Optimal. Leaf size=25 \[ 3+x-\left (-3+\frac {1}{2} (3-2 x)\right )^2 (-5+x+\log (x))^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(25)=50\).
time = 0.08, antiderivative size = 74, normalized size of antiderivative = 2.96, number of steps
used = 17, number of rules used = 9, integrand size = 59, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {12, 14, 2404,
2332, 2338, 2341, 2367, 2333, 2342} \begin {gather*} -x^4+7 x^3-2 x^3 \log (x)+\frac {11 x^2}{4}-x^2 \log ^2(x)+4 x^2 \log (x)-\frac {103 x}{2}-3 x \log ^2(x)-\frac {9 \log ^2(x)}{4}+\frac {51}{2} x \log (x)+\frac {45 \log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2332
Rule 2333
Rule 2338
Rule 2341
Rule 2342
Rule 2367
Rule 2404
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {45-52 x+19 x^2+38 x^3-8 x^4+\left (-9+39 x+12 x^2-12 x^3\right ) \log (x)+\left (-6 x-4 x^2\right ) \log ^2(x)}{x} \, dx\\ &=\frac {1}{2} \int \left (\frac {45-52 x+19 x^2+38 x^3-8 x^4}{x}-\frac {3 (3+2 x) \left (1-5 x+2 x^2\right ) \log (x)}{x}-2 (3+2 x) \log ^2(x)\right ) \, dx\\ &=\frac {1}{2} \int \frac {45-52 x+19 x^2+38 x^3-8 x^4}{x} \, dx-\frac {3}{2} \int \frac {(3+2 x) \left (1-5 x+2 x^2\right ) \log (x)}{x} \, dx-\int (3+2 x) \log ^2(x) \, dx\\ &=\frac {1}{2} \int \left (-52+\frac {45}{x}+19 x+38 x^2-8 x^3\right ) \, dx-\frac {3}{2} \int \left (-13 \log (x)+\frac {3 \log (x)}{x}-4 x \log (x)+4 x^2 \log (x)\right ) \, dx-\int \left (3 \log ^2(x)+2 x \log ^2(x)\right ) \, dx\\ &=-26 x+\frac {19 x^2}{4}+\frac {19 x^3}{3}-x^4+\frac {45 \log (x)}{2}-2 \int x \log ^2(x) \, dx-3 \int \log ^2(x) \, dx-\frac {9}{2} \int \frac {\log (x)}{x} \, dx+6 \int x \log (x) \, dx-6 \int x^2 \log (x) \, dx+\frac {39}{2} \int \log (x) \, dx\\ &=-\frac {91 x}{2}+\frac {13 x^2}{4}+7 x^3-x^4+\frac {45 \log (x)}{2}+\frac {39}{2} x \log (x)+3 x^2 \log (x)-2 x^3 \log (x)-\frac {9 \log ^2(x)}{4}-3 x \log ^2(x)-x^2 \log ^2(x)+2 \int x \log (x) \, dx+6 \int \log (x) \, dx\\ &=-\frac {103 x}{2}+\frac {11 x^2}{4}+7 x^3-x^4+\frac {45 \log (x)}{2}+\frac {51}{2} x \log (x)+4 x^2 \log (x)-2 x^3 \log (x)-\frac {9 \log ^2(x)}{4}-3 x \log ^2(x)-x^2 \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(25)=50\).
time = 0.03, size = 74, normalized size = 2.96 \begin {gather*} -\frac {103 x}{2}+\frac {11 x^2}{4}+7 x^3-x^4+\frac {45 \log (x)}{2}+\frac {51}{2} x \log (x)+4 x^2 \log (x)-2 x^3 \log (x)-\frac {9 \log ^2(x)}{4}-3 x \log ^2(x)-x^2 \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs.
\(2(19)=38\).
time = 0.19, size = 65, normalized size = 2.60
method | result | size |
risch | \(\frac {\left (-2 x^{2}-6 x -\frac {9}{2}\right ) \ln \left (x \right )^{2}}{2}+\frac {\left (-4 x^{3}+8 x^{2}+51 x \right ) \ln \left (x \right )}{2}-x^{4}+7 x^{3}+\frac {11 x^{2}}{4}-\frac {103 x}{2}+\frac {45 \ln \left (x \right )}{2}\) | \(58\) |
default | \(-x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+\frac {11 x^{2}}{4}-2 x^{3} \ln \left (x \right )+7 x^{3}-x^{4}-3 x \ln \left (x \right )^{2}+\frac {51 x \ln \left (x \right )}{2}-\frac {103 x}{2}-\frac {9 \ln \left (x \right )^{2}}{4}+\frac {45 \ln \left (x \right )}{2}\) | \(65\) |
norman | \(-x^{2} \ln \left (x \right )^{2}+4 x^{2} \ln \left (x \right )+\frac {11 x^{2}}{4}-2 x^{3} \ln \left (x \right )+7 x^{3}-x^{4}-3 x \ln \left (x \right )^{2}+\frac {51 x \ln \left (x \right )}{2}-\frac {103 x}{2}-\frac {9 \ln \left (x \right )^{2}}{4}+\frac {45 \ln \left (x \right )}{2}\) | \(65\) |
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. 78 vs.
\(2 (19) = 38\).
time = 0.35, size = 78, normalized size = 3.12 \begin {gather*} -x^{4} - 2 \, x^{3} \log \left (x\right ) - \frac {1}{2} \, {\left (2 \, \log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 1\right )} x^{2} + 7 \, x^{3} + 3 \, x^{2} \log \left (x\right ) - 3 \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + \frac {13}{4} \, x^{2} + \frac {39}{2} \, x \log \left (x\right ) - \frac {9}{4} \, \log \left (x\right )^{2} - \frac {91}{2} \, x + \frac {45}{2} \, \log \left (x\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. 54 vs.
\(2 (19) = 38\).
time = 0.40, size = 54, normalized size = 2.16 \begin {gather*} -x^{4} + 7 \, x^{3} - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x\right )^{2} + \frac {11}{4} \, x^{2} - \frac {1}{2} \, {\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x - 45\right )} \log \left (x\right ) - \frac {103}{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. 61 vs.
\(2 (19) = 38\).
time = 0.10, size = 61, normalized size = 2.44 \begin {gather*} - x^{4} + 7 x^{3} + \frac {11 x^{2}}{4} - \frac {103 x}{2} + \left (- x^{2} - 3 x - \frac {9}{4}\right ) \log {\left (x \right )}^{2} + \left (- 2 x^{3} + 4 x^{2} + \frac {51 x}{2}\right ) \log {\left (x \right )} + \frac {45 \log {\left (x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 57 vs.
\(2 (19) = 38\).
time = 0.40, size = 57, normalized size = 2.28 \begin {gather*} -x^{4} + 7 \, x^{3} - \frac {1}{4} \, {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (x\right )^{2} + \frac {11}{4} \, x^{2} - \frac {1}{2} \, {\left (4 \, x^{3} - 8 \, x^{2} - 51 \, x\right )} \log \left (x\right ) - \frac {103}{2} \, x + \frac {45}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.41, size = 64, normalized size = 2.56 \begin {gather*} -x^4-2\,x^3\,\ln \left (x\right )+7\,x^3-x^2\,{\ln \left (x\right )}^2+4\,x^2\,\ln \left (x\right )+\frac {11\,x^2}{4}-3\,x\,{\ln \left (x\right )}^2+\frac {51\,x\,\ln \left (x\right )}{2}-\frac {103\,x}{2}-\frac {9\,{\ln \left (x\right )}^2}{4}+\frac {45\,\ln \left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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