Optimal. Leaf size=25 \[ -2-\frac {4}{x}-\frac {\log (3)}{3+2 x+e^{3/2} x} \]
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Rubi [A]
time = 0.17, antiderivative size = 33, normalized size of antiderivative = 1.32, number of steps
used = 6, number of rules used = 5, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 1694, 12,
1828, 8} \begin {gather*} -\frac {x \left (8+4 e^{3/2}+\log (3)\right )+12}{x \left (\left (2+e^{3/2}\right ) x+3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 8
Rule 12
Rule 1694
Rule 1828
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {36+48 x+16 x^2+4 e^3 x^2+e^{3/2} \left (24 x+16 x^2\right )+\left (2 x^2+e^{3/2} x^2\right ) \log (3)}{9 x^2+12 x^3+\left (4+e^3\right ) x^4+e^{3/2} \left (6 x^3+4 x^4\right )} \, dx\\ &=\int \frac {36+48 x+\left (16+4 e^3\right ) x^2+e^{3/2} \left (24 x+16 x^2\right )+\left (2 x^2+e^{3/2} x^2\right ) \log (3)}{9 x^2+12 x^3+\left (4+e^3\right ) x^4+e^{3/2} \left (6 x^3+4 x^4\right )} \, dx\\ &=\text {Subst}\left (\int \frac {4 \left (2+e^{3/2}\right ) \left (12 \left (2+e^{3/2}\right ) x \left (8+4 e^{3/2}-\log (3)\right )+9 \left (8+4 e^{3/2}+\log (3)\right )+4 \left (2+e^{3/2}\right )^2 x^2 \left (8+4 e^{3/2}+\log (3)\right )\right )}{\left (9-4 \left (2+e^{3/2}\right )^2 x^2\right )^2} \, dx,x,\frac {12+6 e^{3/2}}{4 \left (4+4 e^{3/2}+e^3\right )}+x\right )\\ &=\left (4 \left (2+e^{3/2}\right )\right ) \text {Subst}\left (\int \frac {12 \left (2+e^{3/2}\right ) x \left (8+4 e^{3/2}-\log (3)\right )+9 \left (8+4 e^{3/2}+\log (3)\right )+4 \left (2+e^{3/2}\right )^2 x^2 \left (8+4 e^{3/2}+\log (3)\right )}{\left (9-4 \left (2+e^{3/2}\right )^2 x^2\right )^2} \, dx,x,\frac {12+6 e^{3/2}}{4 \left (4+4 e^{3/2}+e^3\right )}+x\right )\\ &=-\frac {12+x \left (8+4 e^{3/2}+\log (3)\right )}{x \left (3+\left (2+e^{3/2}\right ) x\right )}-\frac {1}{9} \left (2 \left (2+e^{3/2}\right )\right ) \text {Subst}\left (\int 0 \, dx,x,\frac {12+6 e^{3/2}}{4 \left (4+4 e^{3/2}+e^3\right )}+x\right )\\ &=-\frac {12+x \left (8+4 e^{3/2}+\log (3)\right )}{x \left (3+\left (2+e^{3/2}\right ) x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.04, size = 41, normalized size = 1.64 \begin {gather*} -\frac {4}{x}-\frac {e^{3/2} \log (3)+\log (9)}{\left (2+e^{3/2}\right ) \left (3+\left (2+e^{3/2}\right ) x\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.68, size = 30, normalized size = 1.20
method | result | size |
risch | \(\frac {\left (-4 \,{\mathrm e}^{\frac {3}{2}}-\ln \left (3\right )-8\right ) x -12}{x \left (3+x \,{\mathrm e}^{\frac {3}{2}}+2 x \right )}\) | \(30\) |
gosper | \(-\frac {x \ln \left (3\right )+4 x \,{\mathrm e}^{\frac {3}{2}}+8 x +12}{x \left (3+x \,{\mathrm e}^{\frac {3}{2}}+2 x \right )}\) | \(31\) |
norman | \(\frac {-12+\left (\frac {\ln \left (3\right ) {\mathrm e}^{\frac {3}{2}}}{3}+\frac {4 \,{\mathrm e}^{3}}{3}+\frac {2 \ln \left (3\right )}{3}+\frac {16 \,{\mathrm e}^{\frac {3}{2}}}{3}+\frac {16}{3}\right ) x^{2}}{x \left (3+x \,{\mathrm e}^{\frac {3}{2}}+2 x \right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 32, normalized size = 1.28 \begin {gather*} -\frac {{\left (e^{\frac {3}{2}} + 2\right )} \log \left (3\right )}{x {\left (e^{3} + 4 \, e^{\frac {3}{2}} + 4\right )} + 3 \, e^{\frac {3}{2}} + 6} - \frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 33, normalized size = 1.32 \begin {gather*} -\frac {4 \, x e^{\frac {3}{2}} + x \log \left (3\right ) + 8 \, x + 12}{x^{2} e^{\frac {3}{2}} + 2 \, x^{2} + 3 \, x} \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. 1023 vs.
