Optimal. Leaf size=32 \[ 5-x-\frac {25}{81} e^{-4+2 x} (5-x)^2 \left (4-x^2\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(97\) vs. \(2(32)=64\).
time = 0.95, antiderivative size = 97, normalized size of antiderivative = 3.03, number of steps
used = 33, number of rules used = 5, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {21, 6820,
2227, 2225, 2207} \begin {gather*} -\frac {25}{81} e^{2 x-4} x^6+\frac {250}{81} e^{2 x-4} x^5-\frac {425}{81} e^{2 x-4} x^4-\frac {2000}{81} e^{2 x-4} x^3+\frac {4600}{81} e^{2 x-4} x^2+\frac {4000}{81} e^{2 x-4} x-x-\frac {10000}{81} e^{2 x-4} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 2207
Rule 2225
Rule 2227
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=-\left (\frac {1}{81} \int e^{-4+2 x} \left (4-x^2\right ) \left (4000-4300 x+200 x^2+350 x^3-50 x^4+\frac {e^{4-2 x} \left (324-81 x^2\right )}{\left (4-x^2\right )^2}\right ) \, dx\right )\\ &=-\left (\frac {1}{81} \int \left (81+50 e^{-4+2 x} \left (320-344 x-64 x^2+114 x^3-8 x^4-7 x^5+x^6\right )\right ) \, dx\right )\\ &=-x-\frac {50}{81} \int e^{-4+2 x} \left (320-344 x-64 x^2+114 x^3-8 x^4-7 x^5+x^6\right ) \, dx\\ &=-x-\frac {50}{81} \int \left (320 e^{-4+2 x}-344 e^{-4+2 x} x-64 e^{-4+2 x} x^2+114 e^{-4+2 x} x^3-8 e^{-4+2 x} x^4-7 e^{-4+2 x} x^5+e^{-4+2 x} x^6\right ) \, dx\\ &=-x-\frac {50}{81} \int e^{-4+2 x} x^6 \, dx+\frac {350}{81} \int e^{-4+2 x} x^5 \, dx+\frac {400}{81} \int e^{-4+2 x} x^4 \, dx+\frac {3200}{81} \int e^{-4+2 x} x^2 \, dx-\frac {1900}{27} \int e^{-4+2 x} x^3 \, dx-\frac {16000}{81} \int e^{-4+2 x} \, dx+\frac {17200}{81} \int e^{-4+2 x} x \, dx\\ &=-\frac {8000}{81} e^{-4+2 x}-x+\frac {8600}{81} e^{-4+2 x} x+\frac {1600}{81} e^{-4+2 x} x^2-\frac {950}{27} e^{-4+2 x} x^3+\frac {200}{81} e^{-4+2 x} x^4+\frac {175}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {50}{27} \int e^{-4+2 x} x^5 \, dx-\frac {800}{81} \int e^{-4+2 x} x^3 \, dx-\frac {875}{81} \int e^{-4+2 x} x^4 \, dx-\frac {3200}{81} \int e^{-4+2 x} x \, dx+\frac {950}{9} \int e^{-4+2 x} x^2 \, dx-\frac {8600}{81} \int e^{-4+2 x} \, dx\\ &=-\frac {4100}{27} e^{-4+2 x}-x+\frac {7000}{81} e^{-4+2 x} x+\frac {5875}{81} e^{-4+2 x} x^2-\frac {3250}{81} e^{-4+2 x} x^3-\frac {475}{162} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6-\frac {125}{27} \int e^{-4+2 x} x^4 \, dx+\frac {400}{27} \int e^{-4+2 x} x^2 \, dx+\frac {1600}{81} \int e^{-4+2 x} \, dx+\frac {1750}{81} \int e^{-4+2 x} x^3 \, dx-\frac {950}{9} \int e^{-4+2 x} x \, dx\\ &=-\frac {11500}{81} e^{-4+2 x}-x+\frac {2725}{81} e^{-4+2 x} x+\frac {6475}{81} e^{-4+2 x} x^2-\frac {2375}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {250}{27} \int e^{-4+2 x} x^3 \, dx-\frac {400}{27} \int e^{-4+2 x} x \, dx-\frac {875}{27} \int e^{-4+2 x} x^2 \, dx+\frac {475}{9} \int e^{-4+2 x} \, dx\\ &=-\frac {18725}{162} e^{-4+2 x}-x+\frac {2125}{81} e^{-4+2 x} x+\frac {10325}{162} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {200}{27} \int e^{-4+2 x} \, dx-\frac {125}{9} \int e^{-4+2 x} x^2 \, dx+\frac {875}{27} \int e^{-4+2 x} x \, dx\\ &=-\frac {18125}{162} e^{-4+2 x}-x+\frac {6875}{162} e^{-4+2 x} x+\frac {4600}{81} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6+\frac {125}{9} \int e^{-4+2 x} x \, dx-\frac {875}{54} \int e^{-4+2 x} \, dx\\ &=-\frac {38875}{324} e^{-4+2 x}-x+\frac {4000}{81} e^{-4+2 x} x+\frac {4600}{81} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6-\frac {125}{18} \int e^{-4+2 x} \, dx\\ &=-\frac {10000}{81} e^{-4+2 x}-x+\frac {4000}{81} e^{-4+2 x} x+\frac {4600}{81} e^{-4+2 x} x^2-\frac {2000}{81} e^{-4+2 x} x^3-\frac {425}{81} e^{-4+2 x} x^4+\frac {250}{81} e^{-4+2 x} x^5-\frac {25}{81} e^{-4+2 x} x^6\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.43, size = 36, normalized size = 1.12 \begin {gather*} -\frac {81 e^4 x+25 e^{2 x} \left (20-4 x-5 x^2+x^3\right )^2}{81 e^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs.
