Optimal. Leaf size=18 \[ \frac {\left (\frac {3}{3+x}-\log (3)\right )^4}{x^4} \]
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Rubi [B] time = 0.20, antiderivative size = 138, normalized size of antiderivative = 7.67, number of steps used = 2, number of rules used = 1, integrand size = 125, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {2074} \begin {gather*} \frac {(1-\log (3))^4}{x^4}-\frac {4 (1-\log (3))^3}{3 x^3}+\frac {2 (1-\log (3))^2 (5-\log (9))}{9 x^2}+\frac {1}{(x+3)^4}+\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 (x+3)}-\frac {4 (1-\log (3)) \left (5+\log ^2(3)-5 \log (3)\right )}{27 x}+\frac {2 (1-\log (3)) (5-\log (27))}{9 (x+3)^2}+\frac {4 (1-\log (3))}{3 (x+3)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4}{(3+x)^5}+\frac {4 (-1+\log (3))}{(3+x)^4}+\frac {4 (5-3 \log (3)) (-1+\log (3))}{9 (3+x)^3}-\frac {4 (-1+\log (3))^3}{x^4}-\frac {4 (-1+\log (3))^4}{x^5}+\frac {4 (1-\log (3)) \left (-5+5 \log (3)-\log ^2(3)\right )}{27 (3+x)^2}+\frac {4 (1-\log (3)) \left (5-5 \log (3)+\log ^2(3)\right )}{27 x^2}+\frac {4 (-1+\log (3))^2 (-5+\log (9))}{9 x^3}\right ) \, dx\\ &=\frac {1}{(3+x)^4}+\frac {4 (1-\log (3))}{3 (3+x)^3}-\frac {4 (1-\log (3))^3}{3 x^3}+\frac {(1-\log (3))^4}{x^4}-\frac {4 (1-\log (3)) \left (5-5 \log (3)+\log ^2(3)\right )}{27 x}+\frac {4 (1-\log (3)) \left (5-5 \log (3)+\log ^2(3)\right )}{27 (3+x)}+\frac {2 (1-\log (3))^2 (5-\log (9))}{9 x^2}+\frac {2 (1-\log (3)) (5-\log (27))}{9 (3+x)^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.23, size = 312, normalized size = 17.33 \begin {gather*} \frac {2916 x (-1+\log (3)) \log (3) (-3+\log (27))^2+729 (-1+\log (3)) (-3+\log (27))^3+324 x^3 \log (3) \left (9 \log ^3(3)-9 \log ^2(3) (-2+\log (27))-(-3+\log (27))^2 \log (27)+3 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )\right )+9 x^4 \log (3) \left (702 \log ^3(3)-125 (-3+\log (27))^2 \log (27)+375 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-675 \log ^2(3) (-5+\log (729))\right )+156 x^5 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+42 x^6 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+4 x^7 \log (3) \left (27 \log ^3(3)-5 (-3+\log (27))^2 \log (27)+15 \log (3) \left (9-9 \log (27)+2 \log ^2(27)\right )-27 \log ^2(3) (-5+\log (729))\right )+486 x^2 \log (3) (-3+\log (27)) \left (9 \log ^2(3)-\log (19683)\right )}{243 x^4 (3+x)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 92, normalized size = 5.11 \begin {gather*} \frac {{\left (x^{4} + 12 \, x^{3} + 54 \, x^{2} + 108 \, x + 81\right )} \log \relax (3)^{4} - 12 \, {\left (x^{3} + 9 \, x^{2} + 27 \, x + 27\right )} \log \relax (3)^{3} + 54 \, {\left (x^{2} + 6 \, x + 9\right )} \log \relax (3)^{2} - 108 \, {\left (x + 3\right )} \log \relax (3) + 81}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 81 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.22, size = 113, normalized size = 6.28 \begin {gather*} \frac {x^{4} \log \relax (3)^{4} + 12 \, x^{3} \log \relax (3)^{4} - 12 \, x^{3} \log \relax (3)^{3} + 54 \, x^{2} \log \relax (3)^{4} - 108 \, x^{2} \log \relax (3)^{3} + 108 \, x \log \relax (3)^{4} + 54 \, x^{2} \log \relax (3)^{2} - 324 \, x \log \relax (3)^{3} + 81 \, \log \relax (3)^{4} + 324 \, x \log \relax (3)^{2} - 324 \, \log \relax (3)^{3} - 108 \, x \log \relax (3) + 486 \, \log \relax (3)^{2} - 324 \, \log \relax (3) + 81}{{\left (x^{2} + 3 \, x\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 107, normalized size = 5.94
method | result | size |
norman | \(\frac {x^{4} \ln \relax (3)^{4}+\left (12 \ln \relax (3)^{4}-12 \ln \relax (3)^{3}\right ) x^{3}+\left (54 \ln \relax (3)^{4}-108 \ln \relax (3)^{3}+54 \ln \relax (3)^{2}\right ) x^{2}+\left (108 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+324 \ln \relax (3)^{2}-108 \ln \relax (3)\right ) x +81+81 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+486 \ln \relax (3)^{2}-324 \ln \relax (3)}{x^{4} \left (3+x \right )^{4}}\) | \(107\) |
