Optimal. Leaf size=21 \[ \left (-1+\log \left (\left (10-\frac {2}{x}+\frac {x^2}{20}\right )^2\right )\right )^2 \]
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Rubi [F] time = 13.28, 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 {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{-40 x+200 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-160-8 x^3+\left (160+8 x^3\right ) \log \left (\frac {1600-16000 x+40000 x^2-80 x^3+400 x^4+x^6}{400 x^2}\right )}{x \left (-40+200 x+x^3\right )} \, dx\\ &=\int \frac {8 \left (20+x^3\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{x \left (40-200 x-x^3\right )} \, dx\\ &=8 \int \frac {\left (20+x^3\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{x \left (40-200 x-x^3\right )} \, dx\\ &=8 \int \left (\frac {1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{2 x}+\frac {\left (-200-3 x^2\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{2 \left (-40+200 x+x^3\right )}\right ) \, dx\\ &=4 \int \frac {1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{x} \, dx+4 \int \frac {\left (-200-3 x^2\right ) \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3} \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )+4 \int \frac {400 x^2 \left (\frac {\left (200+3 x^2\right ) \left (-40+200 x+x^3\right )}{200 x^2}-\frac {\left (-40+200 x+x^3\right )^2}{200 x^3}\right ) \log (x)}{\left (-40+200 x+x^3\right )^2} \, dx+4 \int \left (-\frac {200 \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3}-\frac {3 x^2 \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3}\right ) \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )-12 \int \frac {x^2 \left (1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right )}{-40+200 x+x^3} \, dx-800 \int \frac {1-\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx+1600 \int \frac {x^2 \left (\frac {\left (200+3 x^2\right ) \left (-40+200 x+x^3\right )}{200 x^2}-\frac {\left (-40+200 x+x^3\right )^2}{200 x^3}\right ) \log (x)}{\left (-40+200 x+x^3\right )^2} \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )-12 \int \left (\frac {x^2}{-40+200 x+x^3}-\frac {x^2 \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3}\right ) \, dx-800 \int \left (\frac {1}{-40+200 x+x^3}-\frac {\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3}\right ) \, dx+1600 \int \left (-\frac {\log (x)}{200 x}+\frac {\left (200+3 x^2\right ) \log (x)}{200 \left (-40+200 x+x^3\right )}\right ) \, dx\\ &=4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )-8 \int \frac {\log (x)}{x} \, dx+8 \int \frac {\left (200+3 x^2\right ) \log (x)}{-40+200 x+x^3} \, dx-12 \int \frac {x^2}{-40+200 x+x^3} \, dx+12 \int \frac {x^2 \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx-800 \int \frac {1}{-40+200 x+x^3} \, dx+800 \int \frac {\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx\\ &=-4 \log ^2(x)+4 \log (x) \left (1-\log \left (\frac {\left (40-200 x-x^3\right )^2}{400 x^2}\right )\right )+8 \int \left (\frac {200 \log (x)}{-40+200 x+x^3}+\frac {3 x^2 \log (x)}{-40+200 x+x^3}\right ) \, dx-12 \int \frac {x^2}{\left (\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )}{3^{2/3}}+x\right ) \left (\frac {2}{9} \left (300+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )-\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right ) x}{3^{2/3}}+x^2\right )} \, dx+12 \int \frac {x^2 \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx-800 \int \frac {1}{\left (\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right )}{3^{2/3}}+x\right ) \left (\frac {2}{9} \left (300+3000 \sqrt [3]{15} \left (\frac {2}{9+\sqrt {60081}}\right )^{2/3}+\sqrt [3]{2} \left (15 \left (9+\sqrt {60081}\right )\right )^{2/3}\right )-\frac {\sqrt [3]{10} \left (20 \sqrt [3]{\frac {15}{9+\sqrt {60081}}}-\sqrt [3]{2 \left (9+\sqrt {60081}\right )}\right ) x}{3^{2/3}}+x^2\right )} \, dx+800 \int \frac {\log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )}{-40+200 x+x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.16, size = 49, normalized size = 2.33 \begin {gather*} 8 \left (-\frac {1}{4} \log \left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )+\frac {1}{8} \log ^2\left (\frac {\left (-40+200 x+x^3\right )^2}{400 x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 63, normalized size = 3.00 \begin {gather*} \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right )^{2} - 2 \, \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left (x^{3} - {\left (x^{3} + 20\right )} \log \left (\frac {x^{6} + 400 \, x^{4} - 80 \, x^{3} + 40000 \, x^{2} - 16000 \, x + 1600}{400 \, x^{2}}\right ) + 20\right )}}{x^{4} + 200 \, x^{2} - 40 \, x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 48, normalized size = 2.29
method | result | size |
norman | \(\ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{400 x^{2}}\right )^{2}-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \relax (x )\) | \(48\) |
default | \(-4 \ln \left (x^{3}+200 x -40\right )+4 \ln \relax (x )+8 \ln \left (20\right ) \ln \relax (x )-8 \ln \left (20\right ) \ln \left (x^{3}+200 x -40\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40\right )}{\sum }\left (\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{x^{2}}\right )}{2}-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-\underline {\hspace {1.25 ex}}\alpha -2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-3 \underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+\underline {\hspace {1.25 ex}}\alpha +2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+3 \underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-\underline {\hspace {1.25 ex}}\alpha -2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}-3 \underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+\underline {\hspace {1.25 ex}}\alpha +2 x}{i \sqrt {3 \underline {\hspace {1.25 ex}}\alpha ^{2}+800}+3 \underline {\hspace {1.25 ex}}\alpha }\right )+\dilog \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )+\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {x}{\underline {\hspace {1.25 ex}}\alpha }\right )-\frac {\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right )^{2}}{2}\right )\right )-4 \ln \relax (x ) \ln \left (\frac {x^{6}+400 x^{4}-80 x^{3}+40000 x^{2}-16000 x +1600}{x^{2}}\right )+8 \ln \relax (x ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )}\right )+8 \dilog \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =1\right )}\right )+8 \ln \relax (x ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )}\right )+8 \dilog \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =2\right )}\right )+8 \ln \relax (x ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )}\right )+8 \dilog \left (\frac {\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )-x}{\RootOf \left (\textit {\_Z}^{3}+200 \textit {\_Z} -40, \mathit {index} =3\right )}\right )-4 \ln \relax (x )^{2}\) | \(536\) |
risch | \(\text {Expression too large to display}\) | \(4105\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 59, normalized size = 2.81 \begin {gather*} -4 \, {\left (2 \, \log \relax (5) + 4 \, \log \relax (2) + 2 \, \log \relax (x) + 1\right )} \log \left (x^{3} + 200 \, x - 40\right ) + 4 \, \log \left (x^{3} + 200 \, x - 40\right )^{2} + 4 \, {\left (2 \, \log \relax (5) + 4 \, \log \relax (2) + 1\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 46, normalized size = 2.19 \begin {gather*} {\ln \left (\frac {\frac {x^6}{400}+x^4-\frac {x^3}{5}+100\,x^2-40\,x+4}{x^2}\right )}^2-4\,\ln \left (x^3+200\,x-40\right )+4\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.21, size = 46, normalized size = 2.19 \begin {gather*} 4 \log {\relax (x )} + \log {\left (\frac {\frac {x^{6}}{400} + x^{4} - \frac {x^{3}}{5} + 100 x^{2} - 40 x + 4}{x^{2}} \right )}^{2} - 4 \log {\left (x^{3} + 200 x - 40 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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