Optimal. Leaf size=19 \[ \frac {x}{-1+\frac {x}{(x+5 (4+x))^4}+\log (3)} \]
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Rubi [F]
time = 2.15, 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 {-25600000000-61440000000 x-64511808000 x^2-38707027200 x^3-14515148160 x^4-3483642816 x^5-522547200 x^6-44789760 x^7-1679616 x^8+\left (25600000000+61440000000 x+64512000000 x^2+38707200000 x^3+14515200000 x^4+3483648000 x^5+522547200 x^6+44789760 x^7+1679616 x^8\right ) \log (3)}{25600000000+61439680000 x+64511616001 x^2+38707027200 x^3+14515165440 x^4+3483645408 x^5+522547200 x^6+44789760 x^7+1679616 x^8+\left (-51200000000-122879680000 x-129023616000 x^2-77414227200 x^3-29030365440 x^4-6967293408 x^5-1045094400 x^6-89579520 x^7-3359232 x^8\right ) \log (3)+\left (25600000000+61440000000 x+64512000000 x^2+38707200000 x^3+14515200000 x^4+3483648000 x^5+522547200 x^6+44789760 x^7+1679616 x^8\right ) \log ^2(3)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {9 x^2 (192001-192000 \log (3))+40 x (143999-144000 \log (3))+6400000 (1-\log (3))+172800 x^3 (1-\log (3))}{3 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2 (1-\log (3))}+\frac {2 (-20+3 x)}{3 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right ) (1-\log (3))}+\frac {1}{-1+\log (3)}\right ) \, dx\\ &=-\frac {x}{1-\log (3)}+\frac {\int \frac {9 x^2 (192001-192000 \log (3))+40 x (143999-144000 \log (3))+6400000 (1-\log (3))+172800 x^3 (1-\log (3))}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{3 (1-\log (3))}+\frac {2 \int \frac {-20+3 x}{x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))} \, dx}{3 (1-\log (3))}\\ &=-\frac {x}{1-\log (3)}-\frac {100}{9 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right ) (1-\log (3))}+\frac {\int \frac {172800 (1-\log (3))-207360 x (1-\log (3))+46656 x^2 (1-\log (3))}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{15552 (1-\log (3))^2}+\frac {2 \int \left (\frac {20}{-x (191999-192000 \log (3))-160000 (1-\log (3))-86400 x^2 (1-\log (3))-17280 x^3 (1-\log (3))-1296 x^4 (1-\log (3))}+\frac {3 x}{x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))}\right ) \, dx}{3 (1-\log (3))}\\ &=-\frac {x}{1-\log (3)}-\frac {100}{9 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right ) (1-\log (3))}+\frac {\int \frac {1728 \left (100-120 x+27 x^2\right ) (1-\log (3))}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{15552 (1-\log (3))^2}+\frac {2 \int \frac {x}{x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))} \, dx}{1-\log (3)}+\frac {40 \int \frac {1}{-x (191999-192000 \log (3))-160000 (1-\log (3))-86400 x^2 (1-\log (3))-17280 x^3 (1-\log (3))-1296 x^4 (1-\log (3))} \, dx}{3 (1-\log (3))}\\ &=-\frac {x}{1-\log (3)}-\frac {100}{9 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right ) (1-\log (3))}+\frac {\int \frac {100-120 x+27 x^2}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{9 (1-\log (3))}+\frac {2 \int \frac {x}{x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))} \, dx}{1-\log (3)}+\frac {40 \int \frac {1}{-x (191999-192000 \log (3))-160000 (1-\log (3))-86400 x^2 (1-\log (3))-17280 x^3 (1-\log (3))-1296 x^4 (1-\log (3))} \, dx}{3 (1-\log (3))}\\ &=-\frac {x}{1-\log (3)}-\frac {100}{9 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right ) (1-\log (3))}+\frac {\int \left (\frac {100}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2}-\frac {120 x}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2}+\frac {27 x^2}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2}\right ) \, dx}{9 (1-\log (3))}+\frac {2 \int \frac {x}{x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))} \, dx}{1-\log (3)}+\frac {40 \int \frac {1}{-x (191999-192000 \log (3))-160000 (1-\log (3))-86400 x^2 (1-\log (3))-17280 x^3 (1-\log (3))-1296 x^4 (1-\log (3))} \, dx}{3 (1-\log (3))}\\ &=-\frac {x}{1-\log (3)}-\frac {100}{9 \left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right ) (1-\log (3))}+\frac {2 \int \frac {x}{x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))} \, dx}{1-\log (3)}+\frac {3 \int \frac {x^2}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{1-\log (3)}+\frac {100 \int \frac {1}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{9 (1-\log (3))}+\frac {40 \int \frac {1}{-x (191999-192000 \log (3))-160000 (1-\log (3))-86400 x^2 (1-\log (3))-17280 x^3 (1-\log (3))-1296 x^4 (1-\log (3))} \, dx}{3 (1-\log (3))}-\frac {40 \int \frac {x}{\left (x (191999-192000 \log (3))+160000 (1-\log (3))+86400 x^2 (1-\log (3))+17280 x^3 (1-\log (3))+1296 x^4 (1-\log (3))\right )^2} \, dx}{3 (1-\log (3))}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(19)=38\).
time = 0.09, size = 46, normalized size = 2.42 \begin {gather*} \frac {10+3 x-\frac {9 x^2}{3 x-48 (10+3 x)^4+48 (10+3 x)^4 \log (3)}}{3 (-1+\log (3))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs.
\(2(17)=34\).
time = 0.76, size = 72, normalized size = 3.79
method | result | size |
default | \(\frac {x}{\ln \left (3\right )-1}-\frac {x^{2}}{1296 \left (\ln \left (3\right )-1\right ) \left (x^{4} \ln \left (3\right )+\frac {40 x^{3} \ln \left (3\right )}{3}-x^{4}+\frac {200 x^{2} \ln \left (3\right )}{3}-\frac {40 x^{3}}{3}+\frac {4000 x \ln \left (3\right )}{27}-\frac {200 x^{2}}{3}+\frac {10000 \ln \left (3\right )}{81}-\frac {191999 x}{1296}-\frac {10000}{81}\right )}\) | \(72\) |
risch | \(\frac {x}{\ln \left (3\right )-1}-\frac {x^{2}}{1296 \left (\ln \left (3\right )-1\right ) \left (x^{4} \ln \left (3\right )+\frac {40 x^{3} \ln \left (3\right )}{3}-x^{4}+\frac {200 x^{2} \ln \left (3\right )}{3}-\frac {40 x^{3}}{3}+\frac {4000 x \ln \left (3\right )}{27}-\frac {200 x^{2}}{3}+\frac {10000 \ln \left (3\right )}{81}-\frac {191999 x}{1296}-\frac {10000}{81}\right )}\) | \(72\) |
norman | \(\frac {-960000 x^{2}-144000 x^{3}-\frac {40 \left (-179999+180000 \ln \left (3\right )\right ) x}{3 \left (\ln \left (3\right )-1\right )}+1296 x^{5}-\frac {6400000}{3}}{1296 x^{4} \ln \left (3\right )+17280 x^{3} \ln \left (3\right )-1296 x^{4}+86400 x^{2} \ln \left (3\right )-17280 x^{3}+192000 x \ln \left (3\right )-86400 x^{2}+160000 \ln \left (3\right )-191999 x -160000}\) | \(86\) |
gosper | \(\frac {1296 x^{5} \ln \left (3\right )-1296 x^{5}-144000 x^{3} \ln \left (3\right )-960000 x^{2} \ln \left (3\right )+144000 x^{3}-2400000 x \ln \left (3\right )+960000 x^{2}-\frac {6400000 \ln \left (3\right )}{3}+\frac {7199960 x}{3}+\frac {6400000}{3}}{\left (1296 x^{4} \ln \left (3\right )+17280 x^{3} \ln \left (3\right )-1296 x^{4}+86400 x^{2} \ln \left (3\right )-17280 x^{3}+192000 x \ln \left (3\right )-86400 x^{2}+160000 \ln \left (3\right )-191999 x -160000\right ) \left (\ln \left (3\right )-1\right )}\) | \(111\) |
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. 87 vs.
