Optimal. Leaf size=27 \[ x \left (-\frac {9}{256}+e^{-2+x}-x+\log \left (-x+\log \left (4 e^{2 x}\right )\right )\right ) \]
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Rubi [F]
time = 0.88, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {265 x+512 x^2+e^{-2+x} \left (-256 x-256 x^2\right )+\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )+\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{-256 x+256 \log \left (4 e^{2 x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-265 x-512 x^2-e^{-2+x} \left (-256 x-256 x^2\right )-\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )-\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{256 \left (x-\log \left (4 e^{2 x}\right )\right )} \, dx\\ &=\frac {1}{256} \int \frac {-265 x-512 x^2-e^{-2+x} \left (-256 x-256 x^2\right )-\left (-9-512 x+e^{-2+x} (256+256 x)\right ) \log \left (4 e^{2 x}\right )-\left (-256 x+256 \log \left (4 e^{2 x}\right )\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{x-\log \left (4 e^{2 x}\right )} \, dx\\ &=\frac {1}{256} \int \left (256 e^{-2+x} (1+x)+\frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )+256 x \log \left (-x+\log \left (4 e^{2 x}\right )\right )-256 \log \left (4 e^{2 x}\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{x-\log \left (4 e^{2 x}\right )}\right ) \, dx\\ &=\frac {1}{256} \int \frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )+256 x \log \left (-x+\log \left (4 e^{2 x}\right )\right )-256 \log \left (4 e^{2 x}\right ) \log \left (-x+\log \left (4 e^{2 x}\right )\right )}{x-\log \left (4 e^{2 x}\right )} \, dx+\int e^{-2+x} (1+x) \, dx\\ &=e^{-2+x} (1+x)+\frac {1}{256} \int \left (\frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )}{x-\log \left (4 e^{2 x}\right )}+256 \log \left (-x+\log \left (4 e^{2 x}\right )\right )\right ) \, dx-\int e^{-2+x} \, dx\\ &=-e^{-2+x}+e^{-2+x} (1+x)+\frac {1}{256} \int \frac {-265 x-512 x^2+9 \log \left (4 e^{2 x}\right )+512 x \log \left (4 e^{2 x}\right )}{x-\log \left (4 e^{2 x}\right )} \, dx+\int \log \left (-x+\log \left (4 e^{2 x}\right )\right ) \, dx\\ &=-e^{-2+x}+e^{-2+x} (1+x)+\frac {1}{256} \int \left (-9-512 x-\frac {256 x}{x-\log \left (4 e^{2 x}\right )}\right ) \, dx+\int \log \left (-x+\log \left (4 e^{2 x}\right )\right ) \, dx\\ &=-e^{-2+x}-\frac {9 x}{256}-x^2+e^{-2+x} (1+x)-\int \frac {x}{x-\log \left (4 e^{2 x}\right )} \, dx+\int \log \left (-x+\log \left (4 e^{2 x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.14, size = 32, normalized size = 1.19 \begin {gather*} -\frac {1}{256} x \left (9-256 e^{-2+x}+256 x-256 \log \left (-x+\log \left (4 e^{2 x}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.77, size = 556, normalized size = 20.59
method | result | size |
risch | \(x \ln \left (2 \ln \left (2\right )+2 \ln \left ({\mathrm e}^{x}\right )-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \left (-\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+\mathrm {csgn}\left (i {\mathrm e}^{x}\right )\right )^{2}}{2}-x \right )-x^{2}-\frac {9 x}{256}+x \,{\mathrm e}^{x -2}\) | \(63\) |
default | \(-x^{2}+\frac {503 x}{256}-\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )-2 \ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+\ln \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-x \right )-\ln \left (4 \,{\mathrm e}^{2 x}\right )-6 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+2 \,{\mathrm e}^{x -2}-5 \left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+{\mathrm e}^{x -2} \left (x -2\right )-2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{x -2}-\left (\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )^{2}+4 \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) \left (\ln \left ({\mathrm e}^{x}\right )-x \right )+4 \left (\ln \left ({\mathrm e}^{x}\right )-x \right )^{2}+4 \ln \left (4 \,{\mathrm e}^{2 x}\right )-8 x +4\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-\left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )-6 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )+2 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )-2 \,{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) x \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )+{\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) \left (\ln \left (4 \,{\mathrm e}^{2 x}\right )-2 \ln \left ({\mathrm e}^{x}\right )\right )^{2}+2 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left ({\mathrm e}^{x -2}-\left (-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right ) {\mathrm e}^{-2+2 x -\ln \left (4 \,{\mathrm e}^{2 x}\right )} \expIntegral \left (1, x -\ln \left (4 \,{\mathrm e}^{2 x}\right )\right )\right )\) | \(556\) |
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. 92 vs.
\(2 (28) = 56\).
time = 0.53, size = 92, normalized size = 3.41 \begin {gather*} 8 \, \log \left (2\right )^{2} \log \left (x + 2 \, \log \left (2\right )\right ) + x^{2} - \frac {1}{128} \, {\left (256 \, x^{2} e^{2} - x {\left (512 \, \log \left (2\right ) - 137\right )} e^{2} - 128 \, x e^{x} + {\left ({\left (1024 \, \log \left (2\right )^{2} - 265 \, \log \left (2\right )\right )} e^{2} - 128 \, x e^{2}\right )} \log \left (x + 2 \, \log \left (2\right )\right )\right )} e^{\left (-2\right )} - 4 \, x \log \left (2\right ) - \frac {265}{128} \, \log \left (2\right ) \log \left (x + 2 \, \log \left (2\right )\right ) + \frac {265}{256} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 24, normalized size = 0.89 \begin {gather*} -x^{2} + x e^{\left (x - 2\right )} + x \log \left (x + 2 \, \log \left (2\right )\right ) - \frac {9}{256} \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 34, normalized size = 1.26 \begin {gather*} -\frac {1}{256} \, {\left (256 \, x^{2} e^{2} - 256 \, x e^{2} \log \left (x + 2 \, \log \left (2\right )\right ) + 9 \, x e^{2} - 256 \, x e^{x}\right )} e^{\left (-2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.09, size = 24, normalized size = 0.89 \begin {gather*} x\,{\mathrm {e}}^{x-2}-\frac {9\,x}{256}-x^2+x\,\ln \left (x+2\,\ln \left (2\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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