Integrand size = 118, antiderivative size = 24 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=3-\left (3 e^{x/24}+x-\frac {3 x^2}{25}\right )^4 \]
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Leaf count is larger than twice the leaf count of optimal. \(161\) vs. \(2(24)=48\).
Time = 0.30 (sec) , antiderivative size = 161, normalized size of antiderivative = 6.71, number of steps used = 54, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {12, 2225, 2227, 2207} \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81 x^8}{390625}+\frac {108 x^7}{15625}+\frac {324 e^{x/24} x^6}{15625}-\frac {54 x^6}{625}-\frac {324}{625} e^{x/24} x^5+\frac {12 x^5}{25}+\frac {108}{25} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4-x^4-12 e^{x/24} x^3+\frac {324}{25} e^{x/12} x^3-54 e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2-108 e^{x/8} x-81 e^{x/6} \]
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Rule 12
Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {\int \left (-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )\right ) \, dx}{781250} \\ & = -x^4+\frac {12 x^5}{25}-\frac {54 x^6}{625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}+\frac {\int e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right ) \, dx}{781250}+\frac {\int e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right ) \, dx}{781250}+\frac {\int e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right ) \, dx}{781250}-\frac {27}{2} \int e^{x/6} \, dx \\ & = -81 e^{x/6}-x^4+\frac {12 x^5}{25}-\frac {54 x^6}{625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}+\frac {\int \left (-84375000 e^{x/8}+9703125 e^{x/8} x+1265625 e^{x/8} x^2\right ) \, dx}{781250}+\frac {\int \left (-84375000 e^{x/12} x+26859375 e^{x/12} x^2-1586250 e^{x/12} x^3-50625 e^{x/12} x^4\right ) \, dx}{781250}+\frac {\int \left (-28125000 e^{x/24} x^2+13109375 e^{x/24} x^3-1884375 e^{x/24} x^4+80325 e^{x/24} x^5+675 e^{x/24} x^6\right ) \, dx}{781250} \\ & = -81 e^{x/6}-x^4+\frac {12 x^5}{25}-\frac {54 x^6}{625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}+\frac {27 \int e^{x/24} x^6 \, dx}{31250}-\frac {81 \int e^{x/12} x^4 \, dx}{1250}+\frac {3213 \int e^{x/24} x^5 \, dx}{31250}+\frac {81}{50} \int e^{x/8} x^2 \, dx-\frac {1269}{625} \int e^{x/12} x^3 \, dx-\frac {603}{250} \int e^{x/24} x^4 \, dx+\frac {621}{50} \int e^{x/8} x \, dx+\frac {839}{50} \int e^{x/24} x^3 \, dx+\frac {1719}{50} \int e^{x/12} x^2 \, dx-36 \int e^{x/24} x^2 \, dx-108 \int e^{x/8} \, dx-108 \int e^{x/12} x \, dx \\ & = -864 e^{x/8}-81 e^{x/6}-1296 e^{x/12} x+\frac {2484}{25} e^{x/8} x-864 e^{x/24} x^2+\frac {10314}{25} e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2+\frac {10068}{25} e^{x/24} x^3-\frac {15228}{625} e^{x/12} x^3-x^4-\frac {7236}{125} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}+\frac {38556 e^{x/24} x^5}{15625}-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}-\frac {1944 \int e^{x/24} x^5 \, dx}{15625}+\frac {1944}{625} \int e^{x/12} x^3 \, dx-\frac {38556 \int e^{x/24} x^4 \, dx}{3125}-\frac {648}{25} \int e^{x/8} x \, dx+\frac {45684}{625} \int e^{x/12} x^2 \, dx-\frac {2484}{25} \int e^{x/8} \, dx+\frac {28944}{125} \int e^{x/24} x^3 \, dx-\frac {20628}{25} \int e^{x/12} x \, dx-\frac {30204}{25} \int e^{x/24} x^2 \, dx+1296 \int e^{x/12} \, dx+1728 \int e^{x/24} x \, dx \\ & = 15552 e^{x/12}-\frac {41472 e^{x/8}}{25}-81 e^{x/6}+41472 e^{x/24} x-\frac {279936}{25} e^{x/12} x-108 e^{x/8} x-\frac {746496}{25} e^{x/24} x^2+\frac {806058}{625} e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2+\frac {744996}{125} e^{x/24} x^3+\frac {324}{25} e^{x/12} x^3-x^4-\frac {1106244 e^{x/24} x^4}{3125}-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}-\frac {324}{625} e^{x/24} x^5-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}+\frac {46656 \int e^{x/24} x^4 \, dx}{3125}-\frac {69984}{625} \int e^{x/12} x^2 \, dx+\frac {5184}{25} \int e^{x/8} \, dx+\frac {3701376 \int e^{x/24} x^3 \, dx}{3125}-\frac {1096416}{625} \int e^{x/12} x \, dx+\frac {247536}{25} \int e^{x/12} \, dx-\frac {2083968}{125} \int e^{x/24} x^2 \, dx-41472 \int e^{x/24} \, dx+\frac {1449792}{25} \int e^{x/24} x \, dx \\ & = -995328 e^{x/24}+\frac {3359232 e^{x/12}}{25}-81 e^{x/6}+\frac {35831808}{25} e^{x/24} x-\frac {20155392}{625} e^{x/12} x-108 e^{x/8} x-\frac {53747712}{125} e^{x/24} x^2-54 e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2+\frac {107457924 e^{x/24} x^3}{3125}+\frac {324}{25} e^{x/12} x^3-x^4+\frac {108}{25} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}-\frac {324}{625} e^{x/24} x^5-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}-\frac {4478976 \int e^{x/24} x^3 \, dx}{3125}+\frac {1679616}{625} \int e^{x/12} x \, dx+\frac {13156992}{625} \int e^{x/12} \, dx-\frac {266499072 \int e^{x/24} x^2 \, dx}{3125}+\frac {100030464}{125} \int e^{x/24} x \, dx-\frac {34795008}{25} \int e^{x/24} \, dx \\ & = -\frac {859963392 e^{x/24}}{25}+\frac {241864704 e^{x/12}}{625}-81 e^{x/6}+\frac {2579890176}{125} e^{x/24} x-108 e^{x/8} x-\frac {7739670528 e^{x/24} x^2}{3125}-54 e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2-12 e^{x/24} x^3+\frac {324}{25} e^{x/12} x^3-x^4+\frac {108}{25} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}-\frac {324}{625} e^{x/24} x^5-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}-\frac {20155392}{625} \int e^{x/12} \, dx+\frac {322486272 \int e^{x/24} x^2 \, dx}{3125}+\frac {12791955456 \int e^{x/24} x \, dx}{3125}-\frac {2400731136}{125} \int e^{x/24} \, dx \\ & = -\frac {61917364224 e^{x/24}}{125}-81 e^{x/6}+\frac {371504185344 e^{x/24} x}{3125}-108 e^{x/8} x-54 e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2-12 e^{x/24} x^3+\frac {324}{25} e^{x/12} x^3-x^4+\frac {108}{25} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}-\frac {324}{625} e^{x/24} x^5-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}-\frac {15479341056 \int e^{x/24} x \, dx}{3125}-\frac {307006930944 \int e^{x/24} \, dx}{3125} \\ & = -\frac {8916100448256 e^{x/24}}{3125}-81 e^{x/6}-108 e^{x/8} x-54 e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2-12 e^{x/24} x^3+\frac {324}{25} e^{x/12} x^3-x^4+\frac {108}{25} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}-\frac {324}{625} e^{x/24} x^5-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625}+\frac {371504185344 \int e^{x/24} \, dx}{3125} \\ & = -81 e^{x/6}-108 e^{x/8} x-54 e^{x/12} x^2+\frac {324}{25} e^{x/8} x^2-12 e^{x/24} x^3+\frac {324}{25} e^{x/12} x^3-x^4+\frac {108}{25} e^{x/24} x^4-\frac {486}{625} e^{x/12} x^4+\frac {12 x^5}{25}-\frac {324}{625} e^{x/24} x^5-\frac {54 x^6}{625}+\frac {324 e^{x/24} x^6}{15625}+\frac {108 x^7}{15625}-\frac {81 x^8}{390625} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {\left (75 e^{x/24}+(25-3 x) x\right )^4}{390625} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(19)=38\).
