Optimal. Leaf size=32 \[ -10+e^{\frac {2-\frac {1}{4} e^{-1-x+x^2}+x}{5 \left (2+x^2\right )^2}} \]
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Rubi [F]
time = 33.22, 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^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} \left (8-32 x-12 x^2+e^{-1-x+x^2} \left (2+x^2-2 x^3\right )\right )}{160+240 x^2+120 x^4+20 x^6} \, 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^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} \left (8-32 x-12 x^2+e^{-1-x+x^2} \left (2+x^2-2 x^3\right )\right )}{20 \left (2+x^2\right )^3} \, dx\\ &=\frac {1}{20} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} \left (8-32 x-12 x^2+e^{-1-x+x^2} \left (2+x^2-2 x^3\right )\right )}{\left (2+x^2\right )^3} \, dx\\ &=\frac {1}{20} \int \left (\frac {8 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3}-\frac {32 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3}-\frac {12 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x^2}{\left (2+x^2\right )^3}-\frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) \left (-2-x^2+2 x^3\right )}{\left (2+x^2\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{20} \int \frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) \left (-2-x^2+2 x^3\right )}{\left (2+x^2\right )^3} \, dx\right )+\frac {2}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3} \, dx-\frac {3}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x^2}{\left (2+x^2\right )^3} \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx\\ &=-\left (\frac {1}{20} \int \left (-\frac {4 \exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) x}{\left (2+x^2\right )^3}+\frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) (-1+2 x)}{\left (2+x^2\right )^2}\right ) \, dx\right )+\frac {2}{5} \int \left (-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}-x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}-x\right )^2}-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}+x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}+x\right )^2}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{32 \left (-2-x^2\right )}\right ) \, dx-\frac {3}{5} \int \left (-\frac {2 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3}+\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2}\right ) \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx\\ &=-\left (\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx\right )-\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {3}{80} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{-2-x^2} \, dx-\frac {1}{20} \int \frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) (-1+2 x)}{\left (2+x^2\right )^2} \, dx+\frac {1}{5} \int \frac {\exp \left (-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}\right ) x}{\left (2+x^2\right )^3} \, dx-\frac {3}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2} \, dx+\frac {6}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^3} \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}\\ &=-\left (\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx\right )-\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {3}{80} \int \left (-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx-\frac {1}{20} \int \left (-\frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2}+\frac {2 e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^2}\right ) \, dx+\frac {1}{5} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {3}{5} \int \left (-\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{4 \left (-2-x^2\right )}\right ) \, dx+\frac {6}{5} \int \left (-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}-x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}-x\right )^2}-\frac {i e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{16 \sqrt {2} \left (i \sqrt {2}+x\right )^3}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{64 \left (i \sqrt {2}+x\right )^2}-\frac {3 e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{32 \left (-2-x^2\right )}\right ) \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}\\ &=-\left (\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx\right )-\frac {3}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {1}{20} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (2+x^2\right )^2} \, dx-\frac {9}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {9}{160} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {3}{40} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {3}{40} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {1}{10} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^2} \, dx-\frac {9}{80} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{-2-x^2} \, dx+\frac {3}{20} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{-2-x^2} \, dx+\frac {1}{5} \int \frac {e^{-1-x+x^2+\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx-\frac {8}{5} \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}} x}{\left (2+x^2\right )^3} \, dx+\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{i \sqrt {2}-x} \, dx}{160 \sqrt {2}}+\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{i \sqrt {2}+x} \, dx}{160 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {i \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}-\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}-x\right )^3} \, dx}{40 \sqrt {2}}-\frac {(3 i) \int \frac {e^{\frac {8-e^{-1-x+x^2}+4 x}{80+80 x^2+20 x^4}}}{\left (i \sqrt {2}+x\right )^3} \, dx}{40 \sqrt {2}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.58, size = 30, normalized size = 0.94 \begin {gather*} e^{\frac {8-e^{-1-x+x^2}+4 x}{20 \left (2+x^2\right )^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 27, normalized size = 0.84
method | result | size |
risch | \({\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 \left (x^{2}+2\right )^{2}}}\) | \(27\) |
norman | \(\frac {x^{4} {\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 x^{4}+80 x^{2}+80}}+4 x^{2} {\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 x^{4}+80 x^{2}+80}}+4 \,{\mathrm e}^{\frac {-{\mathrm e}^{x^{2}-x -1}+4 x +8}{20 x^{4}+80 x^{2}+80}}}{\left (x^{2}+2\right )^{2}}\) | \(117\) |
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. 61 vs.
\(2 (28) = 56\).
time = 0.43, size = 61, normalized size = 1.91 \begin {gather*} e^{\left (\frac {x}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} - \frac {e^{\left (x^{2} - x\right )}}{20 \, {\left (x^{4} e + 4 \, x^{2} e + 4 \, e\right )}} + \frac {2}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 31, normalized size = 0.97 \begin {gather*} e^{\left (\frac {4 \, x - e^{\left (x^{2} - x - 1\right )} + 8}{20 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.30, size = 26, normalized size = 0.81 \begin {gather*} e^{\frac {4 x - e^{x^{2} - x - 1} + 8}{20 x^{4} + 80 x^{2} + 80}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 54, normalized size = 1.69 \begin {gather*} e^{\left (\frac {x}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} - \frac {e^{\left (x^{2} - x - 1\right )}}{20 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}} + \frac {2}{5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.98, size = 63, normalized size = 1.97 \begin {gather*} {\mathrm {e}}^{\frac {4\,x}{20\,x^4+80\,x^2+80}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-1}}{20\,x^4+80\,x^2+80}}\,{\mathrm {e}}^{\frac {8}{20\,x^4+80\,x^2+80}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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