Optimal. Leaf size=18 \[ x \left (-2+\log (x)+\log \left (x+\frac {144 x^2}{\log ^2(2)}\right )\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps
used = 8, number of rules used = 6, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {6820, 45, 2332,
2579, 31, 8} \begin {gather*} -2 x+x \log \left (x \left (\frac {144 x}{\log ^2(2)}+1\right )\right )+x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 31
Rule 45
Rule 2332
Rule 2579
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {144 x}{144 x+\log ^2(2)}+\log (x)+\log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right )\right ) \, dx\\ &=144 \int \frac {x}{144 x+\log ^2(2)} \, dx+\int \log (x) \, dx+\int \log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right ) \, dx\\ &=-x+x \log (x)+x \log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right )-2 \int 1 \, dx+144 \int \left (\frac {1}{144}-\frac {\log ^2(2)}{144 \left (144 x+\log ^2(2)\right )}\right ) \, dx+\int \frac {1}{1+\frac {144 x}{\log ^2(2)}} \, dx\\ &=-2 x+x \log (x)+x \log \left (x \left (1+\frac {144 x}{\log ^2(2)}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.22 \begin {gather*} -2 x+x \log (x)+x \log \left (x+\frac {144 x^2}{\log ^2(2)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.91, size = 28, normalized size = 1.56
method | result | size |
default | \(-2 x \ln \left (\ln \left (2\right )\right )+x \ln \left (x \right )+x \ln \left (x \left (\ln \left (2\right )^{2}+144 x \right )\right )-2 x\) | \(28\) |
norman | \(x \ln \left (x \right )+x \ln \left (\frac {x \ln \left (2\right )^{2}+144 x^{2}}{\ln \left (2\right )^{2}}\right )-2 x\) | \(29\) |
risch | \(x \ln \left (\ln \left (2\right )^{2}+144 x \right )+2 x \ln \left (x \right )-\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \left (2\right )^{2}+144 x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i \left (\ln \left (2\right )^{2}+144 x \right )\right ) \mathrm {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )^{2}}{2}-\frac {i \pi x \mathrm {csgn}\left (i x \left (\ln \left (2\right )^{2}+144 x \right )\right )^{3}}{2}-2 x \ln \left (\ln \left (2\right )\right )-2 x\) | \(139\) |
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. 51 vs.
\(2 (18) = 36\).
time = 0.55, size = 51, normalized size = 2.83 \begin {gather*} -\frac {1}{144} \, \log \left (2\right )^{2} \log \left (\log \left (2\right )^{2} + 144 \, x\right ) - x {\left (2 \, \log \left (\log \left (2\right )\right ) + 3\right )} + \frac {1}{144} \, {\left (\log \left (2\right )^{2} + 144 \, x\right )} \log \left (\log \left (2\right )^{2} + 144 \, x\right ) + 2 \, x \log \left (x\right ) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 28, normalized size = 1.56 \begin {gather*} x \log \left (x\right ) + x \log \left (\frac {x \log \left (2\right )^{2} + 144 \, x^{2}}{\log \left (2\right )^{2}}\right ) - 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs.
\(2 (19) = 38\).
time = 0.44, size = 54, normalized size = 3.00 \begin {gather*} x \log {\left (x \right )} - 2 x + \left (x + \frac {\log {\left (2 \right )}^{2}}{864}\right ) \log {\left (\frac {144 x^{2} + x \log {\left (2 \right )}^{2}}{\log {\left (2 \right )}^{2}} \right )} - \frac {\log {\left (2 \right )}^{2} \log {\left (144 x^{2} + x \log {\left (2 \right )}^{2} \right )}}{864} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 25, normalized size = 1.39 \begin {gather*} -2 \, x {\left (\log \left (\log \left (2\right )\right ) + 1\right )} + x \log \left (\log \left (2\right )^{2} + 144 \, x\right ) + 2 \, x \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.39, size = 29, normalized size = 1.61 \begin {gather*} x\,\ln \left (144\,x^2+{\ln \left (2\right )}^2\,x\right )-2\,x-2\,x\,\ln \left (\ln \left (2\right )\right )+x\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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