risch | \(\frac {x^{4} \ln \relax (3)^{4}+\left (12 \ln \relax (3)^{4}-12 \ln \relax (3)^{3}\right ) x^{3}+\left (54 \ln \relax (3)^{4}-108 \ln \relax (3)^{3}+54 \ln \relax (3)^{2}\right ) x^{2}+\left (108 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+324 \ln \relax (3)^{2}-108 \ln \relax (3)\right ) x +81+81 \ln \relax (3)^{4}-324 \ln \relax (3)^{3}+486 \ln \relax (3)^{2}-324 \ln \relax (3)}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\) | \(122\) |
gosper | \(\frac {x^{4} \ln \relax (3)^{4}+12 \ln \relax (3)^{4} x^{3}+54 x^{2} \ln \relax (3)^{4}-12 x^{3} \ln \relax (3)^{3}+108 x \ln \relax (3)^{4}-108 \ln \relax (3)^{3} x^{2}+81 \ln \relax (3)^{4}-324 x \ln \relax (3)^{3}+54 x^{2} \ln \relax (3)^{2}-324 \ln \relax (3)^{3}+324 x \ln \relax (3)^{2}+486 \ln \relax (3)^{2}-108 x \ln \relax (3)-324 \ln \relax (3)+81}{x^{4} \left (x^{4}+12 x^{3}+54 x^{2}+108 x +81\right )}\) | \(128\) |
default | \(-\frac {4 \left (\ln \relax (3)-1\right )}{3 \left (3+x \right )^{3}}+\frac {1}{\left (3+x \right )^{4}}-\frac {2 \left (-\frac {5}{9}+\frac {8 \ln \relax (3)}{9}-\frac {\ln \relax (3)^{2}}{3}\right )}{\left (3+x \right )^{2}}-\frac {4 \left (-\frac {5}{27}+\frac {10 \ln \relax (3)}{27}-\frac {2 \ln \relax (3)^{2}}{9}+\frac {\ln \relax (3)^{3}}{27}\right )}{3+x}-\frac {4 \left (-\ln \relax (3)^{3}+3 \ln \relax (3)^{2}-3 \ln \relax (3)+1\right )}{3 x^{3}}-\frac {-\ln \relax (3)^{4}+4 \ln \relax (3)^{3}-6 \ln \relax (3)^{2}+4 \ln \relax (3)-1}{x^{4}}-\frac {4 \left (-\frac {\ln \relax (3)^{3}}{27}+\frac {2 \ln \relax (3)^{2}}{9}-\frac {10 \ln \relax (3)}{27}+\frac {5}{27}\right )}{x}-\frac {2 \left (\frac {2 \ln \relax (3)^{3}}{9}-\ln \relax (3)^{2}+\frac {4 \ln \relax (3)}{3}-\frac {5}{9}\right )}{x^{2}}\) | \(160\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 119, normalized size = 6.61 \begin {gather*} \frac {x^{4} \log \relax (3)^{4} + 12 \, {\left (\log \relax (3)^{4} - \log \relax (3)^{3}\right )} x^{3} + 81 \, \log \relax (3)^{4} + 54 \, {\left (\log \relax (3)^{4} - 2 \, \log \relax (3)^{3} + \log \relax (3)^{2}\right )} x^{2} - 324 \, \log \relax (3)^{3} + 108 \, {\left (\log \relax (3)^{4} - 3 \, \log \relax (3)^{3} + 3 \, \log \relax (3)^{2} - \log \relax (3)\right )} x + 486 \, \log \relax (3)^{2} - 324 \, \log \relax (3) + 81}{x^{8} + 12 \, x^{7} + 54 \, x^{6} + 108 \, x^{5} + 81 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 494, normalized size = 27.44 \begin {gather*} \frac {81}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,\ln \relax (3)}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {486\,{\ln \relax (3)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {81\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {324\,x\,{\ln \relax (3)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {324\,x\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {108\,x\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {54\,x^2\,{\ln \relax (3)}^2}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {108\,x^2\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {54\,x^2\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {12\,x^3\,{\ln \relax (3)}^3}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {12\,x^3\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}+\frac {x^4\,{\ln \relax (3)}^4}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4}-\frac {108\,x\,\ln \relax (3)}{x^8+12\,x^7+54\,x^6+108\,x^5+81\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 7.54, size = 128, normalized size = 7.11 \begin {gather*} - \frac {- x^{4} \log {\relax (3 )}^{4} + x^{3} \left (- 12 \log {\relax (3 )}^{4} + 12 \log {\relax (3 )}^{3}\right ) + x^{2} \left (- 54 \log {\relax (3 )}^{4} - 54 \log {\relax (3 )}^{2} + 108 \log {\relax (3 )}^{3}\right ) + x \left (- 324 \log {\relax (3 )}^{2} - 108 \log {\relax (3 )}^{4} + 108 \log {\relax (3 )} + 324 \log {\relax (3 )}^{3}\right ) - 486 \log {\relax (3 )}^{2} - 81 \log {\relax (3 )}^{4} - 81 + 324 \log {\relax (3 )} + 324 \log {\relax (3 )}^{3}}{x^{8} + 12 x^{7} + 54 x^{6} + 108 x^{5} + 81 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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