Optimal. Leaf size=30 \[ x+\frac {1}{4} \left (-x^2-\frac {e^x x^2}{\left (1+x-x^2\right )^2}\right ) \]
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Rubi [C] time = 2.45, antiderivative size = 774, normalized size of antiderivative = 25.80, number of steps used = 57, number of rules used = 6, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {1594, 6728, 6742, 2177, 2178, 2268} \begin {gather*} \frac {3}{200} \left (5+\sqrt {5}\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (2 x-\sqrt {5}-1\right )\right )-\frac {3}{100} \left (3+\sqrt {5}\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (2 x-\sqrt {5}-1\right )\right )+\frac {1}{40} \left (1+\sqrt {5}\right ) e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (2 x-\sqrt {5}-1\right )\right )-\frac {e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (2 x-\sqrt {5}-1\right )\right )}{20 \sqrt {5}}-\frac {1}{100} e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (2 x-\sqrt {5}-1\right )\right )+\frac {3}{200} \left (5-\sqrt {5}\right ) e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (2 x+\sqrt {5}-1\right )\right )-\frac {3}{100} \left (3-\sqrt {5}\right ) e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (2 x+\sqrt {5}-1\right )\right )+\frac {1}{40} \left (1-\sqrt {5}\right ) e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (2 x+\sqrt {5}-1\right )\right )+\frac {e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (2 x+\sqrt {5}-1\right )\right )}{20 \sqrt {5}}-\frac {1}{100} e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (2 x+\sqrt {5}-1\right )\right )-\frac {1}{4} (2-x)^2+\frac {3 \left (5-\sqrt {5}\right ) e^x}{100 \left (-2 x-\sqrt {5}+1\right )}-\frac {3 \left (3-\sqrt {5}\right ) e^x}{50 \left (-2 x-\sqrt {5}+1\right )}+\frac {\left (1-\sqrt {5}\right ) e^x}{20 \left (-2 x-\sqrt {5}+1\right )}-\frac {e^x}{10 \sqrt {5} \left (-2 x-\sqrt {5}+1\right )}-\frac {e^x}{50 \left (-2 x-\sqrt {5}+1\right )}+\frac {3 \left (5+\sqrt {5}\right ) e^x}{100 \left (-2 x+\sqrt {5}+1\right )}-\frac {3 \left (3+\sqrt {5}\right ) e^x}{50 \left (-2 x+\sqrt {5}+1\right )}+\frac {\left (1+\sqrt {5}\right ) e^x}{20 \left (-2 x+\sqrt {5}+1\right )}+\frac {e^x}{10 \sqrt {5} \left (-2 x+\sqrt {5}+1\right )}-\frac {e^x}{50 \left (-2 x+\sqrt {5}+1\right )}-\frac {3 \left (5-\sqrt {5}\right ) e^x}{50 \left (-2 x-\sqrt {5}+1\right )^2}+\frac {e^x}{5 \sqrt {5} \left (-2 x-\sqrt {5}+1\right )^2}-\frac {3 \left (5+\sqrt {5}\right ) e^x}{50 \left (-2 x+\sqrt {5}+1\right )^2}-\frac {e^x}{5 \sqrt {5} \left (-2 x+\sqrt {5}+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1594
Rule 2177
Rule 2178
Rule 2268
Rule 6728
