Optimal. Leaf size=24 \[ e^{1+e^{x^2} \left (x+\log \left (\frac {x^2}{256+x^2}\right )\right )} \]
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Rubi [F]
time = 11.52, 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 {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{1+e^{x^2} x+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (e^{x^2} \left (512+256 x+513 x^3+2 x^5\right )+e^{x^2} \left (512 x^2+2 x^4\right ) \log \left (\frac {x^2}{256+x^2}\right )\right )}{x \left (256+x^2\right )} \, dx\\ &=\int \left (\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 x}-\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{256 \left (256+x^2\right )}\right ) \, dx\\ &=\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{x} \, dx-\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5+512 x^2 \log \left (\frac {x^2}{256+x^2}\right )+2 x^4 \log \left (\frac {x^2}{256+x^2}\right )\right )}{256+x^2} \, dx\\ &=-\left (\frac {1}{256} \int \left (\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5\right )}{256+x^2}+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right )\right ) \, dx\right )+\frac {1}{256} \int \left (\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5\right )}{x}+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (256+x^2\right ) \log \left (\frac {x^2}{256+x^2}\right )\right ) \, dx\\ &=\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \left (512+256 x+513 x^3+2 x^5\right )}{x} \, dx-\frac {1}{256} \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (512+256 x+513 x^3+2 x^5\right )}{256+x^2} \, dx-\frac {1}{128} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {1}{128} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \left (256+x^2\right ) \log \left (\frac {x^2}{256+x^2}\right ) \, dx\\ &=\frac {1}{256} \int \left (256 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}+\frac {512 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x}+513 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^4\right ) \, dx-\frac {1}{256} \int \left (e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2+2 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^4+\frac {512 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x}{256+x^2}\right ) \, dx-\frac {1}{128} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {1}{128} \int \left (256 e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right )+e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^3 \log \left (\frac {x^2}{256+x^2}\right )\right ) \, dx\\ &=-\left (\frac {1}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx\right )+2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x} \, dx-2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x}{256+x^2} \, dx+2 \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {513}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx+\int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \, dx\\ &=-\left (\frac {1}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx\right )+2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x} \, dx-2 \int \left (-\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{2 (16 i-x)}+\frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{2 (16 i+x)}\right ) \, dx+2 \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {513}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx+\int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \, dx\\ &=-\left (\frac {1}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx\right )+2 \int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{x} \, dx+2 \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x \log \left (\frac {x^2}{256+x^2}\right ) \, dx+\frac {513}{256} \int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} x^2 \, dx+\int e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )} \, dx+\int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{16 i-x} \, dx-\int \frac {e^{1+e^{x^2} x+x^2+e^{x^2} \log \left (\frac {x^2}{256+x^2}\right )}}{16 i+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.82, size = 29, normalized size = 1.21 \begin {gather*} e^{1+e^{x^2} x} \left (\frac {x^2}{256+x^2}\right )^{e^{x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.97, size = 222, normalized size = 9.25
method | result | size |
risch | \(x^{2 \,{\mathrm e}^{x^{2}}} \left (x^{2}+256\right )^{-{\mathrm e}^{x^{2}}} {\mathrm e}^{1-\frac {i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (i x^{2}\right )^{3}}{2}+i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-\frac {i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}}{2}+\frac {i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{2}}{2}-\frac {i {\mathrm e}^{x^{2}} \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right ) \mathrm {csgn}\left (\frac {i}{x^{2}+256}\right )}{2}-\frac {i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{3}}{2}+\frac {i {\mathrm e}^{x^{2}} \pi \mathrm {csgn}\left (\frac {i x^{2}}{x^{2}+256}\right )^{2} \mathrm {csgn}\left (\frac {i}{x^{2}+256}\right )}{2}+{\mathrm e}^{x^{2}} x}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.40, size = 29, normalized size = 1.21 \begin {gather*} e^{\left (x e^{\left (x^{2}\right )} - e^{\left (x^{2}\right )} \log \left (x^{2} + 256\right ) + 2 \, e^{\left (x^{2}\right )} \log \left (x\right ) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 26, normalized size = 1.08 \begin {gather*} e^{\left (x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )} \log \left (\frac {x^{2}}{x^{2} + 256}\right ) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.41, size = 24, normalized size = 1.00 \begin {gather*} e^{x e^{x^{2}} + e^{x^{2}} \log {\left (\frac {x^{2}}{x^{2} + 256} \right )} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 26, normalized size = 1.08 \begin {gather*} e^{\left (x e^{\left (x^{2}\right )} + e^{\left (x^{2}\right )} \log \left (\frac {x^{2}}{x^{2} + 256}\right ) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.47, size = 26, normalized size = 1.08 \begin {gather*} \mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{x^2}}\,{\left (\frac {x^2}{x^2+256}\right )}^{{\mathrm {e}}^{x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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