Optimal. Leaf size=15 \[ \left (x+\frac {(3+x) \log (2) \log (x)}{x^2}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(15)=30\).
time = 0.28, antiderivative size = 165, normalized size of antiderivative = 11.00, number of steps
used = 18, number of rules used = 4, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {14, 2404,
2341, 2342} \begin {gather*} \frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {9 \log ^2(2)}{8 x^4}-\frac {\log (2) \log (512) \log (x)}{2 x^4}-\frac {\log (2) \log (512)}{8 x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {4 \log ^2(2) \log (x)}{x^3}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {2 \log (2) \log (64)}{9 x^3}+x^2+\frac {\log ^2(2) \log ^2(x)}{x^2}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {2 \log (8)}{x}+\frac {6 \log (2)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2341
Rule 2342
Rule 2404
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 \left (x^3+x \log (2)+\log (8)\right )}{x^2}-\frac {2 \log (2) \left (3 x^3-x^2 \log (2)-x \log (64)-\log (512)\right ) \log (x)}{x^5}-\frac {2 (3+x) (6+x) \log ^2(2) \log ^2(x)}{x^5}\right ) \, dx\\ &=2 \int \frac {x^3+x \log (2)+\log (8)}{x^2} \, dx-(2 \log (2)) \int \frac {\left (3 x^3-x^2 \log (2)-x \log (64)-\log (512)\right ) \log (x)}{x^5} \, dx-\left (2 \log ^2(2)\right ) \int \frac {(3+x) (6+x) \log ^2(x)}{x^5} \, dx\\ &=2 \int \left (x+\frac {\log (2)}{x}+\frac {\log (8)}{x^2}\right ) \, dx-(2 \log (2)) \int \left (\frac {3 \log (x)}{x^2}-\frac {\log (2) \log (x)}{x^3}-\frac {\log (64) \log (x)}{x^4}-\frac {\log (512) \log (x)}{x^5}\right ) \, dx-\left (2 \log ^2(2)\right ) \int \left (\frac {18 \log ^2(x)}{x^5}+\frac {9 \log ^2(x)}{x^4}+\frac {\log ^2(x)}{x^3}\right ) \, dx\\ &=x^2-\frac {2 \log (8)}{x}+2 \log (2) \log (x)-(6 \log (2)) \int \frac {\log (x)}{x^2} \, dx+\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (2 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^3} \, dx-\left (18 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^4} \, dx-\left (36 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^5} \, dx+(2 \log (2) \log (64)) \int \frac {\log (x)}{x^4} \, dx+(2 \log (2) \log (512)) \int \frac {\log (x)}{x^5} \, dx\\ &=x^2+\frac {6 \log (2)}{x}-\frac {\log ^2(2)}{2 x^2}-\frac {2 \log (8)}{x}-\frac {2 \log (2) \log (64)}{9 x^3}-\frac {\log (2) \log (512)}{8 x^4}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {\log ^2(2) \log (x)}{x^2}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {\log (2) \log (512) \log (x)}{2 x^4}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2}-\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (12 \log ^2(2)\right ) \int \frac {\log (x)}{x^4} \, dx-\left (18 \log ^2(2)\right ) \int \frac {\log (x)}{x^5} \, dx\\ &=x^2+\frac {6 \log (2)}{x}+\frac {9 \log ^2(2)}{8 x^4}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (8)}{x}-\frac {2 \log (2) \log (64)}{9 x^3}-\frac {\log (2) \log (512)}{8 x^4}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {4 \log ^2(2) \log (x)}{x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {\log (2) \log (512) \log (x)}{2 x^4}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(15)=30\).
time = 0.03, size = 57, normalized size = 3.80 \begin {gather*} x^2+2 \log (2) \log (x)+\frac {2 \log (8) \log (x)}{x}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2} \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. \(175\) vs.
