Optimal. Leaf size=30 \[ -4-4 x-x^2-\log (3)+\frac {1}{25} \left (1+\frac {2}{x+\log (x)}\right )^2 \]
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Rubi [A]
time = 0.56, antiderivative size = 28, normalized size of antiderivative = 0.93, number of steps
used = 6, number of rules used = 4, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {6820, 12,
6874, 6818} \begin {gather*} -(x+2)^2+\frac {4}{25 (x+\log (x))}+\frac {4}{25 (x+\log (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6818
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-4-6 x-2 x^2-50 x^4-25 x^5-\left (2+2 x+150 x^3+75 x^4\right ) \log (x)-75 x^2 (2+x) \log ^2(x)-25 x (2+x) \log ^3(x)\right )}{25 x (x+\log (x))^3} \, dx\\ &=\frac {2}{25} \int \frac {-4-6 x-2 x^2-50 x^4-25 x^5-\left (2+2 x+150 x^3+75 x^4\right ) \log (x)-75 x^2 (2+x) \log ^2(x)-25 x (2+x) \log ^3(x)}{x (x+\log (x))^3} \, dx\\ &=\frac {2}{25} \int \left (-25 (2+x)-\frac {4 (1+x)}{x (x+\log (x))^3}-\frac {2 (1+x)}{x (x+\log (x))^2}\right ) \, dx\\ &=-(2+x)^2-\frac {4}{25} \int \frac {1+x}{x (x+\log (x))^2} \, dx-\frac {8}{25} \int \frac {1+x}{x (x+\log (x))^3} \, dx\\ &=-(2+x)^2+\frac {4}{25 (x+\log (x))^2}+\frac {4}{25 (x+\log (x))}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 28, normalized size = 0.93 \begin {gather*} -\frac {2}{25} \left (50 x+\frac {25 x^2}{2}-\frac {2 (1+x+\log (x))}{(x+\log (x))^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 58, normalized size = 1.93
method | result | size |
risch | \(-x^{2}-4 x +\frac {\frac {4}{25}+\frac {4 x}{25}+\frac {4 \ln \left (x \right )}{25}}{\left (x +\ln \left (x \right )\right )^{2}}\) | \(23\) |
default | \(-\frac {2 \left (-2-2 \ln \left (x \right )+50 x^{3}+\frac {25 x^{4}}{2}+50 x \ln \left (x \right )^{2}+100 x^{2} \ln \left (x \right )+\frac {25 x^{2} \ln \left (x \right )^{2}}{2}+25 x^{3} \ln \left (x \right )-2 x \right )}{25 \left (x +\ln \left (x \right )\right )^{2}}\) | \(58\) |
norman | \(\frac {\frac {4}{25}+2 \ln \left (x \right )^{3}-6 x^{2} \ln \left (x \right )+\frac {4 x}{25}-4 x^{3}-x^{4}-x^{2} \ln \left (x \right )^{2}-2 x^{3} \ln \left (x \right )+\frac {4 \ln \left (x \right )}{25}}{\left (x +\ln \left (x \right )\right )^{2}}-2 \ln \left (x \right )\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs.
\(2 (28) = 56\).
time = 0.32, size = 61, normalized size = 2.03 \begin {gather*} -\frac {25 \, x^{4} + 100 \, x^{3} + 25 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} + 2 \, {\left (25 \, x^{3} + 100 \, x^{2} - 2\right )} \log \left (x\right ) - 4 \, x - 4}{25 \, {\left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \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. 61 vs.
\(2 (28) = 56\).
time = 0.44, size = 61, normalized size = 2.03 \begin {gather*} -\frac {25 \, x^{4} + 100 \, x^{3} + 25 \, {\left (x^{2} + 4 \, x\right )} \log \left (x\right )^{2} + 2 \, {\left (25 \, x^{3} + 100 \, x^{2} - 2\right )} \log \left (x\right ) - 4 \, x - 4}{25 \, {\left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 34, normalized size = 1.13 \begin {gather*} - x^{2} - 4 x + \frac {4 x + 4 \log {\left (x \right )} + 4}{25 x^{2} + 50 x \log {\left (x \right )} + 25 \log {\left (x \right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 31, normalized size = 1.03 \begin {gather*} -x^{2} - 4 \, x + \frac {4 \, {\left (x + \log \left (x\right ) + 1\right )}}{25 \, {\left (x^{2} + 2 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.43, size = 25, normalized size = 0.83 \begin {gather*} \frac {\frac {4\,x}{25}+\frac {4\,\ln \left (x\right )}{25}+\frac {4}{25}}{{\left (x+\ln \left (x\right )\right )}^2}-x^2-4\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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