\(2 (20) = 40\).
time = 3.73, size = 1023, normalized size = 40.92 \begin {gather*} \frac {x \left (- 41287833600 e^{33} - 181666467840 e^{\frac {63}{2}} - 8078054400 e^{\frac {69}{2}} - 693635604480 e^{30} - 1346342400 e^{36} - 2312118681600 e^{\frac {57}{2}} - 188487936 e^{\frac {75}{2}} - 6758500761600 e^{27} - 17379001958400 e^{\frac {51}{2}} - 21748608 e^{39} - 39392404439040 e^{24} - 2013760 e^{\frac {81}{2}} - 78784808878080 e^{\frac {45}{2}} - 7325260800 e^{\frac {63}{2}} \log {\left (3 \right )} - 30766095360 e^{30} \log {\left (3 \right )} - 1498348800 e^{33} \log {\left (3 \right )} - 111876710400 e^{\frac {57}{2}} \log {\left (3 \right )} - 260582400 e^{\frac {69}{2}} \log {\left (3 \right )} - 143840 e^{42} - 354276249600 e^{27} \log {\left (3 \right )} - 139032015667200 e^{21} - 38001600 e^{36} \log {\left (3 \right )} - 981072691200 e^{\frac {51}{2}} \log {\left (3 \right )} - 4560192 e^{\frac {75}{2}} \log {\left (3 \right )} - 2382605107200 e^{24} \log {\left (3 \right )} - 216272024371200 e^{\frac {39}{2}} - 7440 e^{\frac {87}{2}} - 438480 e^{39} \log {\left (3 \right )} - 5082890895360 e^{\frac {45}{2}} \log {\left (3 \right )} - 295951191244800 e^{18} - 32480 e^{\frac {81}{2}} \log {\left (3 \right )} - 9530420428800 e^{21} \log {\left (3 \right )} - 248 e^{45} - 355141429493760 e^{\frac {33}{2}} - 15697163059200 e^{\frac {39}{2}} \log {\left (3 \right )} - 1740 e^{42} \log {\left (3 \right )} - 22673679974400 e^{18} \log {\left (3 \right )} - 372052926136320 e^{15} - 4 e^{\frac {93}{2}} - 60 e^{\frac {87}{2}} \log {\left (3 \right )} - 28640437862400 e^{\frac {33}{2}} \log {\left (3 \right )} - 338229932851200 e^{\frac {27}{2}} - 31504481648640 e^{15} \log {\left (3 \right )} - 264701686579200 e^{12} - e^{45} \log {\left (3 \right )} - 30004268236800 e^{\frac {27}{2}} \log {\left (3 \right )} - 176467791052800 e^{\frac {21}{2}} - 24548946739200 e^{12} \log {\left (3 \right )} - 98821962989568 e^{9} - 17077528166400 e^{\frac {21}{2}} \log {\left (3 \right )} - 9961891430400 e^{9} \log {\left (3 \right )} - 45610136764416 e^{\frac {15}{2}} - 4781707886592 e^{\frac {15}{2}} \log {\left (3 \right )} - 16892643246080 e^{6} - 1839118417920 e^{6} \log {\left (3 \right )} - 4826469498880 e^{\frac {9}{2}} - 544923975680 e^{\frac {9}{2}} \log {\left (3 \right )} - 998579896320 e^{3} - 116769423360 e^{3} \log {\left (3 \right )} - 133143986176 e^{\frac {3}{2}} - 16106127360 e^{\frac {3}{2}} \log {\left (3 \right )} - 8589934592 - 1073741824 \log {\left (3 \right )}\right ) - 87903129600 e^{\frac {63}{2}} - 369193144320 e^{30} - 17980185600 e^{33} - 1342520524800 e^{\frac {57}{2}} - 3126988800 e^{\frac {69}{2}} - 4251314995200 e^{27} - 