Optimal. Leaf size=13 \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]
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Rubi [F] time = 0.625373, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt{x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt{x+\log (x)}}{x^3+2 x^2 \log (x)+x \log ^2(x)} \, dx &=\int \frac{x^2+2 x \log (x)+\log ^2(x)+(1+x) \sqrt{x+\log (x)}}{x (x+\log (x))^2} \, dx\\ &=\int \left (\frac{1}{x}+\frac{1}{(x+\log (x))^{3/2}}-\frac{1}{\log (x) (x+\log (x))^{3/2}}-\frac{1}{\log ^2(x) \sqrt{x+\log (x)}}+\frac{\sqrt{x+\log (x)}}{x \log ^2(x)}\right ) \, dx\\ &=\log (x)+\int \frac{1}{(x+\log (x))^{3/2}} \, dx-\int \frac{1}{\log (x) (x+\log (x))^{3/2}} \, dx-\int \frac{1}{\log ^2(x) \sqrt{x+\log (x)}} \, dx+\int \frac{\sqrt{x+\log (x)}}{x \log ^2(x)} \, dx\\ \end{align*}
Mathematica [A] time = 0.150842, size = 13, normalized size = 1. \[ \log (x)-\frac{2}{\sqrt{x+\log (x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.012, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}+2\,{x}^{2}\ln \left ( x \right ) +x \left ( \ln \left ( x \right ) \right ) ^{2}} \left ({x}^{2}+2\,x\ln \left ( x \right ) + \left ( \ln \left ( x \right ) \right ) ^{2}+ \left ( 1+x \right ) \sqrt{x+\ln \left ( x \right ) } \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x + \log \left (x\right )}{\left (x + 1\right )}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2}}\,{d x} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20676, size = 77, normalized size = 5.92 \begin{align*} \frac{x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt{x + \log \left (x\right )}}{x + \log \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + x \sqrt{x + \log{\left (x \right )}} + 2 x \log{\left (x \right )} + \sqrt{x + \log{\left (x \right )}} + \log{\left (x \right )}^{2}}{x \left (x + \log{\left (x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} + \sqrt{x + \log \left (x\right )}{\left (x + 1\right )}}{x^{3} + 2 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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