Optimal. Leaf size=26 \[ 1+x^2 \left (x+\frac {e^{16} (-x+\log (x))}{3 x}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(26)=52\).
time = 0.10, antiderivative size = 72, normalized size of antiderivative = 2.77, number of steps
used = 9, number of rules used = 6, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {12, 14, 2404,
2332, 2338, 2341} \begin {gather*} x^4-\frac {2 e^{16} x^3}{3}+\frac {1}{9} e^{16} \left (3+e^{16}\right ) x^2-\frac {e^{16} x^2}{3}+\frac {2}{3} e^{16} x^2 \log (x)+\frac {1}{9} e^{32} \log ^2(x)-\frac {2}{9} e^{32} x \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 2332
Rule 2338
Rule 2341
Rule 2404
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \frac {36 x^4+e^{32} \left (-2 x+2 x^2\right )+e^{16} \left (6 x^2-18 x^3\right )+\left (e^{32} (2-2 x)+12 e^{16} x^2\right ) \log (x)}{x} \, dx\\ &=\frac {1}{9} \int \left (2 \left (-e^{32}+e^{16} \left (3+e^{16}\right ) x-9 e^{16} x^2+18 x^3\right )+\frac {2 e^{16} \left (e^{16}-e^{16} x+6 x^2\right ) \log (x)}{x}\right ) \, dx\\ &=\frac {2}{9} \int \left (-e^{32}+e^{16} \left (3+e^{16}\right ) x-9 e^{16} x^2+18 x^3\right ) \, dx+\frac {1}{9} \left (2 e^{16}\right ) \int \frac {\left (e^{16}-e^{16} x+6 x^2\right ) \log (x)}{x} \, dx\\ &=-\frac {2 e^{32} x}{9}+\frac {1}{9} e^{16} \left (3+e^{16}\right ) x^2-\frac {2 e^{16} x^3}{3}+x^4+\frac {1}{9} \left (2 e^{16}\right ) \int \left (-e^{16} \log (x)+\frac {e^{16} \log (x)}{x}+6 x \log (x)\right ) \, dx\\ &=-\frac {2 e^{32} x}{9}+\frac {1}{9} e^{16} \left (3+e^{16}\right ) x^2-\frac {2 e^{16} x^3}{3}+x^4+\frac {1}{3} \left (4 e^{16}\right ) \int x \log (x) \, dx-\frac {1}{9} \left (2 e^{32}\right ) \int \log (x) \, dx+\frac {1}{9} \left (2 e^{32}\right ) \int \frac {\log (x)}{x} \, dx\\ &=-\frac {1}{3} e^{16} x^2+\frac {1}{9} e^{16} \left (3+e^{16}\right ) x^2-\frac {2 e^{16} x^3}{3}+x^4-\frac {2}{9} e^{32} x \log (x)+\frac {2}{3} e^{16} x^2 \log (x)+\frac {1}{9} e^{32} \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.88 \begin {gather*} \frac {1}{9} \left (\left (e^{16}-3 x\right ) x-e^{16} \log (x)\right )^2 \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. \(75\) vs.
\(2(23)=46\).
time = 0.05, size = 76, normalized size = 2.92
method | result | size |
risch | \(\frac {{\mathrm e}^{32} \ln \left (x \right )^{2}}{9}-\frac {2 \,{\mathrm e}^{16} x \left ({\mathrm e}^{16}-3 x \right ) \ln \left (x \right )}{9}+\frac {{\mathrm e}^{32} x^{2}}{9}-\frac {2 x^{3} {\mathrm e}^{16}}{3}+x^{4}\) | \(40\) |
norman | \(x^{4}-\frac {2 x^{3} {\mathrm e}^{16}}{3}+\frac {{\mathrm e}^{32} x^{2}}{9}+\frac {{\mathrm e}^{32} \ln \left (x \right )^{2}}{9}+\frac {2 x^{2} {\mathrm e}^{16} \ln \left (x \right )}{3}-\frac {2 \,{\mathrm e}^{32} \ln \left (x \right ) x}{9}\) | \(49\) |
default | \(-\frac {2 \,{\mathrm e}^{32} \left (x \ln \left (x \right )-x \right )}{9}+\frac {4 \,{\mathrm e}^{16} \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )}{3}+\frac {{\mathrm e}^{32} x^{2}}{9}-\frac {2 x^{3} {\mathrm e}^{16}}{3}+x^{4}+\frac {{\mathrm e}^{32} \ln \left (x \right )^{2}}{9}-\frac {2 x \,{\mathrm e}^{32}}{9}+\frac {x^{2} {\mathrm e}^{16}}{3}\) | \(76\) |
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. 67 vs.
\(2 (26) = 52\).
time = 0.26, size = 67, normalized size = 2.58 \begin {gather*} x^{4} - \frac {2}{3} \, x^{3} e^{16} + \frac {1}{9} \, x^{2} e^{32} + \frac {1}{3} \, x^{2} e^{16} + \frac {1}{9} \, e^{32} \log \left (x\right )^{2} - \frac {2}{9} \, {\left (x \log \left (x\right ) - x\right )} e^{32} - \frac {2}{9} \, x e^{32} + \frac {1}{3} \, {\left (2 \, x^{2} \log \left (x\right ) - x^{2}\right )} e^{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 43, normalized size = 1.65 \begin {gather*} x^{4} - \frac {2}{3} \, x^{3} e^{16} + \frac {1}{9} \, x^{2} e^{32} + \frac {1}{9} \, e^{32} \log \left (x\right )^{2} + \frac {2}{9} \, {\left (3 \, x^{2} e^{16} - x e^{32}\right )} \log \left (x\right ) \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. 53 vs.
\(2 (20) = 40\).
time = 0.08, size = 53, normalized size = 2.04 \begin {gather*} x^{4} - \frac {2 x^{3} e^{16}}{3} + \frac {x^{2} e^{32}}{9} + \left (\frac {2 x^{2} e^{16}}{3} - \frac {2 x e^{32}}{9}\right ) \log {\left (x \right )} + \frac {e^{32} \log {\left (x \right )}^{2}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 42, normalized size = 1.62 \begin {gather*} x^{4} - \frac {2}{3} \, x^{3} e^{16} + \frac {2}{3} \, x^{2} e^{16} \log \left (x\right ) + \frac {1}{9} \, x^{2} e^{32} - \frac {2}{9} \, x e^{32} \log \left (x\right ) + \frac {1}{9} \, e^{32} \log \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.86, size = 20, normalized size = 0.77 \begin {gather*} \frac {{\left ({\mathrm {e}}^{16}\,\ln \left (x\right )-x\,{\mathrm {e}}^{16}+3\,x^2\right )}^2}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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