Optimal. Leaf size=25 \[ 2 x+e^4 (-2+3 x)^2 \log ^2\left (\frac {1}{2 x}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(25)=50\).
time = 0.07, antiderivative size = 62, normalized size of antiderivative = 2.48, number of steps
used = 11, number of rules used = 9, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {14, 45, 2372,
2338, 2367, 2333, 2332, 2342, 2341} \begin {gather*} 9 e^4 x^2 \log ^2\left (\frac {1}{2 x}\right )+2 x-12 e^4 x \log ^2\left (\frac {1}{2 x}\right )-4 e^4 \log ^2(x)-8 e^4 \log (x) \log \left (\frac {1}{2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2341
Rule 2342
Rule 2367
Rule 2372
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2-\frac {2 e^4 (-2+3 x)^2 \log \left (\frac {1}{2 x}\right )}{x}+6 e^4 (-2+3 x) \log ^2\left (\frac {1}{2 x}\right )\right ) \, dx\\ &=2 x-\left (2 e^4\right ) \int \frac {(-2+3 x)^2 \log \left (\frac {1}{2 x}\right )}{x} \, dx+\left (6 e^4\right ) \int (-2+3 x) \log ^2\left (\frac {1}{2 x}\right ) \, dx\\ &=2 x+e^4 \log \left (\frac {1}{2 x}\right ) \left (24 x-9 x^2-8 \log (x)\right )-\left (2 e^4\right ) \int \left (-12+\frac {9 x}{2}+\frac {4 \log (x)}{x}\right ) \, dx+\left (6 e^4\right ) \int \left (-2 \log ^2\left (\frac {1}{2 x}\right )+3 x \log ^2\left (\frac {1}{2 x}\right )\right ) \, dx\\ &=2 x+24 e^4 x-\frac {9 e^4 x^2}{2}+e^4 \log \left (\frac {1}{2 x}\right ) \left (24 x-9 x^2-8 \log (x)\right )-\left (8 e^4\right ) \int \frac {\log (x)}{x} \, dx-\left (12 e^4\right ) \int \log ^2\left (\frac {1}{2 x}\right ) \, dx+\left (18 e^4\right ) \int x \log ^2\left (\frac {1}{2 x}\right ) \, dx\\ &=2 x+24 e^4 x-\frac {9 e^4 x^2}{2}-12 e^4 x \log ^2\left (\frac {1}{2 x}\right )+9 e^4 x^2 \log ^2\left (\frac {1}{2 x}\right )+e^4 \log \left (\frac {1}{2 x}\right ) \left (24 x-9 x^2-8 \log (x)\right )-4 e^4 \log ^2(x)+\left (18 e^4\right ) \int x \log \left (\frac {1}{2 x}\right ) \, dx-\left (24 e^4\right ) \int \log \left (\frac {1}{2 x}\right ) \, dx\\ &=2 x-24 e^4 x \log \left (\frac {1}{2 x}\right )+9 e^4 x^2 \log \left (\frac {1}{2 x}\right )-12 e^4 x \log ^2\left (\frac {1}{2 x}\right )+9 e^4 x^2 \log ^2\left (\frac {1}{2 x}\right )+e^4 \log \left (\frac {1}{2 x}\right ) \left (24 x-9 x^2-8 \log (x)\right )-4 e^4 \log ^2(x)\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(25)=50\).
time = 0.02, size = 53, normalized size = 2.12 \begin {gather*} 2 x+4 e^4 \log ^2\left (\frac {1}{2 x}\right )-12 e^4 x \log ^2\left (\frac {1}{2 x}\right )+9 e^4 x^2 \log ^2\left (\frac {1}{2 x}\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. \(126\) vs.
\(2(24)=48\).
time = 0.23, size = 127, normalized size = 5.08
method | result | size |
risch | \(2 x +4 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2}-12 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2} x +9 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2} x^{2}\) | \(45\) |
norman | \(2 x +4 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2}-12 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2} x +9 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2} x^{2}\) | \(51\) |
derivativedivides | \(6 \,{\mathrm e}^{4} \left (-2 x \ln \left (\frac {1}{2 x}\right )^{2}-4 \ln \left (\frac {1}{2 x}\right ) x -4 x \right )+4 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2}-\frac {9 \,{\mathrm e}^{4} \left (-2 \ln \left (\frac {1}{2 x}\right )^{2} x^{2}-2 \ln \left (\frac {1}{2 x}\right ) x^{2}-x^{2}\right )}{2}-12 \,{\mathrm e}^{4} \left (-2 \ln \left (\frac {1}{2 x}\right ) x -2 x \right )+\frac {9 \,{\mathrm e}^{4} \left (-2 \ln \left (\frac {1}{2 x}\right ) x^{2}-x^{2}\right )}{2}+2 x\) | \(127\) |
default | \(6 \,{\mathrm e}^{4} \left (-2 x \ln \left (\frac {1}{2 x}\right )^{2}-4 \ln \left (\frac {1}{2 x}\right ) x -4 x \right )+4 \,{\mathrm e}^{4} \ln \left (\frac {1}{2 x}\right )^{2}-\frac {9 \,{\mathrm e}^{4} \left (-2 \ln \left (\frac {1}{2 x}\right )^{2} x^{2}-2 \ln \left (\frac {1}{2 x}\right ) x^{2}-x^{2}\right )}{2}-12 \,{\mathrm e}^{4} \left (-2 \ln \left (\frac {1}{2 x}\right ) x -2 x \right )+\frac {9 \,{\mathrm e}^{4} \left (-2 \ln \left (\frac {1}{2 x}\right ) x^{2}-x^{2}\right )}{2}+2 x\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 44, normalized size = 1.76 \begin {gather*} 9 \, x^{2} e^{4} \log \left (\frac {1}{2 \, x}\right )^{2} - 12 \, x e^{4} \log \left (\frac {1}{2 \, x}\right )^{2} + 4 \, e^{4} \log \left (\frac {1}{2 \, x}\right )^{2} + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 25, normalized size = 1.00 \begin {gather*} {\left (9 \, x^{2} - 12 \, x + 4\right )} e^{4} \log \left (\frac {1}{2 \, x}\right )^{2} + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 31, normalized size = 1.24 \begin {gather*} 2 x + \left (9 x^{2} e^{4} - 12 x e^{4} + 4 e^{4}\right ) \log {\left (\frac {1}{2 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. 46 vs.
\(2 (22) = 44\).
time = 0.40, size = 46, normalized size = 1.84 \begin {gather*} {\left (9 \, e^{4} \log \left (2 \, x\right )^{2} - \frac {12 \, e^{4} \log \left (2 \, x\right )^{2}}{x} + \frac {4 \, e^{4} \log \left (2 \, x\right )^{2}}{x^{2}} + \frac {2}{x}\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.66, size = 45, normalized size = 1.80 \begin {gather*} 4\,{\mathrm {e}}^4\,{\ln \left (\frac {1}{2\,x}\right )}^2-x\,\left (12\,{\mathrm {e}}^4\,{\ln \left (\frac {1}{2\,x}\right )}^2-2\right )+9\,x^2\,{\mathrm {e}}^4\,{\ln \left (\frac {1}{2\,x}\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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