\(2(32)=64\).
time = 1.16, size = 80, normalized size = 2.50
method | result | size |
risch | \(-x +\left (-\frac {10000}{81}+\frac {4000}{81} x +\frac {4600}{81} x^{2}-\frac {2000}{81} x^{3}-\frac {425}{81} x^{4}+\frac {250}{81} x^{5}-\frac {25}{81} x^{6}\right ) {\mathrm e}^{2 x -4}\) | \(42\) |
default | \(-x -\frac {100 \,{\mathrm e}^{2 x -4} \left (2 x -4\right )^{2}}{9}+\frac {25 \,{\mathrm e}^{2 x -4} \left (2 x -4\right )^{3}}{27}+\frac {575 \,{\mathrm e}^{2 x -4} \left (2 x -4\right )^{4}}{1296}-\frac {25 \left (2 x -4\right )^{5} {\mathrm e}^{2 x -4}}{1296}-\frac {25 \left (2 x -4\right )^{6} {\mathrm e}^{2 x -4}}{5184}\) | \(80\) |
meijerg | \(-\frac {250 \,2^{2+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} x \hypergeom \left (\left [\frac {1}{2}, 1-2 \,{\mathrm e}^{-4}\right ], \left [\frac {3}{2}\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{81}-\frac {2 \,2^{-4 \,{\mathrm e}^{4}-1+4 \,{\mathrm e}^{-4}} {\mathrm e}^{4} \left (-x^{2}+4\right )^{2 \,{\mathrm e}^{4}+2-2 \,{\mathrm e}^{-4}} x \hypergeom \left (\left [\frac {1}{2}, -2 \,{\mathrm e}^{-4}+2 \,{\mathrm e}^{4}+1\right ], \left [\frac {3}{2}\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{\left (x^{2}-4\right )^{2}}+\frac {2 \,2^{-3-4 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{-4}} {\mathrm e}^{4} \left (-x^{2}+4\right )^{2 \,{\mathrm e}^{4}+2-2 \,{\mathrm e}^{-4}} x^{3} \hypergeom \left (\left [\frac {3}{2}, -2 \,{\mathrm e}^{-4}+2 \,{\mathrm e}^{4}+1\right ], \left [\frac {5}{2}\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{3 \left (x^{2}-4\right )^{2}}-\frac {50 \,{\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} 2^{4 \,{\mathrm e}^{-4}} x^{3} \hypergeom \left (\left [\frac {3}{2}, 1-2 \,{\mathrm e}^{-4}\right ], \left [\frac {5}{2}\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{243}+\frac {10 \,2^{-2+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} x^{5} \hypergeom \left (\left [\frac {5}{2}, 1-2 \,{\mathrm e}^{-4}\right ], \left [\frac {7}{2}\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{81}-\frac {350 \,2^{-4+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} x^{4} \hypergeom \left (\left [2, 1-2 \,{\mathrm e}^{-4}\right ], \left [3\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{81}+\frac {2150 \,2^{-2+4 \,{\mathrm e}^{-4}} {\mathrm e}^{2 x} \left (-x^{2}+4\right )^{2-2 \,{\mathrm e}^{-4}} x^{2} \hypergeom \left (\left [1, 1-2 \,{\mathrm e}^{-4}\right ], \left [2\right ], \frac {x^{2}}{4}\right ) {\mathrm e}^{-4}}{81}\) | \(373\) |
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. 130 vs.
\(2 (28) = 56\).
time = 0.41, size = 130, normalized size = 4.06 \begin {gather*} -\frac {25}{81} \, {\left (x^{6} - 10 \, x^{5} + 17 \, x^{4} + 80 \, x^{3} - 184 \, x^{2} - 160 \, x + 400\right )} e^{\left (2 \, x - 4\right )} - x + \frac {9 \, x^{3} - 28 \, x}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} - \frac {3 \, {\left (5 \, x^{3} - 12 \, x\right )}}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} + \frac {3 \, x^{3} - 20 \, x}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} + \frac {3 \, {\left (x^{3} + 4 \, x\right )}}{2 \, {\left (x^{4} - 8 \, x^{2} + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 30, normalized size = 0.94 \begin {gather*} -\frac {25}{81} \, {\left (x^{2} - 10 \, x + 25\right )} e^{\left (2 \, x + 2 \, \log \left (-x^{2} + 4\right ) - 4\right )} - x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 39, normalized size = 1.22 \begin {gather*} - x + \frac {\left (- 25 x^{6} + 250 x^{5} - 425 x^{4} - 2000 x^{3} + 4600 x^{2} + 4000 x - 10000\right ) e^{2 x - 4}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (28) = 56\).
time = 0.41, size = 68, normalized size = 2.12 \begin {gather*} -\frac {1}{81} \, {\left (25 \, x^{6} e^{\left (2 \, x\right )} - 250 \, x^{5} e^{\left (2 \, x\right )} + 425 \, x^{4} e^{\left (2 \, x\right )} + 2000 \, x^{3} e^{\left (2 \, x\right )} - 4600 \, x^{2} e^{\left (2 \, x\right )} + 81 \, x e^{4} - 4000 \, x e^{\left (2 \, x\right )} + 10000 \, e^{\left (2 \, x\right )}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.26, size = 76, normalized size = 2.38 \begin {gather*} \frac {4000\,x\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {10000\,{\mathrm {e}}^{2\,x-4}}{81}-x+\frac {4600\,x^2\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {2000\,x^3\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {425\,x^4\,{\mathrm {e}}^{2\,x-4}}{81}+\frac {250\,x^5\,{\mathrm {e}}^{2\,x-4}}{81}-\frac {25\,x^6\,{\mathrm {e}}^{2\,x-4}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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