Optimal. Leaf size=27 \[ \frac {1}{8} \left (x-(3-e) \left (x-\left (-5+\frac {\log (x)}{x}\right )^2\right )\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(27)=54\).
time = 0.08, antiderivative size = 73, normalized size of antiderivative = 2.70, number of steps
used = 11, number of rules used = 7, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used = {12, 14, 37,
2372, 45, 2342, 2341} \begin {gather*} \frac {(3-e) \log ^2(x)}{8 x^2}-\frac {(3-e) (5 x+1)^2 \log (x)}{8 x^2}+\frac {(3-e) \log (x)}{8 x^2}-\frac {1}{8} (2-e) x+\frac {25}{8} (3-e) \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 37
Rule 45
Rule 2341
Rule 2342
Rule 2372
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {-30 x-2 x^3+e \left (10 x+x^3\right )+(6+e (-2-10 x)+30 x) \log (x)+(-6+2 e) \log ^2(x)}{x^3} \, dx\\ &=\frac {1}{8} \int \left (\frac {-10 (3-e)-(2-e) x^2}{x^2}-\frac {2 (-3+e) (1+5 x) \log (x)}{x^3}+\frac {2 (-3+e) \log ^2(x)}{x^3}\right ) \, dx\\ &=\frac {1}{8} \int \frac {-10 (3-e)-(2-e) x^2}{x^2} \, dx+\frac {1}{4} (3-e) \int \frac {(1+5 x) \log (x)}{x^3} \, dx+\frac {1}{4} (-3+e) \int \frac {\log ^2(x)}{x^3} \, dx\\ &=-\frac {(3-e) (1+5 x)^2 \log (x)}{8 x^2}+\frac {(3-e) \log ^2(x)}{8 x^2}+\frac {1}{8} \int \left (-2 \left (1-\frac {e}{2}\right )+\frac {10 (-3+e)}{x^2}\right ) \, dx+\frac {1}{4} (-3+e) \int -\frac {(1+5 x)^2}{2 x^3} \, dx+\frac {1}{4} (-3+e) \int \frac {\log (x)}{x^3} \, dx\\ &=\frac {3-e}{16 x^2}+\frac {5 (3-e)}{4 x}-\frac {1}{8} (2-e) x+\frac {(3-e) \log (x)}{8 x^2}-\frac {(3-e) (1+5 x)^2 \log (x)}{8 x^2}+\frac {(3-e) \log ^2(x)}{8 x^2}+\frac {1}{8} (3-e) \int \frac {(1+5 x)^2}{x^3} \, dx\\ &=\frac {3-e}{16 x^2}+\frac {5 (3-e)}{4 x}-\frac {1}{8} (2-e) x+\frac {(3-e) \log (x)}{8 x^2}-\frac {(3-e) (1+5 x)^2 \log (x)}{8 x^2}+\frac {(3-e) \log ^2(x)}{8 x^2}+\frac {1}{8} (3-e) \int \left (\frac {1}{x^3}+\frac {10}{x^2}+\frac {25}{x}\right ) \, dx\\ &=-\frac {1}{8} (2-e) x+\frac {25}{8} (3-e) \log (x)+\frac {(3-e) \log (x)}{8 x^2}-\frac {(3-e) (1+5 x)^2 \log (x)}{8 x^2}+\frac {(3-e) \log ^2(x)}{8 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 33, normalized size = 1.22 \begin {gather*} \frac {1}{8} \left ((-2+e) x+\frac {10 (-3+e) \log (x)}{x}+\frac {(3-e) \log ^2(x)}{x^2}\right ) \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. \(92\) vs.
\(2(26)=52\).
time = 0.03, size = 93, normalized size = 3.44
method | result | size |
risch | \(-\frac {\left ({\mathrm e}-3\right ) \ln \left (x \right )^{2}}{8 x^{2}}+\frac {5 \left ({\mathrm e}-3\right ) \ln \left (x \right )}{4 x}-\frac {x}{4}+\frac {x \,{\mathrm e}}{8}\) | \(34\) |
norman | \(\frac {\left (-\frac {1}{4}+\frac {{\mathrm e}}{8}\right ) x^{3}+\left (-\frac {{\mathrm e}}{8}+\frac {3}{8}\right ) \ln \left (x \right )^{2}+\left (\frac {5 \,{\mathrm e}}{4}-\frac {15}{4}\right ) x \ln \left (x \right )}{x^{2}}\) | \(37\) |
default | \(\frac {x \,{\mathrm e}}{8}+\frac {{\mathrm e} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )}{4}-\frac {5 \,{\mathrm e} \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )}{4}-\frac {x}{4}+\frac {3 \ln \left (x \right )^{2}}{8 x^{2}}-\frac {{\mathrm e} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )}{4}-\frac {15 \ln \left (x \right )}{4 x}-\frac {5 \,{\mathrm e}}{4 x}\) | \(93\) |
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. 100 vs.
\(2 (24) = 48\).
time = 0.28, size = 100, normalized size = 3.70 \begin {gather*} \frac {1}{8} \, x e + \frac {5}{4} \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} e + \frac {1}{16} \, {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} e - \frac {1}{4} \, x - \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} e}{16 \, x^{2}} - \frac {5 \, e}{4 \, x} - \frac {15 \, \log \left (x\right )}{4 \, x} + \frac {3 \, {\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )}}{16 \, x^{2}} - \frac {3 \, \log \left (x\right )}{8 \, x^{2}} - \frac {3}{16 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 39, normalized size = 1.44 \begin {gather*} \frac {x^{3} e - 2 \, x^{3} - {\left (e - 3\right )} \log \left (x\right )^{2} + 10 \, {\left (x e - 3 \, x\right )} \log \left (x\right )}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 37, normalized size = 1.37 \begin {gather*} x \left (- \frac {1}{4} + \frac {e}{8}\right ) + \frac {\left (-15 + 5 e\right ) \log {\left (x \right )}}{4 x} + \frac {\left (3 - e\right ) \log {\left (x \right )}^{2}}{8 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 43, normalized size = 1.59 \begin {gather*} \frac {x^{3} e - 2 \, x^{3} + 10 \, x e \log \left (x\right ) - e \log \left (x\right )^{2} - 30 \, x \log \left (x\right ) + 3 \, \log \left (x\right )^{2}}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 38, normalized size = 1.41 \begin {gather*} \frac {\frac {x^2\,\ln \left (x\right )\,\left (10\,\mathrm {e}-30\right )}{8}-\frac {x\,{\ln \left (x\right )}^2\,\left (\mathrm {e}-3\right )}{8}}{x^3}+x\,\left (\frac {\mathrm {e}}{8}-\frac {1}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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