Optimal. Leaf size=26 \[ -x+\frac {x^2 \left (2 x+x^2\right )^2}{3 (x+\log (5))} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(96\) vs. \(2(26)=52\).
time = 0.13, antiderivative size = 96, normalized size of antiderivative = 3.69, number of steps
used = 4, number of rules used = 3, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {27, 12, 1864}
\begin {gather*} \frac {x^5}{3}+\frac {1}{3} x^4 (4-\log (5))+\frac {1}{3} x^3 (2-\log (5))^2-\frac {1}{3} x^2 (2-\log (5))^2 \log (5)+\frac {(2-\log (5))^2 \log ^4(5)}{3 (x+\log (5))}-\frac {1}{3} x \left (3-(2-\log (5))^2 \log ^2(5)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 1864
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{3 (x+\log (5))^2} \, dx\\ &=\frac {1}{3} \int \frac {-3 x^2+12 x^4+16 x^5+5 x^6+\left (-6 x+16 x^3+20 x^4+6 x^5\right ) \log (5)-3 \log ^2(5)}{(x+\log (5))^2} \, dx\\ &=\frac {1}{3} \int \left (5 x^4-4 x^3 (-4+\log (5))+3 x^2 (-2+\log (5))^2-2 x (-2+\log (5))^2 \log (5)-\frac {(-2+\log (5))^2 \log ^4(5)}{(x+\log (5))^2}-3 \left (1-\frac {1}{3} (-2+\log (5))^2 \log ^2(5)\right )\right ) \, dx\\ &=\frac {x^5}{3}+\frac {1}{3} x^3 (2-\log (5))^2+\frac {1}{3} x^4 (4-\log (5))-\frac {1}{3} x^2 (2-\log (5))^2 \log (5)+\frac {(2-\log (5))^2 \log ^4(5)}{3 (x+\log (5))}-\frac {1}{3} x \left (3-(2-\log (5))^2 \log ^2(5)\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(26)=52\).
time = 0.04, size = 111, normalized size = 4.27 \begin {gather*} \frac {24 x^5+6 x^6+x^4 \left (24+15 \log ^2(5)-5 \log (5) \log (125)\right )+6 x^2 \left (-3+15 \log ^4(5)-5 \log ^3(5) \log (125)\right )+6 \log ^4(5) \left (4+25 \log ^2(5)-4 \log (5) (1+\log (15625))\right )+6 x \log (5) \left (-3+4 \log ^2(5)+25 \log ^4(5)-4 \log ^3(5) (1+\log (15625))\right )}{18 (x+\log (5))} \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. \(110\) vs.
\(2(24)=48\).
time = 7.17, size = 111, normalized size = 4.27
method | result | size |
norman | \(\frac {-x^{2}+\frac {4 x^{4}}{3}+\frac {4 x^{5}}{3}+\frac {x^{6}}{3}+\ln \left (5\right )^{2}}{\ln \left (5\right )+x}\) | \(33\) |
gosper | \(\frac {x^{6}+4 x^{5}+4 x^{4}+3 \ln \left (5\right )^{2}-3 x^{2}}{3 \ln \left (5\right )+3 x}\) | \(34\) |
default | \(\frac {x \ln \left (5\right )^{4}}{3}-\frac {x^{2} \ln \left (5\right )^{3}}{3}+\frac {x^{3} \ln \left (5\right )^{2}}{3}-\frac {x^{4} \ln \left (5\right )}{3}+\frac {x^{5}}{3}-\frac {4 \ln \left (5\right )^{3} x}{3}+\frac {4 x^{2} \ln \left (5\right )^{2}}{3}-\frac {4 x^{3} \ln \left (5\right )}{3}+\frac {4 x^{4}}{3}+\frac {4 x \ln \left (5\right )^{2}}{3}-\frac {4 x^{2} \ln \left (5\right )}{3}+\frac {4 x^{3}}{3}-x +\frac {\ln \left (5\right )^{4} \left (\ln \left (5\right )^{2}-4 \ln \left (5\right )+4\right )}{3 \ln \left (5\right )+3 x}\) | \(111\) |
