^2 \log (x)} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {442}{9 x}-2 x-\frac {8 \log (x)}{3}+\frac {40}{3} \text {Ei}(-\log (x)) (1+\log (x))-\frac {40 \log (x \log (x))}{3 x}+4 \log (x \log (x)) \text {li}(x)-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \left (\frac {\text {li}(x)}{x}+\frac {\text {li}(x)}{x \log (x)}\right ) \, dx-\frac {40}{3} \int \frac {\text {Ei}(-\log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {442}{9 x}-2 x-\frac {8 \log (x)}{3}+\frac {40}{3} \text {Ei}(-\log (x)) (1+\log (x))-\frac {40 \log (x \log (x))}{3 x}+4 \log (x \log (x)) \text {li}(x)-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \frac {\text {li}(x)}{x} \, dx-4 \int \frac {\text {li}(x)}{x \log (x)} \, dx-\frac {40}{3} \operatorname {Subst}(\int \text {Ei}(-x) \, dx,x,\log (x))+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ &=-\frac {562}{9 x}+2 x-\frac {8 \log (x)}{3}-\frac {40}{3} \text {Ei}(-\log (x)) \log (x)+\frac {40}{3} \text {Ei}(-\log (x)) (1+\log (x))-\frac {40 \log (x \log (x))}{3 x}-4 \log (x) \text {li}(x)+4 \log (x \log (x)) \text {li}(x)-\frac {4}{3} \int \frac {(5+x) (-13+3 x)}{x^2 \log (x)} \, dx+2 \int \log ^2(x \log (x)) \, dx-4 \int \frac {\text {li}(x)}{x \log (x)} \, dx+40 \int \frac {\log (x \log (x))}{x} \, dx+40 \int \frac {\log (x \log (x))}{x \log (x)} \, dx-50 \int \frac {\log ^2(x \log (x))}{x^2} \, dx+100 \int \frac {\log (x \log (x))}{x^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.18, size = 53, normalized size = 2.30 \begin {gather*} \frac {2 \left (169+9 x^2-12 x \log (x)-12 x \log (\log (x))+6 \left (65-3 x^2\right ) \log (x \log (x))+9 (5+x)^2 \log ^2(x \log (x))\right )}{9 x} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 46, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (9 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (x \log \relax (x)\right )^{2} + 9 \, x^{2} - 6 \, {\left (3 \, x^{2} + 2 \, x - 65\right )} \log \left (x \log \relax (x)\right ) + 169\right )}}{9 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (9 \, {\left (x^{2} - 25\right )} \log \left (x \log \relax (x)\right )^{2} \log \relax (x) - 18 \, x^{2} + 6 \, {\left (3 \, x^{2} + 10 \, {\left (3 \, x + 1\right )} \log \relax (x) + 30 \, x + 75\right )} \log \left (x \log \relax (x)\right ) - {\left (9 \, x^{2} + 12 \, x - 221\right )} \log \relax (x) - 12 \, x + 390\right )}}{9 \, x^{2} \log \relax (x)}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 1198, normalized size = 52.09
method | result | size |
risch | \(\frac {2 \left (x^{2}+10 x +25\right ) \ln \left (\ln \relax (x )\right )^{2}}{x}+\frac {2 \left (-3 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+3 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+3 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-3 i \pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-75 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+75 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+75 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-75 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+6 x^{2} \ln \relax (x )-6 x^{2}+60 x \ln \relax (x )+150 \ln \relax (x )+130\right ) \ln \left (\ln \relax (x )\right )}{3 x}+\frac {676-72 x^{2} \ln \relax (x )-48 x \ln \left (\ln \relax (x )\right )+36 x^{2} \ln \relax (x )^{2}+1560 \ln \relax (x )+900 \ln \relax (x )^{2}+36 x^{2}-36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-36 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-780 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} x +360 x \ln \relax (x )^{2}-48 x \ln \relax (x )-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+18 \pi ^{2} x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}-225 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+450 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+450 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}-900 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+36 i \pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+780 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+780 i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-225 \pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}-360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} x +900 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-36 i \pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \ln \relax (x )+900 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-36 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}-360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right ) x -36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right ) \ln \relax (x )-360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right ) x -780 i \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-225 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+450 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}-9 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}-225 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4}+450 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5}+360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x +360 i \pi \ln \left (\ln \relax (x )\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x -900 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \ln \relax (x )+36 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \ln \relax (x )+36 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )+360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x +360 i \pi \ln \relax (x ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} x}{18 x}\) | \(1198\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, x + \frac {2 \, {\left (3 \, {\left (x^{2} + 10 \, x + 25\right )} \log \relax (x)^{2} + 3 \, {\left (x^{2} + 10 \, x + 25\right )} \log \left (\log \relax (x)\right )^{2} + 6 \, x^{2} - 2 \, {\left (3 \, x^{2} - 65\right )} \log \relax (x) - 2 \, {\left (3 \, x^{2} - 3 \, {\left (x^{2} + 10 \, x + 25\right )} \log \relax (x) - 65\right )} \log \left (\log \relax (x)\right ) + 130\right )}}{3 \, x} - \frac {442}{9 \, x} + \frac {260}{3} \, {\rm Ei}\left (-\log \relax (x)\right ) - 4 \, {\rm Ei}\left (\log \relax (x)\right ) + \frac {2}{9} \, \int \frac {6 \, {\left (3 \, x^{2} - 65\right )}}{x^{2} \log \relax (x)}\,{d x} - \frac {8}{3} \, \log \relax (x) - \frac {8}{3} \, \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 69, normalized size = 3.00 \begin {gather*} 2\,x+\frac {112\,\ln \left (\ln \relax (x)\right )}{3}+\frac {112\,\ln \relax (x)}{3}-\ln \left (x\,\ln \relax (x)\right )\,\left (8\,x-\frac {4\,x^2-40\,x+\frac {260}{3}}{x}\right )+\frac {338}{9\,x}+{\ln \left (x\,\ln \relax (x)\right )}^2\,\left (4\,x-\frac {2\,x^2-50}{x}+20\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.52, size = 60, normalized size = 2.61 \begin {gather*} 2 x - \frac {8 \log {\relax (x )}}{3} - \frac {8 \log {\left (\log {\relax (x )} \right )}}{3} + \frac {\left (260 - 12 x^{2}\right ) \log {\left (x \log {\relax (x )} \right )}}{3 x} + \frac {\left (2 x^{2} + 20 x + 50\right ) \log {\left (x \log {\relax (x )} \right )}^{2}}{x} + \frac {338}{9 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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