\(2(15)=30\).
time = 0.28, size = 176, normalized size = 11.73
method | result | size |
risch | \(\frac {\ln \left (2\right )^{2} \left (x^{2}+6 x +9\right ) \ln \left (x \right )^{2}}{x^{4}}+\frac {6 \ln \left (2\right ) \ln \left (x \right )}{x}+x^{2}+2 \ln \left (2\right ) \ln \left (x \right )\) | \(40\) |
norman | \(\frac {x^{6}+2 x^{4} \ln \left (x \right ) \ln \left (2\right )+\ln \left (2\right )^{2} \ln \left (x \right )^{2} x^{2}+9 \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 x \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 x^{3} \ln \left (x \right ) \ln \left (2\right )}{x^{4}}\) | \(60\) |
default | \(-2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+x^{2}-18 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{3 x^{3}}-\frac {2 \ln \left (x \right )}{9 x^{3}}-\frac {2}{27 x^{3}}\right )+2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-6 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )+2 \ln \left (2\right ) \ln \left (x \right )-36 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{4 x^{4}}-\frac {\ln \left (x \right )}{8 x^{4}}-\frac {1}{32 x^{4}}\right )+12 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )-\frac {6 \ln \left (2\right )}{x}+18 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )\) | \(176\) |
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. 145 vs.
\(2 (15) = 30\).
time = 0.29, size = 145, normalized size = 9.67 \begin {gather*} -\frac {1}{2} \, {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (2\right )^{2} - \frac {4}{3} \, {\left (\frac {3 \, \log \left (x\right )}{x^{3}} + \frac {1}{x^{3}}\right )} \log \left (2\right )^{2} - \frac {9}{8} \, {\left (\frac {4 \, \log \left (x\right )}{x^{4}} + \frac {1}{x^{4}}\right )} \log \left (2\right )^{2} + x^{2} + 6 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} \log \left (2\right ) + 2 \, \log \left (2\right ) \log \left (x\right ) + \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{2 \, x^{2}} - \frac {6 \, \log \left (2\right )}{x} + \frac {2 \, {\left (9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) + 2\right )} \log \left (2\right )^{2}}{3 \, x^{3}} + \frac {9 \, {\left (8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{8 \, x^{4}} \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. 40 vs.
\(2 (15) = 30\).
time = 0.38, size = 40, normalized size = 2.67 \begin {gather*} \frac {x^{6} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (2\right )^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{4} + 3 \, x^{3}\right )} \log \left (2\right ) \log \left (x\right )}{x^{4}} \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. 53 vs.
\(2 (15) = 30\).
time = 0.14, size = 53, normalized size = 3.53 \begin {gather*} x^{2} + 2 \log {\left (2 \right )} \log {\left (x \right )} + \frac {6 \log {\left (2 \right )} \log {\left (x \right )}}{x} + \frac {\left (x^{2} \log {\left (2 \right )}^{2} + 6 x \log {\left (2 \right )}^{2} + 9 \log {\left (2 \right )}^{2}\right ) \log {\left (x \right )}^{2}}{x^{4}} \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. 49 vs.
\(2 (15) = 30\).
time = 0.41, size = 49, normalized size = 3.27 \begin {gather*} x^{2} + 2 \, \log \left (2\right ) \log \left (x\right ) + \frac {6 \, \log \left (2\right ) \log \left (x\right )}{x} + \frac {{\left (x^{2} \log \left (2\right )^{2} + 6 \, x \log \left (2\right )^{2} + 9 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2}}{x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.41, size = 60, normalized size = 4.00 \begin {gather*} \frac {x^7+\ln \left (4\right )\,x^5\,\ln \left (x\right )+\ln \left (64\right )\,x^4\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^3\,{\ln \left (x\right )}^2+6\,{\ln \left (2\right )}^2\,x^2\,{\ln \left (x\right )}^2+9\,{\ln \left (2\right )}^2\,x\,{\ln \left (x\right )}^2}{x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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