risch | \(\frac {x \ln \left (5\right )^{4}}{3}-\frac {x^{2} \ln \left (5\right )^{3}}{3}+\frac {x^{3} \ln \left (5\right )^{2}}{3}-\frac {x^{4} \ln \left (5\right )}{3}+\frac {x^{5}}{3}-\frac {4 \ln \left (5\right )^{3} x}{3}+\frac {4 x^{2} \ln \left (5\right )^{2}}{3}-\frac {4 x^{3} \ln \left (5\right )}{3}+\frac {4 x^{4}}{3}+\frac {4 x \ln \left (5\right )^{2}}{3}-\frac {4 x^{2} \ln \left (5\right )}{3}+\frac {4 x^{3}}{3}-x +\frac {\ln \left (5\right )^{6}}{3 \ln \left (5\right )+3 x}-\frac {4 \ln \left (5\right )^{5}}{3 \left (\ln \left (5\right )+x \right )}+\frac {4 \ln \left (5\right )^{4}}{3 \left (\ln \left (5\right )+x \right )}\) | \(125\) |
meijerg | \(-\frac {x}{1+\frac {x}{\ln \left (5\right )}}+\ln \left (5\right )^{4} \left (2 \ln \left (5\right )+\frac {16}{3}\right ) \left (-\frac {x \left (-\frac {3 x^{4}}{\ln \left (5\right )^{4}}+\frac {5 x^{3}}{\ln \left (5\right )^{3}}-\frac {10 x^{2}}{\ln \left (5\right )^{2}}+\frac {30 x}{\ln \left (5\right )}+60\right )}{12 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}+5 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )+\ln \left (5\right )^{3} \left (\frac {20 \ln \left (5\right )}{3}+4\right ) \left (\frac {x \left (\frac {5 x^{3}}{\ln \left (5\right )^{3}}-\frac {10 x^{2}}{\ln \left (5\right )^{2}}+\frac {30 x}{\ln \left (5\right )}+60\right )}{15 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}-4 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )+\frac {16 \ln \left (5\right )^{3} \left (-\frac {x \left (-\frac {2 x^{2}}{\ln \left (5\right )^{2}}+\frac {6 x}{\ln \left (5\right )}+12\right )}{4 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}+3 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )}{3}-2 \ln \left (5\right ) \left (-\frac {x}{\ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}+\ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )+\frac {5 \ln \left (5\right )^{5} \left (\frac {x \left (\frac {14 x^{5}}{\ln \left (5\right )^{5}}-\frac {21 x^{4}}{\ln \left (5\right )^{4}}+\frac {35 x^{3}}{\ln \left (5\right )^{3}}-\frac {70 x^{2}}{\ln \left (5\right )^{2}}+\frac {210 x}{\ln \left (5\right )}+420\right )}{70 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}-6 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )}{3}-\ln \left (5\right ) \left (\frac {x \left (\frac {3 x}{\ln \left (5\right )}+6\right )}{3 \ln \left (5\right ) \left (1+\frac {x}{\ln \left (5\right )}\right )}-2 \ln \left (1+\frac {x}{\ln \left (5\right )}\right )\right )\) | \(364\) |
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. 96 vs.
\(2 (24) = 48\).
time = 0.27, size = 96, normalized size = 3.69 \begin {gather*} \frac {1}{3} \, x^{5} - \frac {1}{3} \, x^{4} {\left (\log \left (5\right ) - 4\right )} + \frac {1}{3} \, {\left (\log \left (5\right )^{2} - 4 \, \log \left (5\right ) + 4\right )} x^{3} - \frac {1}{3} \, {\left (\log \left (5\right )^{3} - 4 \, \log \left (5\right )^{2} + 4 \, \log \left (5\right )\right )} x^{2} + \frac {1}{3} \, {\left (\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} + 4 \, \log \left (5\right )^{2} - 3\right )} x + \frac {\log \left (5\right )^{6} - 4 \, \log \left (5\right )^{5} + 4 \, \log \left (5\right )^{4}}{3 \, {\left (x + \log \left (5\right )\right )}} \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. 60 vs.
