Optimal. Leaf size=28 \[ \log (x)+\frac {\log \left (\frac {7}{5}-x^2\right )}{x \log \left (1+\log \left (\frac {5}{x}\right )\right )} \]
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Rubi [F] time = 3.85, 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 {\left (-7+5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )+\log \left (\frac {5}{x}\right ) \left (10 x^2+\left (7-5 x^2\right ) \log \left (\frac {1}{5} \left (7-5 x^2\right )\right )\right )\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )+\left (-7 x+5 x^3+\left (-7 x+5 x^3\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}{\left (-7 x^2+5 x^4+\left (-7 x^2+5 x^4\right ) \log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log \left (\frac {7}{5}-x^2\right ) \left (\frac {1}{1+\log \left (\frac {5}{x}\right )}-\log \left (1+\log \left (\frac {5}{x}\right )\right )\right )+x \log \left (1+\log \left (\frac {5}{x}\right )\right ) \left (\frac {10 x}{-7+5 x^2}+\log \left (1+\log \left (\frac {5}{x}\right )\right )\right )}{x^2 \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\int \left (\frac {1}{x}+\frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {10 x^2+7 \log \left (\frac {7}{5}-x^2\right )-5 x^2 \log \left (\frac {7}{5}-x^2\right )}{x^2 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx\\ &=\log (x)+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {10 x^2+7 \log \left (\frac {7}{5}-x^2\right )-5 x^2 \log \left (\frac {7}{5}-x^2\right )}{x^2 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\int \left (\frac {-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )}{7 x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {5 \left (-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )\right )}{7 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {1}{7} \int \frac {-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {5}{7} \int \frac {-10 x^2-7 \log \left (\frac {7}{5}-x^2\right )+5 x^2 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {1}{7} \int \frac {-10 x^2+\left (-7+5 x^2\right ) \log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {5}{7} \int \left (-\frac {10 x^2}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {7 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {5 x^2 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {1}{7} \int \left (-\frac {10}{\log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {5 \log \left (\frac {7}{5}-x^2\right )}{\log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {7 \log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx-\frac {25}{7} \int \frac {x^2 \log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\frac {50}{7} \int \frac {x^2}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)+\frac {5}{7} \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {10}{7} \int \frac {1}{\log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {25}{7} \int \left (\frac {\log \left (\frac {7}{5}-x^2\right )}{5 \log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {7 \log \left (\frac {7}{5}-x^2\right )}{5 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+5 \int \left (-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\frac {50}{7} \int \left (\frac {1}{5 \log \left (1+\log \left (\frac {5}{x}\right )\right )}+\frac {7}{5 \left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)-5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx+10 \int \frac {1}{\left (-7+5 x^2\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)-5 \int \left (-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {\log \left (\frac {7}{5}-x^2\right )}{2 \sqrt {7} \left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx+10 \int \left (-\frac {1}{2 \sqrt {7} \left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}-\frac {1}{2 \sqrt {7} \left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )}\right ) \, dx-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}-\frac {5 \int \frac {\log \left (\frac {7}{5}-x^2\right )}{\left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{2 \sqrt {7}}+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ &=\log (x)-\frac {5 \int \frac {1}{\left (\sqrt {7}-\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{\sqrt {7}}-\frac {5 \int \frac {1}{\left (\sqrt {7}+\sqrt {5} x\right ) \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx}{\sqrt {7}}+\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \left (1+\log \left (\frac {5}{x}\right )\right ) \log ^2\left (1+\log \left (\frac {5}{x}\right )\right )} \, dx-\int \frac {\log \left (\frac {7}{5}-x^2\right )}{x^2 \log \left (1+\log \left (\frac {5}{x}\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.13, size = 28, normalized size = 1.00 \begin {gather*} \log (x)+\frac {\log \left (\frac {7}{5}-x^2\right )}{x \log \left (1+\log \left (\frac {5}{x}\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 44, normalized size = 1.57 \begin {gather*} -\frac {x \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right ) + 1\right ) - \log \left (-x^{2} + \frac {7}{5}\right )}{x \log \left (\log \left (\frac {5}{x}\right ) + 1\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 {{\left (5 \, x^{3} + {\left (5 \, x^{3} - 7 \, x\right )} \log \left (\frac {5}{x}\right ) - 7 \, x\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )^{2} + {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right ) + {\left (10 \, x^{2} - {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right ) + {\left (10 \, x^{2} - {\left (5 \, x^{2} - 7\right )} \log \left (-x^{2} + \frac {7}{5}\right )\right )} \log \left (\frac {5}{x}\right )\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )}{{\left (5 \, x^{4} - 7 \, x^{2} + {\left (5 \, x^{4} - 7 \, x^{2}\right )} \log \left (\frac {5}{x}\right )\right )} \log \left (\log \left (\frac {5}{x}\right ) + 1\right )^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 27, normalized size = 0.96
method | result | size |
risch | \(\ln \relax (x )+\frac {\ln \left (-x^{2}+\frac {7}{5}\right )}{x \ln \left (\ln \relax (5)-\ln \relax (x )+1\right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 32, normalized size = 1.14 \begin {gather*} -\frac {\log \relax (5) - \log \left (-5 \, x^{2} + 7\right )}{x \log \left (\log \relax (5) - \log \relax (x) + 1\right )} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.43, size = 139, normalized size = 4.96 \begin {gather*} \ln \relax (x)+\ln \left (\frac {5}{x}\right )\,\left (\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}-\frac {2\,x}{x^2-\frac {7}{5}}\right )+\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}-\frac {2\,x}{x^2-\frac {7}{5}}+\frac {\frac {\ln \left (\frac {7}{5}-x^2\right )}{x}+\frac {\ln \left (\ln \left (\frac {5}{x}\right )+1\right )\,\left (\ln \left (\frac {5}{x}\right )+1\right )\,\left (7\,\ln \left (\frac {7}{5}-x^2\right )-5\,x^2\,\ln \left (\frac {7}{5}-x^2\right )+10\,x^2\right )}{x\,\left (5\,x^2-7\right )}}{\ln \left (\ln \left (\frac {5}{x}\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 20, normalized size = 0.71 \begin {gather*} \log {\relax (x )} + \frac {\log {\left (\frac {7}{5} - x^{2} \right )}}{x \log {\left (\log {\left (\frac {5}{x} \right )} + 1 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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