Optimal. Leaf size=27 \[ e^{-x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \]
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Rubi [F]
time = 7.66, 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 {\exp \left (\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (\frac {1}{13} (65+5 x \log (2))\right )}\right ) \left (-e^{2 x} \log (2)+e^{2 x} (26+2 x \log (2)) \log \left (\frac {1}{13} (65+5 x \log (2))\right )+\left (-13+26 x+\left (-x+2 x^2\right ) \log (2)\right ) \log ^2\left (\frac {1}{13} (65+5 x \log (2))\right )\right )}{(13+x \log (2)) \log ^2\left (\frac {1}{13} (65+5 x \log (2))\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 {\exp \left (\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) \left (-e^{2 x} \log (2)+e^{2 x} (26+2 x \log (2)) \log \left (\frac {1}{13} (65+5 x \log (2))\right )+\left (-13+26 x+\left (-x+2 x^2\right ) \log (2)\right ) \log ^2\left (\frac {1}{13} (65+5 x \log (2))\right )\right )}{(13+x \log (2)) \log ^2\left (5+\frac {5}{13} x \log (2)\right )} \, dx\\ &=\int \left (-\exp \left (\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right )+2 \exp \left (\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) x+\frac {\exp \left (2 x+\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) \left (-\log (2)+26 \log \left (5+\frac {5}{13} x \log (2)\right )+2 x \log (2) \log \left (5+\frac {5}{13} x \log (2)\right )\right )}{(13+x \log (2)) \log ^2\left (5+\frac {5}{13} x \log (2)\right )}\right ) \, dx\\ &=2 \int \exp \left (\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) x \, dx-\int \exp \left (\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) \, dx+\int \frac {\exp \left (2 x+\frac {e^{2 x}+\left (-x+x^2\right ) \log \left (\frac {1}{13} (65+5 x \log (2))\right )}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) \left (-\log (2)+26 \log \left (5+\frac {5}{13} x \log (2)\right )+2 x \log (2) \log \left (5+\frac {5}{13} x \log (2)\right )\right )}{(13+x \log (2)) \log ^2\left (5+\frac {5}{13} x \log (2)\right )} \, dx\\ &=2 \int e^{(-1+x) x+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} x \, dx-\int e^{(-1+x) x+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \, dx+\int \frac {e^{x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \left (-\log (2)+2 (13+x \log (2)) \log \left (5+\frac {5}{13} x \log (2)\right )\right )}{(13+x \log (2)) \log ^2\left (5+\frac {5}{13} x \log (2)\right )} \, dx\\ &=2 \int e^{(-1+x) x+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} x \, dx-\int e^{(-1+x) x+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \, dx+\int \left (-\frac {e^{x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \log (2)}{(13+x \log (2)) \log ^2\left (5+\frac {5}{13} x \log (2)\right )}+\frac {2 e^{x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}}}{\log \left (5+\frac {5}{13} x \log (2)\right )}\right ) \, dx\\ &=2 \int e^{(-1+x) x+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} x \, dx+2 \int \frac {e^{x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}}}{\log \left (5+\frac {5}{13} x \log (2)\right )} \, dx-\log (2) \int \frac {e^{x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}}}{(13+x \log (2)) \log ^2\left (5+\frac {5}{13} x \log (2)\right )} \, dx-\int e^{(-1+x) x+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.20, size = 27, normalized size = 1.00 \begin {gather*} e^{-x+x^2+\frac {e^{2 x}}{\log \left (5+\frac {5}{13} x \log (2)\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.84, size = 41, normalized size = 1.52
method | result | size |
risch | \({\mathrm e}^{\frac {\ln \left (\frac {5 x \ln \left (2\right )}{13}+5\right ) x^{2}-\ln \left (\frac {5 x \ln \left (2\right )}{13}+5\right ) x +{\mathrm e}^{2 x}}{\ln \left (\frac {5 x \ln \left (2\right )}{13}+5\right )}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 33, normalized size = 1.22 \begin {gather*} e^{\left (\frac {{\left (x^{2} - x\right )} \log \left (\frac {5}{13} \, x \log \left (2\right ) + 5\right ) + e^{\left (2 \, x\right )}}{\log \left (\frac {5}{13} \, x \log \left (2\right ) + 5\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.48, size = 34, normalized size = 1.26 \begin {gather*} e^{\frac {\left (x^{2} - x\right ) \log {\left (\frac {5 x \log {\left (2 \right )}}{13} + 5 \right )} + e^{2 x}}{\log {\left (\frac {5 x \log {\left (2 \right )}}{13} + 5 \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.62, size = 23, normalized size = 0.85 \begin {gather*} e^{\left (x^{2} - x + \frac {e^{\left (2 \, x\right )}}{\log \left (\frac {5}{13} \, x \log \left (2\right ) + 5\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.19, size = 25, normalized size = 0.93 \begin {gather*} {\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}}{\ln \left (\frac {5\,x\,\ln \left (2\right )}{13}+5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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