\(2 (20) = 40\).
time = 0.29, size = 87, normalized size = 4.58 \begin {gather*} -\frac {x^{2}}{1296 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) + 1\right )} x^{4} + 17280 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) + 1\right )} x^{3} + 86400 \, {\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) + 1\right )} x^{2} + {\left (192000 \, \log \left (3\right )^{2} - 383999 \, \log \left (3\right ) + 191999\right )} x + 160000 \, \log \left (3\right )^{2} - 320000 \, \log \left (3\right ) + 160000} + \frac {x}{\log \left (3\right ) - 1} \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. 72 vs.
\(2 (20) = 40\).
time = 0.35, size = 72, normalized size = 3.79 \begin {gather*} -\frac {16 \, {\left (81 \, x^{5} + 1080 \, x^{4} + 5400 \, x^{3} + 12000 \, x^{2} + 10000 \, x\right )}}{1296 \, x^{4} + 17280 \, x^{3} + 86400 \, x^{2} - 16 \, {\left (81 \, x^{4} + 1080 \, x^{3} + 5400 \, x^{2} + 12000 \, x + 10000\right )} \log \left (3\right ) + 191999 \, x + 160000} \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. 88 vs.
\(2 (14) = 28\).
time = 4.61, size = 88, normalized size = 4.63 \begin {gather*} - \frac {x^{2}}{x^{4} \left (- 2592 \log {\left (3 \right )} + 1296 + 1296 \log {\left (3 \right )}^{2}\right ) + x^{3} \left (- 34560 \log {\left (3 \right )} + 17280 + 17280 \log {\left (3 \right )}^{2}\right ) + x^{2} \left (- 172800 \log {\left (3 \right )} + 86400 + 86400 \log {\left (3 \right )}^{2}\right ) + x \left (- 383999 \log {\left (3 \right )} + 191999 + 192000 \log {\left (3 \right )}^{2}\right ) - 320000 \log {\left (3 \right )} + 160000 + 160000 \log {\left (3 \right )}^{2}} + \frac {x}{-1 + \log {\left (3 \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. 85 vs.
\(2 (20) = 40\).
time = 0.42, size = 85, normalized size = 4.47 \begin {gather*} \frac {x \log \left (3\right ) - x}{\log \left (3\right )^{2} - 2 \, \log \left (3\right ) + 1} - \frac {x^{2}}{{\left (1296 \, x^{4} \log \left (3\right ) - 1296 \, x^{4} + 17280 \, x^{3} \log \left (3\right ) - 17280 \, x^{3} + 86400 \, x^{2} \log \left (3\right ) - 86400 \, x^{2} + 192000 \, x \log \left (3\right ) - 191999 \, x + 160000 \, \log \left (3\right ) - 160000\right )} {\left (\log \left (3\right ) - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.74, size = 66, normalized size = 3.47 \begin {gather*} \frac {x}{\ln \left (3\right )-1}-\frac {x^2}{\left (\ln \left (3\right )-1\right )\,\left (\left (1296\,\ln \left (3\right )-1296\right )\,x^4+\left (17280\,\ln \left (3\right )-17280\right )\,x^3+\left (86400\,\ln \left (3\right )-86400\right )\,x^2+\left (192000\,\ln \left (3\right )-191999\right )\,x+160000\,\ln \left (3\right )-160000\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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