Optimal. Leaf size=24 \[ 2+\frac {1}{e^3}-x+\frac {x}{\left (-2+e^2+2 x\right ) \log (4)} \]
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Rubi [A]
time = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps
used = 5, number of rules used = 4, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.066, Rules used = {12, 2009, 27,
697} \begin {gather*} -x-\frac {2-e^2}{2 \left (-2 x-e^2+2\right ) \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 27
Rule 697
Rule 2009
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-2+e^2+\left (-4-e^4+e^2 (4-4 x)+8 x-4 x^2\right ) \log (4)}{4+e^4-8 x+4 x^2+e^2 (-4+4 x)} \, dx}{\log (4)}\\ &=\frac {\int \frac {4 \left (2-e^2\right ) x \log (4)-4 x^2 \log (4)-\left (2-e^2\right ) \left (1+2 \log (4)-e^2 \log (4)\right )}{\left (-2+e^2\right )^2-4 \left (2-e^2\right ) x+4 x^2} \, dx}{\log (4)}\\ &=\frac {\int \frac {4 \left (2-e^2\right ) x \log (4)-4 x^2 \log (4)-\left (2-e^2\right ) \left (1+2 \log (4)-e^2 \log (4)\right )}{\left (-2+e^2+2 x\right )^2} \, dx}{\log (4)}\\ &=\frac {\int \left (\frac {-2+e^2}{\left (-2+e^2+2 x\right )^2}-\log (4)\right ) \, dx}{\log (4)}\\ &=-x-\frac {2-e^2}{2 \left (2-e^2-2 x\right ) \log (4)}\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(24)=48\).
time = 0.02, size = 49, normalized size = 2.04 \begin {gather*} -\frac {-2+e^2-4 e^2 \log (4)+\left (-2+e^2+2 x\right )^2 \log (4)+e^2 \log (256)}{2 \left (-2+e^2+2 x\right ) \log (4)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.11, size = 37, normalized size = 1.54
method | result | size |
risch | \(-x -\frac {{\mathrm e}^{2}}{4 \ln \left (2\right ) \left (2 x -2+{\mathrm e}^{2}\right )}+\frac {1}{2 \ln \left (2\right ) \left (2 x -2+{\mathrm e}^{2}\right )}\) | \(37\) |
gosper | \(\frac {2 \,{\mathrm e}^{4} \ln \left (2\right )-8 x^{2} \ln \left (2\right )-8 \,{\mathrm e}^{2} \ln \left (2\right )-{\mathrm e}^{2}+8 \ln \left (2\right )+2}{4 \ln \left (2\right ) \left (2 x -2+{\mathrm e}^{2}\right )}\) | \(47\) |
meijerg | \(-\frac {x}{2 \left (\frac {{\mathrm e}^{2}}{2}-1\right ) \ln \left (2\right ) \left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}-\frac {{\mathrm e}^{4} x}{2 \left (\frac {{\mathrm e}^{2}}{2}-1\right ) \left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}+\frac {2 \,{\mathrm e}^{2} x}{\left (\frac {{\mathrm e}^{2}}{2}-1\right ) \left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}+\frac {{\mathrm e}^{2} x}{4 \left (\frac {{\mathrm e}^{2}}{2}-1\right ) \ln \left (2\right ) \left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}-\frac {2 x}{\left (\frac {{\mathrm e}^{2}}{2}-1\right ) \left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}+\left (2-{\mathrm e}^{2}\right ) \left (-\frac {2 x}{\left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}+\ln \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )\right )-\frac {\left ({\mathrm e}^{2}-2\right )^{2} \left (\frac {2 x \left (\frac {6 x}{{\mathrm e}^{2}-2}+6\right )}{3 \left ({\mathrm e}^{2}-2\right ) \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )}-2 \ln \left (1+\frac {2 x}{{\mathrm e}^{2}-2}\right )\right )}{4 \left (\frac {{\mathrm e}^{2}}{2}-1\right )}\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 26, normalized size = 1.08 \begin {gather*} -\frac {4 \, x \log \left (2\right ) + \frac {e^{2} - 2}{2 \, x + e^{2} - 2}}{4 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 36, normalized size = 1.50 \begin {gather*} -\frac {4 \, {\left (2 \, x^{2} + x e^{2} - 2 \, x\right )} \log \left (2\right ) + e^{2} - 2}{4 \, {\left (2 \, x + e^{2} - 2\right )} \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 27, normalized size = 1.12 \begin {gather*} - x - \frac {-2 + e^{2}}{8 x \log {\left (2 \right )} - 8 \log {\left (2 \right )} + 4 e^{2} \log {\left (2 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 26, normalized size = 1.08 \begin {gather*} -\frac {4 \, x \log \left (2\right ) + \frac {e^{2} - 2}{2 \, x + e^{2} - 2}}{4 \, \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 30, normalized size = 1.25 \begin {gather*} -x-\frac {\frac {{\mathrm {e}}^2}{2}-1}{2\,{\mathrm {e}}^2\,\ln \left (2\right )-4\,\ln \left (2\right )+4\,x\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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