\(2 (24) = 48\).
time = 0.37, size = 60, normalized size = 2.31 \begin {gather*} \frac {x^{6} + {\left (x - 4\right )} \log \left (5\right )^{5} + \log \left (5\right )^{6} + 4 \, x^{5} - 4 \, {\left (x - 1\right )} \log \left (5\right )^{4} + 4 \, x^{4} + 4 \, x \log \left (5\right )^{3} - 3 \, x^{2} - 3 \, x \log \left (5\right )}{3 \, {\left (x + \log \left (5\right )\right )}} \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. 116 vs.
\(2 (20) = 40\).
time = 0.21, size = 116, normalized size = 4.46 \begin {gather*} \frac {x^{5}}{3} + x^{4} \cdot \left (\frac {4}{3} - \frac {\log {\left (5 \right )}}{3}\right ) + x^{3} \left (- \frac {4 \log {\left (5 \right )}}{3} + \frac {\log {\left (5 \right )}^{2}}{3} + \frac {4}{3}\right ) + x^{2} \left (- \frac {4 \log {\left (5 \right )}}{3} - \frac {\log {\left (5 \right )}^{3}}{3} + \frac {4 \log {\left (5 \right )}^{2}}{3}\right ) + x \left (- \frac {4 \log {\left (5 \right )}^{3}}{3} - 1 + \frac {\log {\left (5 \right )}^{4}}{3} + \frac {4 \log {\left (5 \right )}^{2}}{3}\right ) + \frac {- 4 \log {\left (5 \right )}^{5} + \log {\left (5 \right )}^{6} + 4 \log {\left (5 \right )}^{4}}{3 x + 3 \log {\left (5 \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. 113 vs.
\(2 (24) = 48\).
time = 0.39, size = 113, normalized size = 4.35 \begin {gather*} \frac {1}{3} \, x^{5} - \frac {1}{3} \, x^{4} \log \left (5\right ) + \frac {1}{3} \, x^{3} \log \left (5\right )^{2} - \frac {1}{3} \, x^{2} \log \left (5\right )^{3} + \frac {1}{3} \, x \log \left (5\right )^{4} + \frac {4}{3} \, x^{4} - \frac {4}{3} \, x^{3} \log \left (5\right ) + \frac {4}{3} \, x^{2} \log \left (5\right )^{2} - \frac {4}{3} \, x \log \left (5\right )^{3} + \frac {4}{3} \, x^{3} - \frac {4}{3} \, x^{2} \log \left (5\right ) + \frac {4}{3} \, x \log \left (5\right )^{2} - x + \frac {\log \left (5\right )^{6} - 4 \, \log \left (5\right )^{5} + 4 \, \log \left (5\right )^{4}}{3 \, {\left (x + \log \left (5\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 196, normalized size = 7.54 \begin {gather*} \frac {4\,{\ln \left (5\right )}^4-4\,{\ln \left (5\right )}^5+{\ln \left (5\right )}^6}{3\,x+3\,\ln \left (5\right )}-x\,\left (2\,\ln \left (5\right )\,\left (\frac {16\,\ln \left (5\right )}{3}-2\,\ln \left (5\right )\,\left (\frac {20\,\ln \left (5\right )}{3}-\frac {5\,{\ln \left (5\right )}^2}{3}+2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )+4\right )+{\ln \left (5\right )}^2\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )\right )+{\ln \left (5\right )}^2\,\left (\frac {20\,\ln \left (5\right )}{3}-\frac {5\,{\ln \left (5\right )}^2}{3}+2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )+4\right )+1\right )-x^4\,\left (\frac {\ln \left (5\right )}{3}-\frac {4}{3}\right )+x^3\,\left (\frac {20\,\ln \left (5\right )}{9}-\frac {5\,{\ln \left (5\right )}^2}{9}+\frac {2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )}{3}+\frac {4}{3}\right )+x^2\,\left (\frac {8\,\ln \left (5\right )}{3}-\ln \left (5\right )\,\left (\frac {20\,\ln \left (5\right )}{3}-\frac {5\,{\ln \left (5\right )}^2}{3}+2\,\ln \left (5\right )\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )+4\right )+\frac {{\ln \left (5\right )}^2\,\left (\frac {4\,\ln \left (5\right )}{3}-\frac {16}{3}\right )}{2}\right )+\frac {x^5}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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