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x-2 x^2+6 x^3-2 x^4+\frac {e^x x^2 \left (2+x+3 x^2-x^3\right )}{1+2 x-x^2-2 x^3+x^4}}{x \left (-4-4 x+4 x^2\right )} \, dx\\ &=\int \left (\frac {2-x}{2}-\frac {e^x x \left (-2-x-3 x^2+x^3\right )}{4 \left (-1-x+x^2\right )^3}\right ) \, dx\\ &=-\frac {1}{4} (2-x)^2-\frac {1}{4} \int \frac {e^x x \left (-2-x-3 x^2+x^3\right )}{\left (-1-x+x^2\right )^3} \, dx\\ &=-\frac {1}{4} (2-x)^2-\frac {1}{4} \int \left (-\frac {2 e^x (1+3 x)}{\left (-1-x+x^2\right )^3}+\frac {e^x (-1-x)}{\left (-1-x+x^2\right )^2}+\frac {e^x}{-1-x+x^2}\right ) \, dx\\ &=-\frac {1}{4} (2-x)^2-\frac {1}{4} \int \frac {e^x (-1-x)}{\left (-1-x+x^2\right )^2} \, dx-\frac {1}{4} \int \frac {e^x}{-1-x+x^2} \, dx+\frac {1}{2} \int \frac {e^x (1+3 x)}{\left (-1-x+x^2\right )^3} \, dx\\ &=-\frac {1}{4} (2-x)^2-\frac {1}{4} \int \left (-\frac {2 e^x}{\sqrt {5} \left (1+\sqrt {5}-2 x\right )}-\frac {2 e^x}{\sqrt {5} \left (-1+\sqrt {5}+2 x\right )}\right ) \, dx-\frac {1}{4} \int \left (-\frac {e^x}{\left (-1-x+x^2\right )^2}-\frac {e^x x}{\left (-1-x+x^2\right )^2}\right ) \, dx+\frac {1}{2} \int \left (\frac {e^x}{\left (-1-x+x^2\right )^3}+\frac {3 e^x x}{\left (-1-x+x^2\right )^3}\right ) \, dx\\ &=-\frac {1}{4} (2-x)^2+\frac {1}{4} \int \frac {e^x}{\left (-1-x+x^2\right )^2} \, dx+\frac {1}{4} \int \frac {e^x x}{\left (-1-x+x^2\right )^2} \, dx+\frac {1}{2} \int \frac {e^x}{\left (-1-x+x^2\right )^3} \, dx+\frac {3}{2} \int \frac {e^x x}{\left (-1-x+x^2\right )^3} \, dx+\frac {\int \frac {e^x}{1+\sqrt {5}-2 x} \, dx}{2 \sqrt {5}}+\frac {\int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx}{2 \sqrt {5}}\\ &=-\frac {1}{4} (2-x)^2-\frac {e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (-1-\sqrt {5}+2 x\right )\right )}{4 \sqrt {5}}+\frac {e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (-1+\sqrt {5}+2 x\right )\right )}{4 \sqrt {5}}+\frac {1}{4} \int \left (\frac {2 \left (1+\sqrt {5}\right ) e^x}{5 \left (1+\sqrt {5}-2 x\right )^2}+\frac {2 e^x}{5 \sqrt {5} \left (1+\sqrt {5}-2 x\right )}+\frac {2 \left (1-\sqrt {5}\right ) e^x}{5 \left (-1+\sqrt {5}+2 x\right )^2}+\frac {2 e^x}{5 \sqrt {5} \left (-1+\sqrt {5}+2 x\right )}\right ) \, dx+\frac {1}{4} \int \left (\frac {4 e^x}{5 \left (1+\sqrt {5}-2 x\right )^2}+\frac {4 e^x}{5 \sqrt {5} \left (1+\sqrt {5}-2 x\right )}+\frac {4 e^x}{5 \left (-1+\sqrt {5}+2 x\right )^2}+\frac {4 e^x}{5 \sqrt {5} \left (-1+\sqrt {5}+2 x\right )}\right ) \, dx+\frac {1}{2} \int \left (-\frac {8 e^x}{5 \sqrt {5} \left (1+\sqrt {5}-2 x\right )^3}-\frac {12 e^x}{25 \left (1+\sqrt {5}-2 x\right )^2}-\frac {12 e^x}{25 \sqrt {5} \left (1+\sqrt {5}-2 x\right )}-\frac {8 e^x}{5 \sqrt {5} \left (-1+\sqrt {5}+2 x\right )^3}-\frac {12 e^x}{25 \left (-1+\sqrt {5}+2 x\right )^2}-\frac {12 e^x}{25 \sqrt {5} \left (-1+\sqrt {5}+2 x\right )}\right ) \, dx+\frac {3}{2} \int \left (\frac {4 \left (-1-\sqrt {5}\right ) e^x}{5 \sqrt {5} \left (1+\sqrt {5}-2 x\right )^3}+\frac {2 \left (-3-\sqrt {5}\right ) e^x}{25 \left (1+\sqrt {5}-2 x\right )^2}-\frac {6 e^x}{25 \sqrt {5} \left (1+\sqrt {5}-2 x\right )}+\frac {4 \left (-1+\sqrt {5}\right ) e^x}{5 \sqrt {5} \left (-1+\sqrt {5}+2 x\right )^3}+\frac {2 \left (-3+\sqrt {5}\right ) e^x}{25 \left (-1+\sqrt {5}+2 x\right )^2}-\frac {6 e^x}{25 \sqrt {5} \left (-1+\sqrt {5}+2 x\right )}\right ) \, dx\\ &=-\frac {1}{4} (2-x)^2-\frac {e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (-1-\sqrt {5}+2 x\right )\right )}{4 \sqrt {5}}+\frac {e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (-1+\sqrt {5}+2 x\right )\right )}{4 \sqrt {5}}+\frac {1}{5} \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^2} \, dx+\frac {1}{5} \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^2} \, dx-\frac {6}{25} \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^2} \, dx-\frac {6}{25} \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^2} \, dx+\frac {\int \frac {e^x}{1+\sqrt {5}-2 x} \, dx}{10 \sqrt {5}}+\frac {\int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx}{10 \sqrt {5}}+\frac {\int \frac {e^x}{1+\sqrt {5}-2 x} \, dx}{5 \sqrt {5}}+\frac {\int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx}{5 \sqrt {5}}-\frac {6 \int \frac {e^x}{1+\sqrt {5}-2 x} \, dx}{25 \sqrt {5}}-\frac {6 \int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx}{25 \sqrt {5}}-\frac {9 \int \frac {e^x}{1+\sqrt {5}-2 x} \, dx}{25 \sqrt {5}}-\frac {9 \int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx}{25 \sqrt {5}}-\frac {4 \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^3} \, dx}{5 \sqrt {5}}-\frac {4 \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^3} \, dx}{5 \sqrt {5}}+\frac {1}{10} \left (1-\sqrt {5}\right ) \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^2} \, dx-\frac {1}{25} \left (3 \left (3-\sqrt {5}\right )\right ) \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^2} \, dx+\frac {1}{25} \left (6 \left (5-\sqrt {5}\right )\right ) \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^3} \, dx+\frac {1}{10} \left (1+\sqrt {5}\right ) \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^2} \, dx-\frac {1}{25} \left (3 \left (3+\sqrt {5}\right )\right ) \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^2} \, dx-\frac {1}{25} \left (6 \left (5+\sqrt {5}\right )\right ) \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^3} \, dx\\ &=\frac {e^x}{5 \sqrt {5} \left (1-\sqrt {5}-2 x\right )^2}-\frac {3 \left (5-\sqrt {5}\right ) e^x}{50 \left (1-\sqrt {5}-2 x\right )^2}-\frac {e^x}{50 \left (1-\sqrt {5}-2 x\right )}+\frac {\left (1-\sqrt {5}\right ) e^x}{20 \left (1-\sqrt {5}-2 x\right )}-\frac {3 \left (3-\sqrt {5}\right ) e^x}{50 \left (1-\sqrt {5}-2 x\right )}-\frac {e^x}{5 \sqrt {5} \left (1+\sqrt {5}-2 x\right )^2}-\frac {3 \left (5+\sqrt {5}\right ) e^x}{50 \left (1+\sqrt {5}-2 x\right )^2}-\frac {e^x}{50 \left (1+\sqrt {5}-2 x\right )}+\frac {\left (1+\sqrt {5}\right ) e^x}{20 \left (1+\sqrt {5}-2 x\right )}-\frac {3 \left (3+\sqrt {5}\right ) e^x}{50 \left (1+\sqrt {5}-2 x\right )}-\frac {1}{4} (2-x)^2-\frac {e^{\frac {1}{2} \left (1+\sqrt {5}\right )} \text {Ei}\left (\frac {1}{2} \left (-1-\sqrt {5}+2 x\right )\right )}{10 \sqrt {5}}+\frac {e^{\frac {1}{2}-\frac {\sqrt {5}}{2}} \text {Ei}\left (\frac {1}{2} \left (-1+\sqrt {5}+2 x\right )\right )}{10 \sqrt {5}}-\frac {1}{10} \int \frac {e^x}{1+\sqrt {5}-2 x} \, dx+\frac {1}{10} \int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx+\frac {3}{25} \int \frac {e^x}{1+\sqrt {5}-2 x} \, dx-\frac {3}{25} \int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx+\frac {\int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^2} \, dx}{5 \sqrt {5}}-\frac {\int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^2} \, dx}{5 \sqrt {5}}+\frac {1}{20} \left (-1-\sqrt {5}\right ) \int \frac {e^x}{1+\sqrt {5}-2 x} \, dx+\frac {1}{20} \left (1-\sqrt {5}\right ) \int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx-\frac {1}{50} \left (3 \left (3-\sqrt {5}\right )\right ) \int \frac {e^x}{-1+\sqrt {5}+2 x} \, dx+\frac {1}{50} \left (3 \left (5-\sqrt {5}\right )\right ) \int \frac {e^x}{\left (-1+\sqrt {5}+2 x\right )^2} \, dx+\frac {1}{50} \left (3 \left (3+\sqrt {5}\right )\right ) \int \frac {e^x}{1+\sqrt {5}-2 x} \, dx+\frac {1}{50} \left (3 \left (5+\sqrt {5}\right )\right ) \int \frac {e^x}{\left (1+\sqrt {5}-2 x\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.76, size = 46, normalized size = 1.53 \begin {gather*} \frac {1}{4} \left (4 x-x^2+e^x \left (\frac {-1-x}{\left (-1-x+x^2\right )^2}-\frac {1}{-1-x+x^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 37, normalized size = 1.23 \begin {gather*} -\frac {1}{4} \, x^{2} + x - \frac {1}{4} \, e^{\left (x + \log \left (\frac {x^{2}}{x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1}\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 55, normalized size = 1.83 \begin {gather*} -\frac {x^{6} - 6 \, x^{5} + 7 \, x^{4} + 6 \, x^{3} + x^{2} e^{x} - 7 \, x^{2} - 4 \, x}{4 \, {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 35, normalized size = 1.17
method | result | size |
risch | \(x -\frac {x^{2}}{4}-\frac {x^{2} {\mathrm e}^{x}}{4 \left (x^{4}-2 x^{3}-x^{2}+2 x +1\right )}\) | \(35\) |
norman | \(x -\frac {x^{2}}{4}-\frac {{\mathrm e}^{\ln \left (\frac {x^{2}}{x^{4}-2 x^{3}-x^{2}+2 x +1}\right )+x}}{4}\) | \(38\) |
default | \(\text {Expression too large to display}\) | \(39308\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 34, normalized size = 1.13 \begin {gather*} -\frac {1}{4} \, x^{2} - \frac {x^{2} e^{x}}{4 \, {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.12, size = 24, normalized size = 0.80 \begin {gather*} x-\frac {x^2}{4}-\frac {x^2\,{\mathrm {e}}^x}{4\,{\left (-x^2+x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 32, normalized size = 1.07 \begin {gather*} - \frac {x^{2}}{4} - \frac {x^{2} e^{x}}{4 x^{4} - 8 x^{3} - 4 x^{2} + 8 x + 4} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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