Optimal. Leaf size=16 \[ \left (2-243 e^{-\frac {2+x}{x}}\right )^8 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(16)=32\).
time = 0.12, antiderivative size = 89, normalized size of antiderivative = 5.56, number of steps
used = 10, number of rules used = 2, integrand size = 156, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {14, 2240}
\begin {gather*} 12157665459056928801 e^{-\frac {16}{x}-8}-800504721583995312 e^{-\frac {14}{x}-7}+23059806794600688 e^{-\frac {12}{x}-6}-379585297030464 e^{-\frac {10}{x}-5}+3905198529120 e^{-\frac {8}{x}-4}-25713241344 e^{-\frac {6}{x}-3}+105815808 e^{-\frac {4}{x}-2}-248832 e^{-\frac {2}{x}-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2240
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {194522647344910860816 e^{-8-\frac {16}{x}}}{x^2}-\frac {11207066102175934368 e^{-7-\frac {14}{x}}}{x^2}+\frac {276717681535208256 e^{-6-\frac {12}{x}}}{x^2}-\frac {3795852970304640 e^{-5-\frac {10}{x}}}{x^2}+\frac {31241588232960 e^{-4-\frac {8}{x}}}{x^2}-\frac {154279448064 e^{-3-\frac {6}{x}}}{x^2}+\frac {423263232 e^{-2-\frac {4}{x}}}{x^2}-\frac {497664 e^{-1-\frac {2}{x}}}{x^2}\right ) \, dx\\ &=-\left (497664 \int \frac {e^{-1-\frac {2}{x}}}{x^2} \, dx\right )+423263232 \int \frac {e^{-2-\frac {4}{x}}}{x^2} \, dx-154279448064 \int \frac {e^{-3-\frac {6}{x}}}{x^2} \, dx+31241588232960 \int \frac {e^{-4-\frac {8}{x}}}{x^2} \, dx-3795852970304640 \int \frac {e^{-5-\frac {10}{x}}}{x^2} \, dx+276717681535208256 \int \frac {e^{-6-\frac {12}{x}}}{x^2} \, dx-11207066102175934368 \int \frac {e^{-7-\frac {14}{x}}}{x^2} \, dx+194522647344910860816 \int \frac {e^{-8-\frac {16}{x}}}{x^2} \, dx\\ &=12157665459056928801 e^{-8-\frac {16}{x}}-800504721583995312 e^{-7-\frac {14}{x}}+23059806794600688 e^{-6-\frac {12}{x}}-379585297030464 e^{-5-\frac {10}{x}}+3905198529120 e^{-4-\frac {8}{x}}-25713241344 e^{-3-\frac {6}{x}}+105815808 e^{-2-\frac {4}{x}}-248832 e^{-1-\frac {2}{x}}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.11, size = 26, normalized size = 1.62 \begin {gather*} e^{-\frac {8 (2+x)}{x}} \left (243-2 e^{1+\frac {2}{x}}\right )^8 \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. \(125\) vs.
\(2(20)=40\).
time = 0.51, size = 126, normalized size = 7.88
method | result | size |
risch | \(12157665459056928801 \,{\mathrm e}^{-\frac {8 \left (2+x \right )}{x}}-800504721583995312 \,{\mathrm e}^{-\frac {7 \left (2+x \right )}{x}}+23059806794600688 \,{\mathrm e}^{-\frac {6 \left (2+x \right )}{x}}-379585297030464 \,{\mathrm e}^{-\frac {5 \left (2+x \right )}{x}}+3905198529120 \,{\mathrm e}^{-\frac {4 \left (2+x \right )}{x}}-25713241344 \,{\mathrm e}^{-\frac {3 \left (2+x \right )}{x}}+105815808 \,{\mathrm e}^{-\frac {2 \left (2+x \right )}{x}}-248832 \,{\mathrm e}^{-\frac {2+x}{x}}\) | \(90\) |
derivativedivides | \(105815808 \,{\mathrm e}^{-\frac {4}{x}-2}-25713241344 \,{\mathrm e}^{-\frac {6}{x}-3}+3905198529120 \,{\mathrm e}^{-\frac {8}{x}-4}-379585297030464 \,{\mathrm e}^{-\frac {10}{x}-5}+23059806794600688 \,{\mathrm e}^{-\frac {12}{x}-6}-800504721583995312 \,{\mathrm e}^{-\frac {14}{x}-7}+12157665459056928801 \,{\mathrm e}^{-\frac {16}{x}-8}-1024 \,{\mathrm e}^{5 \ln \left (3\right )-1-\frac {2}{x}}\) | \(126\) |
default | \(105815808 \,{\mathrm e}^{-\frac {4}{x}-2}-25713241344 \,{\mathrm e}^{-\frac {6}{x}-3}+3905198529120 \,{\mathrm e}^{-\frac {8}{x}-4}-379585297030464 \,{\mathrm e}^{-\frac {10}{x}-5}+23059806794600688 \,{\mathrm e}^{-\frac {12}{x}-6}-800504721583995312 \,{\mathrm e}^{-\frac {14}{x}-7}+12157665459056928801 \,{\mathrm e}^{-\frac {16}{x}-8}-1024 \,{\mathrm e}^{5 \ln \left (3\right )-1-\frac {2}{x}}\) | \(126\) |
norman | \(\frac {x \,{\mathrm e}^{\frac {40 x \ln \left (3\right )-8 x -16}{x}}+1792 x \,{\mathrm e}^{\frac {10 x \ln \left (3\right )-2 x -4}{x}}-1792 x \,{\mathrm e}^{\frac {15 x \ln \left (3\right )-3 x -6}{x}}+1120 x \,{\mathrm e}^{\frac {20 x \ln \left (3\right )-4 x -8}{x}}-448 x \,{\mathrm e}^{\frac {25 x \ln \left (3\right )-5 x -10}{x}}+112 x \,{\mathrm e}^{\frac {30 x \ln \left (3\right )-6 x -12}{x}}-16 x \,{\mathrm e}^{\frac {35 x \ln \left (3\right )-7 x -14}{x}}-1024 \,{\mathrm e}^{\frac {5 x \ln \left (3\right )-x -2}{x}} x}{x}\) | \(163\) |
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. 81 vs.
