Optimal. Leaf size=31 \[ \frac {16}{25} \left (5-\frac {2}{x}-x-x^2\right )^2 \left (x-\frac {2}{\log (x)}\right )^2 \]
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Rubi [F]
time = 0.84, 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 {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{25 x^3 \log ^3(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {-512+2560 x-3712 x^2+768 x^3+1152 x^4-256 x^5-128 x^6+\left (-512+1536 x-1280 x^2+1472 x^3-1536 x^4-192 x^5+384 x^6+64 x^7\right ) \log (x)+\left (256 x-1856 x^3+768 x^4+1728 x^5-512 x^6-320 x^7\right ) \log ^2(x)+\left (-320 x^3+928 x^4-288 x^5-576 x^6+160 x^7+96 x^8\right ) \log ^3(x)}{x^3 \log ^3(x)} \, dx\\ &=\frac {1}{25} \int \frac {32 \left (2-5 x+x^2+x^3\right ) \left (-4 \left (2-5 x+x^2+x^3\right )+2 \left (-4+2 x-3 x^2+5 x^3+x^4\right ) \log (x)-2 x \left (-2-5 x+3 x^2+5 x^3\right ) \log ^2(x)+x^3 \left (-5+2 x+3 x^2\right ) \log ^3(x)\right )}{x^3 \log ^3(x)} \, dx\\ &=\frac {32}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-4 \left (2-5 x+x^2+x^3\right )+2 \left (-4+2 x-3 x^2+5 x^3+x^4\right ) \log (x)-2 x \left (-2-5 x+3 x^2+5 x^3\right ) \log ^2(x)+x^3 \left (-5+2 x+3 x^2\right ) \log ^3(x)\right )}{x^3 \log ^3(x)} \, dx\\ &=\frac {32}{25} \int \left ((-1+x) (5+3 x) \left (2-5 x+x^2+x^3\right )-\frac {4 \left (2-5 x+x^2+x^3\right )^2}{x^3 \log ^3(x)}+\frac {2 \left (2-5 x+x^2+x^3\right ) \left (-4+2 x-3 x^2+5 x^3+x^4\right )}{x^3 \log ^2(x)}-\frac {2 \left (2-5 x+x^2+x^3\right ) \left (-2-5 x+3 x^2+5 x^3\right )}{x^2 \log (x)}\right ) \, dx\\ &=\frac {32}{25} \int (-1+x) (5+3 x) \left (2-5 x+x^2+x^3\right ) \, dx+\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-4+2 x-3 x^2+5 x^3+x^4\right )}{x^3 \log ^2(x)} \, dx-\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-2-5 x+3 x^2+5 x^3\right )}{x^2 \log (x)} \, dx-\frac {128}{25} \int \frac {\left (2-5 x+x^2+x^3\right )^2}{x^3 \log ^3(x)} \, dx\\ &=\frac {16}{25} \left (2-5 x+x^2+x^3\right )^2+\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-4+2 x-3 x^2+5 x^3+x^4\right )}{x^3 \log ^2(x)} \, dx-\frac {64}{25} \int \frac {\left (2-5 x+x^2+x^3\right ) \left (-2-5 x+3 x^2+5 x^3\right )}{x^2 \log (x)} \, dx-\frac {128}{25} \int \frac {\left (2-5 x+x^2+x^3\right )^2}{x^3 \log ^3(x)} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(31)=62\).
time = 0.03, size = 77, normalized size = 2.48 \begin {gather*} \frac {32}{25} \left (\frac {1}{2} x \left (-20+29 x-6 x^2-9 x^3+2 x^4+x^5\right )+\frac {2 \left (2-5 x+x^2+x^3\right )^2}{x^2 \log ^2(x)}-\frac {2 \left (2-5 x+x^2+x^3\right )^2}{x \log (x)}\right ) \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. \(145\) vs.
\(2(29)=58\).
