Optimal. Leaf size=22 \[ \left (-1-x+\frac {2 (-14+x) \left (-x+x^2\right )}{e^2}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(76\) vs. \(2(22)=44\).
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 3.45, number of steps
used = 3, number of rules used = 1, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {12}
\begin {gather*} \frac {4 x^6}{e^4}-\frac {120 x^5}{e^4}-\frac {4 x^4}{e^2}+\frac {1012 x^4}{e^4}+\frac {56 x^3}{e^2}-\frac {1680 x^3}{e^4}+\frac {4 x^2}{e^2}+\frac {784 x^2}{e^4}-\frac {56 x}{e^2}+(x+1)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \left (1568 x-5040 x^2+4048 x^3-600 x^4+24 x^5+e^4 (2+2 x)+e^2 \left (-56+8 x+168 x^2-16 x^3\right )\right ) \, dx}{e^4}\\ &=\frac {784 x^2}{e^4}-\frac {1680 x^3}{e^4}+\frac {1012 x^4}{e^4}-\frac {120 x^5}{e^4}+\frac {4 x^6}{e^4}+(1+x)^2+\frac {\int \left (-56+8 x+168 x^2-16 x^3\right ) \, dx}{e^2}\\ &=-\frac {56 x}{e^2}+\frac {784 x^2}{e^4}+\frac {4 x^2}{e^2}-\frac {1680 x^3}{e^4}+\frac {56 x^3}{e^2}+\frac {1012 x^4}{e^4}-\frac {4 x^4}{e^2}-\frac {120 x^5}{e^4}+\frac {4 x^6}{e^4}+(1+x)^2\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(22)=44\).
time = 0.01, size = 81, normalized size = 3.68 \begin {gather*} 2 \left (x-\frac {28 x}{e^2}+\frac {x^2}{2}+\frac {392 x^2}{e^4}+\frac {2 x^2}{e^2}-\frac {840 x^3}{e^4}+\frac {28 x^3}{e^2}+\frac {506 x^4}{e^4}-\frac {2 x^4}{e^2}-\frac {60 x^5}{e^4}+\frac {2 x^6}{e^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 28, normalized size = 1.27
method | result | size |
default | \({\mathrm e}^{-4} \left (-2 x^{3}+{\mathrm e}^{2} x +30 x^{2}+{\mathrm e}^{2}-28 x \right )^{2}\) | \(28\) |
gosper | \(x \left (4 x^{5}-4 x^{3} {\mathrm e}^{2}-120 x^{4}+x \,{\mathrm e}^{4}+56 x^{2} {\mathrm e}^{2}+1012 x^{3}+2 \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2} x -1680 x^{2}-56 \,{\mathrm e}^{2}+784 x \right ) {\mathrm e}^{-4}\) | \(66\) |
norman | \(\left (\left (2 \,{\mathrm e}^{2}-56\right ) x +\left ({\mathrm e}^{4}+4 \,{\mathrm e}^{2}+784\right ) {\mathrm e}^{-2} x^{2}-120 \,{\mathrm e}^{-2} x^{5}+4 \,{\mathrm e}^{-2} x^{6}-4 \left ({\mathrm e}^{2}-253\right ) {\mathrm e}^{-2} x^{4}+56 \left ({\mathrm e}^{2}-30\right ) {\mathrm e}^{-2} x^{3}\right ) {\mathrm e}^{-2}\) | \(77\) |
risch | \(4 \,{\mathrm e}^{-4} x^{6}-4 \,{\mathrm e}^{2} {\mathrm e}^{-4} x^{4}-120 \,{\mathrm e}^{-4} x^{5}+{\mathrm e}^{4} {\mathrm e}^{-4} x^{2}+56 \,{\mathrm e}^{2} {\mathrm e}^{-4} x^{3}+1012 \,{\mathrm e}^{-4} x^{4}+2 \,{\mathrm e}^{4} {\mathrm e}^{-4} x +4 \,{\mathrm e}^{2} {\mathrm e}^{-4} x^{2}-1680 \,{\mathrm e}^{-4} x^{3}+{\mathrm e}^{4} {\mathrm e}^{-4}-56 \,{\mathrm e}^{2} {\mathrm e}^{-4} x +784 x^{2} {\mathrm e}^{-4}\) | \(97\) |
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. 60 vs.
\(2 (21) = 42\).
time = 0.27, size = 60, normalized size = 2.73 \begin {gather*} {\left (4 \, x^{6} - 120 \, x^{5} + 1012 \, x^{4} - 1680 \, x^{3} + 784 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{4} - 4 \, {\left (x^{4} - 14 \, x^{3} - x^{2} + 14 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (21) = 42\).
time = 0.30, size = 60, normalized size = 2.73 \begin {gather*} {\left (4 \, x^{6} - 120 \, x^{5} + 1012 \, x^{4} - 1680 \, x^{3} + 784 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{4} - 4 \, {\left (x^{4} - 14 \, x^{3} - x^{2} + 14 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (19) = 38\).
time = 0.02, size = 71, normalized size = 3.23 \begin {gather*} \frac {4 x^{6}}{e^{4}} - \frac {120 x^{5}}{e^{4}} + \frac {x^{4} \cdot \left (1012 - 4 e^{2}\right )}{e^{4}} + \frac {x^{3} \left (-1680 + 56 e^{2}\right )}{e^{4}} + \frac {x^{2} \cdot \left (4 e^{2} + e^{4} + 784\right )}{e^{4}} + \frac {x \left (-56 + 2 e^{2}\right )}{e^{2}} \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. 60 vs.
\(2 (21) = 42\).
time = 0.41, size = 60, normalized size = 2.73 \begin {gather*} {\left (4 \, x^{6} - 120 \, x^{5} + 1012 \, x^{4} - 1680 \, x^{3} + 784 \, x^{2} + {\left (x^{2} + 2 \, x\right )} e^{4} - 4 \, {\left (x^{4} - 14 \, x^{3} - x^{2} + 14 \, x\right )} e^{2}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.33, size = 67, normalized size = 3.05 \begin {gather*} 4\,{\mathrm {e}}^{-4}\,x^6-120\,{\mathrm {e}}^{-4}\,x^5-\frac {{\mathrm {e}}^{-4}\,\left (16\,{\mathrm {e}}^2-4048\right )\,x^4}{4}+\frac {{\mathrm {e}}^{-4}\,\left (168\,{\mathrm {e}}^2-5040\right )\,x^3}{3}+\frac {{\mathrm {e}}^{-4}\,\left (8\,{\mathrm {e}}^2+2\,{\mathrm {e}}^4+1568\right )\,x^2}{2}+2\,{\mathrm {e}}^{-2}\,\left ({\mathrm {e}}^2-28\right )\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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