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.04
method | result | size |
risch | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {\left (10125000 x^{2}-84375000 x \right ) {\mathrm e}^{\frac {x}{8}}}{781250}+\frac {\left (-607500 x^{4}+10125000 x^{3}-42187500 x^{2}\right ) {\mathrm e}^{\frac {x}{12}}}{781250}+\frac {\left (16200 x^{6}-405000 x^{5}+3375000 x^{4}-9375000 x^{3}\right ) {\mathrm e}^{\frac {x}{24}}}{781250}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(97\) |
derivativedivides | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
default | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
parallelrisch | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
parts | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81}{390625} \, x^{8} + \frac {108}{15625} \, x^{7} - \frac {54}{625} \, x^{6} + \frac {12}{25} \, x^{5} - x^{4} + \frac {108}{25} \, {\left (3 \, x^{2} - 25 \, x\right )} e^{\left (\frac {1}{8} \, x\right )} - \frac {54}{625} \, {\left (9 \, x^{4} - 150 \, x^{3} + 625 \, x^{2}\right )} e^{\left (\frac {1}{12} \, x\right )} + \frac {12}{15625} \, {\left (27 \, x^{6} - 675 \, x^{5} + 5625 \, x^{4} - 15625 \, x^{3}\right )} e^{\left (\frac {1}{24} \, x\right )} - 81 \, e^{\left (\frac {1}{6} \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=- \frac {81 x^{8}}{390625} + \frac {108 x^{7}}{15625} - \frac {54 x^{6}}{625} + \frac {12 x^{5}}{25} - x^{4} + \frac {\left (3164062500 x^{2} - 26367187500 x\right ) e^{\frac {x}{8}}}{244140625} + \frac {\left (- 189843750 x^{4} + 3164062500 x^{3} - 13183593750 x^{2}\right ) e^{\frac {x}{12}}}{244140625} + \frac {\left (5062500 x^{6} - 126562500 x^{5} + 1054687500 x^{4} - 2929687500 x^{3}\right ) e^{\frac {x}{24}}}{244140625} - 81 e^{\frac {x}{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81}{390625} \, x^{8} + \frac {108}{15625} \, x^{7} - \frac {54}{625} \, x^{6} + \frac {12}{25} \, x^{5} - x^{4} + \frac {108}{25} \, {\left (3 \, x^{2} - 25 \, x\right )} e^{\left (\frac {1}{8} \, x\right )} - \frac {54}{625} \, {\left (9 \, x^{4} - 150 \, x^{3} + 625 \, x^{2}\right )} e^{\left (\frac {1}{12} \, x\right )} + \frac {12}{15625} \, {\left (27 \, x^{6} - 675 \, x^{5} + 5625 \, x^{4} - 15625 \, x^{3}\right )} e^{\left (\frac {1}{24} \, x\right )} - 81 \, e^{\left (\frac {1}{6} \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81}{390625} \, x^{8} + \frac {108}{15625} \, x^{7} - \frac {54}{625} \, x^{6} + \frac {12}{25} \, x^{5} - x^{4} + \frac {108}{25} \, {\left (3 \, x^{2} - 25 \, x\right )} e^{\left (\frac {1}{8} \, x\right )} - \frac {54}{625} \, {\left (9 \, x^{4} - 150 \, x^{3} + 625 \, x^{2}\right )} e^{\left (\frac {1}{12} \, x\right )} + \frac {12}{15625} \, {\left (27 \, x^{6} - 675 \, x^{5} + 5625 \, x^{4} - 15625 \, x^{3}\right )} e^{\left (\frac {1}{24} \, x\right )} - 81 \, e^{\left (\frac {1}{6} \, x\right )} \]
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Time = 8.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {{\left (25\,x+75\,{\mathrm {e}}^{x/24}-3\,x^2\right )}^4}{390625} \]
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