time = 0.30, size = 146, normalized size = 4.71
method | result | size |
risch | \(\frac {16 x^{6}}{25}+\frac {32 x^{5}}{25}-\frac {144 x^{4}}{25}-\frac {96 x^{3}}{25}+\frac {464 x^{2}}{25}-\frac {64 x}{5}+\frac {64}{25}-\frac {64 \left (x^{7} \ln \left (x \right )+2 x^{6} \ln \left (x \right )-x^{6}-9 x^{5} \ln \left (x \right )-2 x^{5}-6 x^{4} \ln \left (x \right )+9 x^{4}+29 x^{3} \ln \left (x \right )+6 x^{3}-20 x^{2} \ln \left (x \right )-29 x^{2}+4 x \ln \left (x \right )+20 x -4\right )}{25 x^{2} \ln \left (x \right )^{2}}\) | \(116\) |
default | \(-\frac {64 x}{5}+\frac {128 x^{3}}{25 \ln \left (x \right )^{2}}+\frac {256}{25 x^{2} \ln \left (x \right )^{2}}-\frac {256}{5 x \ln \left (x \right )^{2}}-\frac {576 x^{2}}{25 \ln \left (x \right )^{2}}+\frac {1856}{25 \ln \left (x \right )^{2}}-\frac {128 x^{4}}{25 \ln \left (x \right )}+\frac {256}{5 \ln \left (x \right )}+\frac {576 x^{3}}{25 \ln \left (x \right )}+\frac {64 x^{4}}{25 \ln \left (x \right )^{2}}-\frac {384 x}{25 \ln \left (x \right )^{2}}-\frac {256}{25 x \ln \left (x \right )}+\frac {384 x^{2}}{25 \ln \left (x \right )}-\frac {1856 x}{25 \ln \left (x \right )}+\frac {16 x^{6}}{25}+\frac {32 x^{5}}{25}-\frac {144 x^{4}}{25}-\frac {96 x^{3}}{25}+\frac {464 x^{2}}{25}-\frac {64 x^{5}}{25 \ln \left (x \right )}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.33, size = 181, normalized size = 5.84 \begin {gather*} \frac {16}{25} \, x^{6} + \frac {32}{25} \, x^{5} - \frac {144}{25} \, x^{4} - \frac {96}{25} \, x^{3} + \frac {464}{25} \, x^{2} - \frac {64}{5} \, x + \frac {256}{5 \, \log \left (x\right )} + \frac {1856}{25 \, \log \left (x\right )^{2}} - \frac {64}{5} \, {\rm Ei}\left (5 \, \log \left (x\right )\right ) - \frac {512}{25} \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) + \frac {1728}{25} \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) + \frac {768}{25} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) + \frac {256}{25} \, {\rm Ei}\left (-\log \left (x\right )\right ) - \frac {1856}{25} \, {\rm Ei}\left (\log \left (x\right )\right ) + \frac {1024}{25} \, \Gamma \left (-1, 2 \, \log \left (x\right )\right ) + \frac {1472}{25} \, \Gamma \left (-1, -\log \left (x\right )\right ) - \frac {3072}{25} \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - \frac {576}{25} \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + \frac {1536}{25} \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + \frac {64}{5} \, \Gamma \left (-1, -5 \, \log \left (x\right )\right ) - \frac {1536}{25} \, \Gamma \left (-1, \log \left (x\right )\right ) + \frac {2048}{25} \, \Gamma \left (-2, 2 \, \log \left (x\right )\right ) - \frac {768}{25} \, \Gamma \left (-2, -\log \left (x\right )\right ) - \frac {4608}{25} \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) + \frac {2304}{25} \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + \frac {2048}{25} \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) - \frac {512}{5} \, \Gamma \left (-2, \log \left (x\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. 109 vs.
\(2 (25) = 50\).
time = 0.38, size = 109, normalized size = 3.52 \begin {gather*} \frac {16 \, {\left (4 \, x^{6} + 8 \, x^{5} - 36 \, x^{4} - 24 \, x^{3} + {\left (x^{8} + 2 \, x^{7} - 9 \, x^{6} - 6 \, x^{5} + 29 \, x^{4} - 20 \, x^{3}\right )} \log \left (x\right )^{2} + 116 \, x^{2} - 4 \, {\left (x^{7} + 2 \, x^{6} - 9 \, x^{5} - 6 \, x^{4} + 29 \, x^{3} - 20 \, x^{2} + 4 \, x\right )} \log \left (x\right ) - 80 \, x + 16\right )}}{25 \, x^{2} \log \left (x\right )^{2}} \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.
\(2 (22) = 44\).
time = 0.09, size = 116, normalized size = 3.74 \begin {gather*} \frac {16 x^{6}}{25} + \frac {32 x^{5}}{25} - \frac {144 x^{4}}{25} - \frac {96 x^{3}}{25} + \frac {464 x^{2}}{25} - \frac {64 x}{5} + \frac {64 x^{6} + 128 x^{5} - 576 x^{4} - 384 x^{3} + 1856 x^{2} - 1280 x + \left (- 64 x^{7} - 128 x^{6} + 576 x^{5} + 384 x^{4} - 1856 x^{3} + 1280 x^{2} - 256 x\right ) \log {\left (x \right )} + 256}{25 x^{2} \log {\left (x \right )}^{2}} \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. 114 vs.
\(2 (25) = 50\).
time = 0.40, size = 114, normalized size = 3.68 \begin {gather*} \frac {16}{25} \, x^{6} + \frac {32}{25} \, x^{5} - \frac {144}{25} \, x^{4} - \frac {96}{25} \, x^{3} + \frac {464}{25} \, x^{2} - \frac {64}{5} \, x - \frac {64 \, {\left (x^{7} \log \left (x\right ) + 2 \, x^{6} \log \left (x\right ) - x^{6} - 9 \, x^{5} \log \left (x\right ) - 2 \, x^{5} - 6 \, x^{4} \log \left (x\right ) + 9 \, x^{4} + 29 \, x^{3} \log \left (x\right ) + 6 \, x^{3} - 20 \, x^{2} \log \left (x\right ) - 29 \, x^{2} + 4 \, x \log \left (x\right ) + 20 \, x - 4\right )}}{25 \, x^{2} \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.36, size = 127, normalized size = 4.10 \begin {gather*} \frac {\frac {16\,{\left (2\,x^3+2\,x^2-10\,x+4\right )}^2}{25}-\ln \left (x\right )\,\left (\frac {16\,\left (x^4+x^3-5\,x^2+4\,x\right )\,\left (2\,x^3+2\,x^2-10\,x+4\right )}{25}+\frac {16\,\left (x^4+x^3-5\,x^2\right )\,\left (2\,x^3+2\,x^2-10\,x+4\right )}{25}\right )}{x^2\,{\ln \left (x\right )}^2}+\frac {16\,\left (x^4+x^3-5\,x^2\right )\,\left (x^4+x^3-5\,x^2+4\,x\right )}{25\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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