Optimal. Leaf size=19 \[ \frac {96+x-x^2-\log (x)}{\log (-5+x)} \]
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Rubi [F] time = 0.90, 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 {-96 x-x^2+x^3+\left (5-6 x+11 x^2-2 x^3\right ) \log (-5+x)+x \log (x)}{\left (-5 x+x^2\right ) \log ^2(-5+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96 x-x^2+x^3+\left (5-6 x+11 x^2-2 x^3\right ) \log (-5+x)+x \log (x)}{(-5+x) x \log ^2(-5+x)} \, dx\\ &=\int \left (\frac {-96 x-x^2+x^3+5 \log (-5+x)-6 x \log (-5+x)+11 x^2 \log (-5+x)-2 x^3 \log (-5+x)}{(-5+x) x \log ^2(-5+x)}+\frac {\log (x)}{(-5+x) \log ^2(-5+x)}\right ) \, dx\\ &=\int \frac {-96 x-x^2+x^3+5 \log (-5+x)-6 x \log (-5+x)+11 x^2 \log (-5+x)-2 x^3 \log (-5+x)}{(-5+x) x \log ^2(-5+x)} \, dx+\int \frac {\log (x)}{(-5+x) \log ^2(-5+x)} \, dx\\ &=\int \left (\frac {-96-x+x^2}{(-5+x) \log ^2(-5+x)}+\frac {-1+x-2 x^2}{x \log (-5+x)}\right ) \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\int \frac {-96-x+x^2}{(-5+x) \log ^2(-5+x)} \, dx+\int \frac {-1+x-2 x^2}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\int \left (\frac {4}{\log ^2(-5+x)}-\frac {76}{(-5+x) \log ^2(-5+x)}+\frac {x}{\log ^2(-5+x)}\right ) \, dx+\int \left (\frac {1}{\log (-5+x)}-\frac {1}{x \log (-5+x)}-\frac {2 x}{\log (-5+x)}\right ) \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=-\left (2 \int \frac {x}{\log (-5+x)} \, dx\right )+4 \int \frac {1}{\log ^2(-5+x)} \, dx-76 \int \frac {1}{(-5+x) \log ^2(-5+x)} \, dx+\int \frac {x}{\log ^2(-5+x)} \, dx+\int \frac {1}{\log (-5+x)} \, dx-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\frac {(5-x) x}{\log (-5+x)}-2 \int \left (\frac {5}{\log (-5+x)}+\frac {-5+x}{\log (-5+x)}\right ) \, dx+2 \int \frac {x}{\log (-5+x)} \, dx+4 \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,-5+x\right )-5 \int \frac {1}{\log (-5+x)} \, dx-76 \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,-5+x\right )-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\frac {4 (5-x)}{\log (-5+x)}+\frac {(5-x) x}{\log (-5+x)}+\text {li}(-5+x)+2 \int \left (\frac {5}{\log (-5+x)}+\frac {-5+x}{\log (-5+x)}\right ) \, dx-2 \int \frac {-5+x}{\log (-5+x)} \, dx+4 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )-5 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )-10 \int \frac {1}{\log (-5+x)} \, dx-76 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (-5+x)\right )-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\frac {76}{\log (-5+x)}+\frac {4 (5-x)}{\log (-5+x)}+\frac {(5-x) x}{\log (-5+x)}+2 \int \frac {-5+x}{\log (-5+x)} \, dx-2 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-5+x\right )+10 \int \frac {1}{\log (-5+x)} \, dx-10 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\frac {76}{\log (-5+x)}+\frac {4 (5-x)}{\log (-5+x)}+\frac {(5-x) x}{\log (-5+x)}-10 \text {li}(-5+x)-2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-5+x)\right )+2 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,-5+x\right )+10 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,-5+x\right )-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=-2 \text {Ei}(2 \log (-5+x))+\frac {76}{\log (-5+x)}+\frac {4 (5-x)}{\log (-5+x)}+\frac {(5-x) x}{\log (-5+x)}+2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (-5+x)\right )-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ &=\frac {76}{\log (-5+x)}+\frac {4 (5-x)}{\log (-5+x)}+\frac {(5-x) x}{\log (-5+x)}-\int \frac {1}{x \log (-5+x)} \, dx+\operatorname {Subst}\left (\int \frac {\log (5+x)}{x \log ^2(x)} \, dx,x,-5+x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 19, normalized size = 1.00 \begin {gather*} \frac {96+x-x^2-\log (x)}{\log (-5+x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x^{2} - x + \log \relax (x) - 96}{\log \left (x - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x^{2} - x + \log \relax (x) - 96}{\log \left (x - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 19, normalized size = 1.00
method | result | size |
risch | \(-\frac {x^{2}-x +\ln \relax (x )-96}{\ln \left (x -5\right )}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 18, normalized size = 0.95 \begin {gather*} -\frac {x^{2} - x + \log \relax (x) - 96}{\log \left (x - 5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.41, size = 38, normalized size = 2.00 \begin {gather*} \frac {x}{\ln \left (x-5\right )}+\frac {96}{\ln \left (x-5\right )}-\frac {x^2}{\ln \left (x-5\right )}-\frac {\ln \relax (x)}{\ln \left (x-5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 14, normalized size = 0.74 \begin {gather*} \frac {- x^{2} + x - \log {\relax (x )} + 96}{\log {\left (x - 5 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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