Optimal. Leaf size=23 \[ \frac {15625 \left (5-e^3\right ) x \log \left (\frac {x}{2}\right )}{x+x^4} \]
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Rubi [A]
time = 0.38, antiderivative size = 37, normalized size of antiderivative = 1.61, number of steps
used = 12, number of rules used = 10, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1608, 28,
6820, 12, 6874, 272, 46, 267, 2373, 266} \begin {gather*} 15625 \left (5-e^3\right ) \log (x)-\frac {15625 \left (5-e^3\right ) x^3 \log \left (\frac {x}{2}\right )}{x^3+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 28
Rule 46
Rule 266
Rule 267
Rule 272
Rule 1608
Rule 2373
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x \left (1+2 x^3+x^6\right )} \, dx\\ &=\int \frac {78125+78125 x^3+e^3 \left (-15625-15625 x^3\right )+\left (-234375 x^3+46875 e^3 x^3\right ) \log \left (\frac {x}{2}\right )}{x \left (1+x^3\right )^2} \, dx\\ &=\int \frac {15625 \left (5-e^3\right ) \left (1+x^3-3 x^3 \log \left (\frac {x}{2}\right )\right )}{x \left (1+x^3\right )^2} \, dx\\ &=\left (15625 \left (5-e^3\right )\right ) \int \frac {1+x^3-3 x^3 \log \left (\frac {x}{2}\right )}{x \left (1+x^3\right )^2} \, dx\\ &=\left (15625 \left (5-e^3\right )\right ) \int \left (\frac {1}{x \left (1+x^3\right )^2}+\frac {x^2}{\left (1+x^3\right )^2}-\frac {3 x^2 \log \left (\frac {x}{2}\right )}{\left (1+x^3\right )^2}\right ) \, dx\\ &=\left (15625 \left (5-e^3\right )\right ) \int \frac {1}{x \left (1+x^3\right )^2} \, dx+\left (15625 \left (5-e^3\right )\right ) \int \frac {x^2}{\left (1+x^3\right )^2} \, dx-\left (46875 \left (5-e^3\right )\right ) \int \frac {x^2 \log \left (\frac {x}{2}\right )}{\left (1+x^3\right )^2} \, dx\\ &=-\frac {15625 \left (5-e^3\right )}{3 \left (1+x^3\right )}-\frac {15625 \left (5-e^3\right ) x^3 \log \left (\frac {x}{2}\right )}{1+x^3}+\frac {1}{3} \left (15625 \left (5-e^3\right )\right ) \text {Subst}\left (\int \frac {1}{x (1+x)^2} \, dx,x,x^3\right )+\left (15625 \left (5-e^3\right )\right ) \int \frac {x^2}{1+x^3} \, dx\\ &=-\frac {15625 \left (5-e^3\right )}{3 \left (1+x^3\right )}-\frac {15625 \left (5-e^3\right ) x^3 \log \left (\frac {x}{2}\right )}{1+x^3}+\frac {15625}{3} \left (5-e^3\right ) \log \left (1+x^3\right )+\frac {1}{3} \left (15625 \left (5-e^3\right )\right ) \text {Subst}\left (\int \left (\frac {1}{-1-x}+\frac {1}{x}-\frac {1}{(1+x)^2}\right ) \, dx,x,x^3\right )\\ &=-\frac {15625 \left (5-e^3\right ) x^3 \log \left (\frac {x}{2}\right )}{1+x^3}+15625 \left (5-e^3\right ) \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 20, normalized size = 0.87 \begin {gather*} -\frac {15625 \left (-5+e^3\right ) \log \left (\frac {x}{2}\right )}{1+x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.43, size = 798, normalized size = 34.70 Too large to display
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. 123 vs.
\(2 (18) = 36\).
time = 0.48, size = 123, normalized size = 5.35 \begin {gather*} -\frac {15625}{3} \, {\left (\frac {3 \, \log \left (\frac {1}{2} \, x\right )}{x^{3} + 1} + \log \left (x^{3} + 1\right ) - \log \left (x^{3}\right )\right )} e^{3} - \frac {15625}{3} \, {\left (\frac {1}{x^{3} + 1} - \log \left (x^{2} - x + 1\right ) - \log \left (x + 1\right ) + 3 \, \log \left (x\right )\right )} e^{3} + \frac {15625 \, e^{3}}{3 \, {\left (x^{3} + 1\right )}} + \frac {78125 \, \log \left (\frac {1}{2} \, x\right )}{x^{3} + 1} + \frac {78125}{3} \, \log \left (x^{3} + 1\right ) - \frac {78125}{3} \, \log \left (x^{3}\right ) - \frac {78125}{3} \, \log \left (x^{2} - x + 1\right ) - \frac {78125}{3} \, \log \left (x + 1\right ) + 78125 \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 17, normalized size = 0.74 \begin {gather*} -\frac {15625 \, {\left (e^{3} - 5\right )} \log \left (\frac {1}{2} \, x\right )}{x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.07, size = 15, normalized size = 0.65 \begin {gather*} \frac {\left (78125 - 15625 e^{3}\right ) \log {\left (\frac {x}{2} \right )}}{x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 23, normalized size = 1.00 \begin {gather*} -\frac {15625 \, {\left (e^{3} \log \left (\frac {1}{2} \, x\right ) - 5 \, \log \left (\frac {1}{2} \, x\right )\right )}}{x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.68, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x^2\,\ln \left (\frac {x}{2}\right )\,\left (15625\,{\mathrm {e}}^3-78125\right )}{x^5+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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