Optimal. Leaf size=25 \[ x^5 (-x+\log (x)+(1+x+4 \log (3)) (-x+\log (\log (x)))) \]
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Rubi [A] time = 0.83, antiderivative size = 39, normalized size of antiderivative = 1.56, number of steps used = 21, number of rules used = 8, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6, 6688, 6742, 2353, 2309, 2178, 2304, 2522} \begin {gather*} -x^7+x^6 \log (\log (x))-2 x^6 (1+\log (9))+x^5 \log (x)+x^5 (1+\log (81)) \log (\log (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 2178
Rule 2304
Rule 2309
Rule 2353
Rule 2522
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^5+x^4 (1+4 \log (3))+\left (x^4-12 x^5-7 x^6-24 x^5 \log (3)\right ) \log (x)+5 x^4 \log ^2(x)+\left (5 x^4+6 x^5+20 x^4 \log (3)\right ) \log (x) \log (\log (x))}{\log (x)} \, dx\\ &=\int \frac {x^4 \left (1+x+\log (81)+5 \log ^2(x)+\log (x) \left (1-7 x^2-12 x (1+\log (9))+(5+6 x+20 \log (3)) \log (\log (x))\right )\right )}{\log (x)} \, dx\\ &=\int \left (\frac {x^4 \left (1+x+\log (81)+\log (x)-7 x^2 \log (x)-12 x (1+\log (9)) \log (x)+5 \log ^2(x)\right )}{\log (x)}+x^4 (5+6 x+20 \log (3)) \log (\log (x))\right ) \, dx\\ &=\int \frac {x^4 \left (1+x+\log (81)+\log (x)-7 x^2 \log (x)-12 x (1+\log (9)) \log (x)+5 \log ^2(x)\right )}{\log (x)} \, dx+\int x^4 (5+6 x+20 \log (3)) \log (\log (x)) \, dx\\ &=\int \left (x^4-7 x^6-12 x^5 (1+\log (9))+\frac {x^4 (1+x+\log (81))}{\log (x)}+5 x^4 \log (x)\right ) \, dx+\int \left (6 x^5 \log (\log (x))+5 x^4 (1+\log (81)) \log (\log (x))\right ) \, dx\\ &=\frac {x^5}{5}-x^7-2 x^6 (1+\log (9))+5 \int x^4 \log (x) \, dx+6 \int x^5 \log (\log (x)) \, dx+(5 (1+\log (81))) \int x^4 \log (\log (x)) \, dx+\int \frac {x^4 (1+x+\log (81))}{\log (x)} \, dx\\ &=-x^7-2 x^6 (1+\log (9))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))+(-1-\log (81)) \int \frac {x^4}{\log (x)} \, dx+\int \left (\frac {x^5}{\log (x)}+\frac {x^4 (1+\log (81))}{\log (x)}\right ) \, dx-\int \frac {x^5}{\log (x)} \, dx\\ &=-x^7-2 x^6 (1+\log (9))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))+(-1-\log (81)) \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+(1+\log (81)) \int \frac {x^4}{\log (x)} \, dx+\int \frac {x^5}{\log (x)} \, dx-\operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )\\ &=-x^7-\text {Ei}(6 \log (x))-2 x^6 (1+\log (9))-\text {Ei}(5 \log (x)) (1+\log (81))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))+(1+\log (81)) \operatorname {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+\operatorname {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )\\ &=-x^7-2 x^6 (1+\log (9))+x^5 \log (x)+x^6 \log (\log (x))+x^5 (1+\log (81)) \log (\log (x))\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.25, size = 24, normalized size = 0.96 \begin {gather*} x^5 (-x (2+x+\log (81))+\log (x)+(1+x+\log (81)) \log (\log (x))) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 42, normalized size = 1.68 \begin {gather*} -x^{7} - 4 \, x^{6} \log \relax (3) - 2 \, x^{6} + x^{5} \log \relax (x) + {\left (x^{6} + 4 \, x^{5} \log \relax (3) + x^{5}\right )} \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 48, normalized size = 1.92 \begin {gather*} -x^{7} - 4 \, x^{6} \log \relax (3) + x^{6} \log \left (\log \relax (x)\right ) + 4 \, x^{5} \log \relax (3) \log \left (\log \relax (x)\right ) - 2 \, x^{6} + x^{5} \log \relax (x) + x^{5} \log \left (\log \relax (x)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 43, normalized size = 1.72
method | result | size |
risch | \(\left (4 x^{5} \ln \relax (3)+x^{6}+x^{5}\right ) \ln \left (\ln \relax (x )\right )-4 x^{6} \ln \relax (3)-x^{7}-2 x^{6}+x^{5} \ln \relax (x )\) | \(43\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.40, size = 66, normalized size = 2.64 \begin {gather*} -x^{7} - 4 \, x^{6} \log \relax (3) + x^{6} \log \left (\log \relax (x)\right ) - 2 \, x^{6} + x^{5} \log \relax (x) + x^{5} \log \left (\log \relax (x)\right ) + 4 \, {\left (x^{5} \log \left (\log \relax (x)\right ) - {\rm Ei}\left (5 \, \log \relax (x)\right )\right )} \log \relax (3) + 4 \, {\rm Ei}\left (5 \, \log \relax (x)\right ) \log \relax (3) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 41, normalized size = 1.64 \begin {gather*} x^5\,\ln \relax (x)+\ln \left (\ln \relax (x)\right )\,\left (x^6+\left (4\,\ln \relax (3)+1\right )\,x^5\right )-x^6\,\left (4\,\ln \relax (3)+2\right )-x^7 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.45, size = 41, normalized size = 1.64 \begin {gather*} - x^{7} + x^{6} \left (- 4 \log {\relax (3 )} - 2\right ) + x^{5} \log {\relax (x )} + \left (x^{6} + x^{5} + 4 x^{5} \log {\relax (3 )}\right ) \log {\left (\log {\relax (x )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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