Optimal. Leaf size=13 \[ \log (x)-\frac {2}{\sqrt {x+\log (x)}} \]
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Rubi [F]
time = 0.39, 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 {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 \end {gather*}
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*}
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Mathematica [A]
time = 0.10, size = 13, normalized size = 1.00 \begin {gather*} \log (x)-\frac {2}{\sqrt {x+\log (x)}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {x^{2}+2 x \ln \left (x \right )+\ln \left (x \right )^{2}+\left (1+x \right ) \sqrt {x +\ln \left (x \right )}}{x^{3}+2 x^{2} \ln \left (x \right )+x \ln \left (x \right )^{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs.
\(2 (11) = 22\).
time = 0.33, size = 24, normalized size = 1.85 \begin {gather*} \frac {x \log \left (x\right ) + \log \left (x\right )^{2} - 2 \, \sqrt {x + \log \left (x\right )}}{x + \log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.02, size = 15, normalized size = 1.15 \begin {gather*} \ln x-\frac {2}{\sqrt {\ln x+x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 11, normalized size = 0.85 \begin {gather*} \ln \left (x\right )-\frac {2}{\sqrt {x+\ln \left (x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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