Optimal. Leaf size=24 \[ e^{-x} \left (-2+\frac {e-x}{3}+x+\log \left (32 x^2\right )\right ) \]
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Rubi [A]
time = 0.18, antiderivative size = 46, normalized size of antiderivative = 1.92, number of steps
used = 13, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {6, 12, 6874,
2230, 2225, 2209, 2207, 2634} \begin {gather*} e^{-x} \log \left (32 x^2\right )+\frac {2 e^{-x} x}{3}-\frac {1}{3} (8-e) e^{-x}+\frac {2 e^{-x}}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2207
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-x} \left (6+(8-e) x-2 x^2-3 x \log \left (32 x^2\right )\right )}{3 x} \, dx\\ &=\frac {1}{3} \int \frac {e^{-x} \left (6+(8-e) x-2 x^2-3 x \log \left (32 x^2\right )\right )}{x} \, dx\\ &=\frac {1}{3} \int \left (\frac {e^{-x} \left (6+(8-e) x-2 x^2\right )}{x}-3 e^{-x} \log \left (32 x^2\right )\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{-x} \left (6+(8-e) x-2 x^2\right )}{x} \, dx-\int e^{-x} \log \left (32 x^2\right ) \, dx\\ &=e^{-x} \log \left (32 x^2\right )+\frac {1}{3} \int \left ((8-e) e^{-x}+\frac {6 e^{-x}}{x}-2 e^{-x} x\right ) \, dx+\int -\frac {2 e^{-x}}{x} \, dx\\ &=e^{-x} \log \left (32 x^2\right )-\frac {2}{3} \int e^{-x} x \, dx+\frac {1}{3} (8-e) \int e^{-x} \, dx\\ &=-\frac {1}{3} (8-e) e^{-x}+\frac {2 e^{-x} x}{3}+e^{-x} \log \left (32 x^2\right )-\frac {2}{3} \int e^{-x} \, dx\\ &=\frac {2 e^{-x}}{3}-\frac {1}{3} (8-e) e^{-x}+\frac {2 e^{-x} x}{3}+e^{-x} \log \left (32 x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 23, normalized size = 0.96 \begin {gather*} \frac {1}{3} e^{-x} \left (-6+e+2 x+3 \log \left (32 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.80, size = 22, normalized size = 0.92
method | result | size |
norman | \(\left (\frac {{\mathrm e}}{3}+\frac {2 x}{3}-2+\ln \left (32 x^{2}\right )\right ) {\mathrm e}^{-x}\) | \(21\) |
default | \(\frac {\left (2 x -6+{\mathrm e}+3 \ln \left (32 x^{2}\right )\right ) {\mathrm e}^{-x}}{3}\) | \(22\) |
risch | \(2 \ln \left (x \right ) {\mathrm e}^{-x}+\frac {\left (-3 i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+6 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-12-3 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \,{\mathrm e}+30 \ln \left (2\right )+4 x \right ) {\mathrm e}^{-x}}{6}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 35, normalized size = 1.46 \begin {gather*} \frac {2}{3} \, {\left (x + 1\right )} e^{\left (-x\right )} + e^{\left (-x\right )} \log \left (32 \, x^{2}\right ) - \frac {8}{3} \, e^{\left (-x\right )} + \frac {1}{3} \, e^{\left (-x + 1\right )} \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 = 1.04 \begin {gather*} \frac {1}{3} \, {\left (2 \, x + e - 6\right )} e^{\left (-x\right )} + e^{\left (-x\right )} \log \left (32 \, x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 20, normalized size = 0.83 \begin {gather*} \frac {\left (2 x + 3 \log {\left (32 x^{2} \right )} - 6 + e\right ) e^{- x}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 33, normalized size = 1.38 \begin {gather*} \frac {2}{3} \, x e^{\left (-x\right )} + e^{\left (-x\right )} \log \left (32 \, x^{2}\right ) - 2 \, e^{\left (-x\right )} + \frac {1}{3} \, e^{\left (-x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.11, size = 21, normalized size = 0.88 \begin {gather*} \frac {{\mathrm {e}}^{-x}\,\left (2\,x+\mathrm {e}+3\,\ln \left (32\,x^2\right )-6\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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