Optimal. Leaf size=26 \[ \frac {e^{-8+2 x^2}}{x^2}+x \log \left (e^{2 x} x^2\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 39, normalized size of antiderivative = 1.50, number of steps used = 5, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {14, 2288, 2548, 9} \begin {gather*} x^2+\frac {e^{2 x^2-8}}{x^2}+x \log \left (e^{2 x} x^2\right )+2 x-(x+1)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 9
Rule 14
Rule 2288
Rule 2548
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2+2 x+\frac {2 e^{-8+2 x^2} \left (-1+2 x^2\right )}{x^3}+\log \left (e^{2 x} x^2\right )\right ) \, dx\\ &=2 x+x^2+2 \int \frac {e^{-8+2 x^2} \left (-1+2 x^2\right )}{x^3} \, dx+\int \log \left (e^{2 x} x^2\right ) \, dx\\ &=\frac {e^{-8+2 x^2}}{x^2}+2 x+x^2+x \log \left (e^{2 x} x^2\right )-\int 2 (1+x) \, dx\\ &=\frac {e^{-8+2 x^2}}{x^2}+2 x+x^2-(1+x)^2+x \log \left (e^{2 x} x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 27, normalized size = 1.04 \begin {gather*} 1+\frac {e^{-8+2 x^2}}{x^2}+x \log \left (e^{2 x} x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 23, normalized size = 0.88 \begin {gather*} 2 \, x^{2} + 2 \, x \log \relax (x) + e^{\left (2 \, x^{2} - 2 \, \log \relax (x) - 8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 30, normalized size = 1.15 \begin {gather*} 2 \, x^{2} \mathrm {sgn}\relax (x)^{2} + 2 \, x \log \left (x \mathrm {sgn}\relax (x)\right ) + \frac {e^{\left (2 \, x^{2} - 8\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 49, normalized size = 1.88
method | result | size |
default | \(-2 \,{\mathrm e}^{-8} \left (-\frac {{\mathrm e}^{2 x^{2}}}{2 x^{2}}-\expIntegralEi \left (1, -2 x^{2}\right )\right )-2 \,{\mathrm e}^{-8} \expIntegralEi \left (1, -2 x^{2}\right )+x \ln \left ({\mathrm e}^{2 x} x^{2}\right )\) | \(49\) |
risch | \(2 x \ln \left ({\mathrm e}^{x}\right )+2 x \ln \relax (x )-\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )}{2}+i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\frac {i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}-\frac {i x \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2}}{2}-\frac {i x \pi \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )}{2}+i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}-\frac {i x \pi \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}}{2}+\frac {i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{2 x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2}}{2}-\frac {i x \pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{3}}{2}+\frac {{\mathrm e}^{2 \left (x -2\right ) \left (2+x \right )}}{x^{2}}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.41, size = 33, normalized size = 1.27 \begin {gather*} 2 \, {\rm Ei}\left (2 \, x^{2}\right ) e^{\left (-8\right )} - 2 \, e^{\left (-8\right )} \Gamma \left (-1, -2 \, x^{2}\right ) + x \log \left (x^{2} e^{\left (2 \, x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.08, size = 24, normalized size = 0.92 \begin {gather*} x\,\ln \left (x^2\right )+2\,x^2+\frac {{\mathrm {e}}^{-8}\,{\mathrm {e}}^{2\,x^2}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 22, normalized size = 0.85 \begin {gather*} x \log {\left (x^{2} e^{2 x} \right )} + \frac {e^{2 x^{2} - 8}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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