\(2 (18) = 36\).
time = 0.28, size = 81, normalized size = 5.06 \begin {gather*} -248832 \, e^{\left (-\frac {2}{x} - 1\right )} + 105815808 \, e^{\left (-\frac {4}{x} - 2\right )} - 25713241344 \, e^{\left (-\frac {6}{x} - 3\right )} + 3905198529120 \, e^{\left (-\frac {8}{x} - 4\right )} - 379585297030464 \, e^{\left (-\frac {10}{x} - 5\right )} + 23059806794600688 \, e^{\left (-\frac {12}{x} - 6\right )} - 800504721583995312 \, e^{\left (-\frac {14}{x} - 7\right )} + 12157665459056928801 \, e^{\left (-\frac {16}{x} - 8\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. 142 vs.
\(2 (18) = 36\).
time = 0.41, size = 142, normalized size = 8.88 \begin {gather*} e^{\left (\frac {8 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} - 16 \, e^{\left (\frac {7 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} + 112 \, e^{\left (\frac {6 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} - 448 \, e^{\left (\frac {5 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} + 1120 \, e^{\left (\frac {4 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} - 1792 \, e^{\left (\frac {3 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} + 1792 \, e^{\left (\frac {2 \, {\left (5 \, x \log \left (3\right ) - x - 2\right )}}{x}\right )} - 1024 \, e^{\left (\frac {5 \, x \log \left (3\right ) - x - 2}{x}\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. 131 vs.
\(2 (10) = 20\).
time = 0.12, size = 131, normalized size = 8.19 \begin {gather*} e^{\frac {8 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} - 16 e^{\frac {7 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} + 112 e^{\frac {6 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} - 448 e^{\frac {5 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} + 1120 e^{\frac {4 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} - 1792 e^{\frac {3 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} + 1792 e^{\frac {2 \left (- x + 5 x \log {\left (3 \right )} - 2\right )}{x}} - 1024 e^{\frac {- x + 5 x \log {\left (3 \right )} - 2}{x}} \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. 85 vs.
\(2 (18) = 36\).
time = 0.41, size = 85, normalized size = 5.31 \begin {gather*} 243 \, {\left (50031545098999707 \, e^{28} - 1024 \, e^{\left (\frac {14}{x} + 35\right )} + 435456 \, e^{\left (\frac {12}{x} + 34\right )} - 105815808 \, e^{\left (\frac {10}{x} + 33\right )} + 16070775840 \, e^{\left (\frac {8}{x} + 32\right )} - 1562079411648 \, e^{\left (\frac {6}{x} + 31\right )} + 94896324257616 \, e^{\left (\frac {4}{x} + 30\right )} - 3294258113514384 \, e^{\left (\frac {2}{x} + 29\right )}\right )} e^{\left (-\frac {16}{x} - 36\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.68, size = 81, normalized size = 5.06 \begin {gather*} 105815808\,{\mathrm {e}}^{-\frac {4}{x}-2}-248832\,{\mathrm {e}}^{-\frac {2}{x}-1}-25713241344\,{\mathrm {e}}^{-\frac {6}{x}-3}+3905198529120\,{\mathrm {e}}^{-\frac {8}{x}-4}-379585297030464\,{\mathrm {e}}^{-\frac {10}{x}-5}+23059806794600688\,{\mathrm {e}}^{-\frac {12}{x}-6}-800504721583995312\,{\mathrm {e}}^{-\frac {14}{x}-7}+12157665459056928801\,{\mathrm {e}}^{-\frac {16}{x}-8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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