Optimal. Leaf size=18 \[ -8+x-x^2 \left (-x+\log ^2(x)\right )^2 \]
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Rubi [A]
time = 0.08, antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps
used = 11, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2341, 2342}
\begin {gather*} -x^4+2 x^3 \log ^2(x)-x^2 \log ^4(x)+x \end {gather*}
Antiderivative was successfully verified.
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Rule 2341
Rule 2342
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=x-x^4-2 \int x \log ^4(x) \, dx+4 \int x^2 \log (x) \, dx-4 \int x \log ^3(x) \, dx+6 \int x^2 \log ^2(x) \, dx\\ &=x-\frac {4 x^3}{9}-x^4+\frac {4}{3} x^3 \log (x)+2 x^3 \log ^2(x)-2 x^2 \log ^3(x)-x^2 \log ^4(x)-4 \int x^2 \log (x) \, dx+4 \int x \log ^3(x) \, dx+6 \int x \log ^2(x) \, dx\\ &=x-x^4+3 x^2 \log ^2(x)+2 x^3 \log ^2(x)-x^2 \log ^4(x)-6 \int x \log (x) \, dx-6 \int x \log ^2(x) \, dx\\ &=x+\frac {3 x^2}{2}-x^4-3 x^2 \log (x)+2 x^3 \log ^2(x)-x^2 \log ^4(x)+6 \int x \log (x) \, dx\\ &=x-x^4+2 x^3 \log ^2(x)-x^2 \log ^4(x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.39 \begin {gather*} x-x^4+2 x^3 \log ^2(x)-x^2 \log ^4(x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 26, normalized size = 1.44
method | result | size |
default | \(-x^{4}+x -x^{2} \ln \left (x \right )^{4}+2 x^{3} \ln \left (x \right )^{2}\) | \(26\) |
norman | \(-x^{4}+x -x^{2} \ln \left (x \right )^{4}+2 x^{3} \ln \left (x \right )^{2}\) | \(26\) |
risch | \(-x^{4}+x -x^{2} \ln \left (x \right )^{4}+2 x^{3} \ln \left (x \right )^{2}\) | \(26\) |
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. 88 vs.
\(2 (18) = 36\).
time = 0.26, size = 88, normalized size = 4.89 \begin {gather*} \frac {2}{9} \, {\left (9 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 2\right )} x^{3} - x^{4} + \frac {4}{3} \, x^{3} \log \left (x\right ) - \frac {1}{2} \, {\left (2 \, \log \left (x\right )^{4} - 4 \, \log \left (x\right )^{3} + 6 \, \log \left (x\right )^{2} - 6 \, \log \left (x\right ) + 3\right )} x^{2} - \frac {1}{2} \, {\left (4 \, \log \left (x\right )^{3} - 6 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) - 3\right )} x^{2} - \frac {4}{9} \, x^{3} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 25, normalized size = 1.39 \begin {gather*} -x^{2} \log \left (x\right )^{4} + 2 \, x^{3} \log \left (x\right )^{2} - x^{4} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 1.22 \begin {gather*} - x^{4} + 2 x^{3} \log {\left (x \right )}^{2} - x^{2} \log {\left (x \right )}^{4} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 25, normalized size = 1.39 \begin {gather*} -x^{2} \log \left (x\right )^{4} + 2 \, x^{3} \log \left (x\right )^{2} - x^{4} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.06, size = 25, normalized size = 1.39 \begin {gather*} -x^4+2\,x^3\,{\ln \left (x\right )}^2-x^2\,{\ln \left (x\right )}^4+x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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