Optimal. Leaf size=27 \[ \frac {\left (-5+2 x^4-\frac {x}{5+x}\right ) \log (4)}{\log \left (\frac {2}{\log (x)}\right )} \]
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Rubi [F] time = 1.65, 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 (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{\left (25 x+10 x^2+x^3\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\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 {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{x \left (25+10 x+x^2\right ) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\\ &=\int \frac {\left (-125-55 x-6 x^2+50 x^4+20 x^5+2 x^6\right ) \log (4)+\left (-5 x+200 x^4+80 x^5+8 x^6\right ) \log (4) \log (x) \log \left (\frac {2}{\log (x)}\right )}{x (5+x)^2 \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\\ &=\int \left (\frac {\left (-25-6 x+10 x^4+2 x^5\right ) \log (4)}{x (5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}+\frac {\left (-5+200 x^3+80 x^4+8 x^5\right ) \log (4)}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )}\right ) \, dx\\ &=\log (4) \int \frac {-25-6 x+10 x^4+2 x^5}{x (5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx+\log (4) \int \frac {-5+200 x^3+80 x^4+8 x^5}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx\\ &=\log (4) \int \left (-\frac {5}{x \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}+\frac {2 x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )}-\frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )}\right ) \, dx+\log (4) \int \left (\frac {8 x^3}{\log \left (\frac {2}{\log (x)}\right )}-\frac {5}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )}\right ) \, dx\\ &=-\left (\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\right )+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{x \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx\\ &=-\left (\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\right )+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \operatorname {Subst}\left (\int \frac {1}{x \log ^2\left (\frac {2}{x}\right )} \, dx,x,\log (x)\right )+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx\\ &=-\left (\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx\right )+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx+(5 \log (4)) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log \left (\frac {2}{\log (x)}\right )\right )+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx\\ &=-\frac {5 \log (4)}{\log \left (\frac {2}{\log (x)}\right )}-\log (4) \int \frac {1}{(5+x) \log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx+(2 \log (4)) \int \frac {x^3}{\log (x) \log ^2\left (\frac {2}{\log (x)}\right )} \, dx-(5 \log (4)) \int \frac {1}{(5+x)^2 \log \left (\frac {2}{\log (x)}\right )} \, dx+(8 \log (4)) \int \frac {x^3}{\log \left (\frac {2}{\log (x)}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.22, size = 32, normalized size = 1.19 \begin {gather*} \frac {\left (-25-6 x+10 x^4+2 x^5\right ) \log (4)}{(5+x) \log \left (\frac {2}{\log (x)}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 33, normalized size = 1.22 \begin {gather*} \frac {2 \, {\left (2 \, x^{5} + 10 \, x^{4} - 6 \, x - 25\right )} \log \relax (2)}{{\left (x + 5\right )} \log \left (\frac {2}{\log \relax (x)}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 48, normalized size = 1.78 \begin {gather*} \frac {2 \, {\left (2 \, x^{5} \log \relax (2) + 10 \, x^{4} \log \relax (2) - 6 \, x \log \relax (2) - 25 \, \log \relax (2)\right )}}{x \log \relax (2) - x \log \left (\log \relax (x)\right ) + 5 \, \log \relax (2) - 5 \, \log \left (\log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 37, normalized size = 1.37
method | result | size |
risch | \(\frac {4 \ln \relax (2) \left (2 x^{5}+10 x^{4}-6 x -25\right )}{\left (5+x \right ) \left (2 \ln \relax (2)-2 \ln \left (\ln \relax (x )\right )\right )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 45, normalized size = 1.67 \begin {gather*} \frac {2 \, {\left (2 \, x^{5} \log \relax (2) + 10 \, x^{4} \log \relax (2) - 6 \, x \log \relax (2) - 25 \, \log \relax (2)\right )}}{x \log \relax (2) - {\left (x + 5\right )} \log \left (\log \relax (x)\right ) + 5 \, \log \relax (2)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.08, size = 33, normalized size = 1.22 \begin {gather*} -\frac {2\,\ln \relax (2)\,\left (-2\,x^5-10\,x^4+6\,x+25\right )}{\ln \left (\frac {2}{\ln \relax (x)}\right )\,\left (x+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 37, normalized size = 1.37 \begin {gather*} \frac {4 x^{5} \log {\relax (2 )} + 20 x^{4} \log {\relax (2 )} - 12 x \log {\relax (2 )} - 50 \log {\relax (2 )}}{\left (x + 5\right ) \log {\left (\frac {2}{\log {\relax (x )}} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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