456019200 e^{36} - 11772872294400 e^{\frac {51}{2}} - 54722304 e^{\frac {75}{2}} - 28591261286400 e^{24} - 5261760 e^{39} - 60994690744320 e^{\frac {45}{2}} - 389760 e^{\frac {81}{2}} - 114365045145600 e^{21} - 188365956710400 e^{\frac {39}{2}} - 20880 e^{42} - 272084159692800 e^{18} - 720 e^{\frac {87}{2}} - 343685254348800 e^{\frac {33}{2}} - 378053779783680 e^{15} - 12 e^{45} - 360051218841600 e^{\frac {27}{2}} - 294587360870400 e^{12} - 204930337996800 e^{\frac {21}{2}} - 119542697164800 e^{9} - 57380494639104 e^{\frac {15}{2}} - 22069421015040 e^{6} - 6539087708160 e^{\frac {9}{2}} - 1401233080320 e^{3} - 193273528320 e^{\frac {3}{2}} - 12884901888}{x^{2} \cdot \left (2147483648 + 33285996544 e^{\frac {3}{2}} + 249644974080 e^{3} + 1206617374720 e^{\frac {9}{2}} + 4223160811520 e^{6} + 11402534191104 e^{\frac {15}{2}} + 24705490747392 e^{9} + 44116947763200 e^{\frac {21}{2}} + 66175421644800 e^{12} + 84557483212800 e^{\frac {27}{2}} + e^{\frac {93}{2}} + 93013231534080 e^{15} + 88785357373440 e^{\frac {33}{2}} + 62 e^{45} + 73987797811200 e^{18} + 1860 e^{\frac {87}{2}} + 54068006092800 e^{\frac {39}{2}} + 34758003916800 e^{21} + 35960 e^{42} + 19696202219520 e^{\frac {45}{2}} + 503440 e^{\frac {81}{2}} + 9848101109760 e^{24} + 5437152 e^{39} + 4344750489600 e^{\frac {51}{2}} + 1689625190400 e^{27} + 47121984 e^{\frac {75}{2}} + 578029670400 e^{\frac {57}{2}} + 336585600 e^{36} + 173408901120 e^{30} + 2019513600 e^{\frac {69}{2}} + 45416616960 e^{\frac {63}{2}} + 10321958400 e^{33}\right ) + x \left (3221225472 + 48318382080 e^{\frac {3}{2}} + 350308270080 e^{3} + 1634771927040 e^{\frac {9}{2}} + 5517355253760 e^{6} + 14345123659776 e^{\frac {15}{2}} + 29885674291200 e^{9} + 51232584499200 e^{\frac {21}{2}} + 73646840217600 e^{12} + 90012804710400 e^{\frac {27}{2}} + 3 e^{45} + 94513444945920 e^{15} + 85921313587200 e^{\frac {33}{2}} + 180 e^{\frac {87}{2}} + 68021039923200 e^{18} + 5220 e^{42} + 47091489177600 e^{\frac {39}{2}} + 28591261286400 e^{21} + 97440 e^{\frac {81}{2}} + 15248672686080 e^{\frac {45}{2}} + 1315440 e^{39} + 7147815321600 e^{24} + 13680576 e^{\frac {75}{2}} + 2943218073600 e^{\frac {51}{2}} + 114004800 e^{36} + 1062828748800 e^{27} + 781747200 e^{\frac {69}{2}} + 335630131200 e^{\frac {57}{2}} + 4495046400 e^{33} + 92298286080 e^{30} + 21975782400 e^{\frac {63}{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 33, normalized size = 1.32 \begin {gather*} -\frac {4 \, x e^{\frac {3}{2}} + x \log \left (3\right ) + 8 \, x + 12}{x^{2} e^{\frac {3}{2}} + 2 \, x^{2} + 3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.21, size = 20, normalized size = 0.80 \begin {gather*} -\frac {\ln \left (3\right )}{x\,\left ({\mathrm {e}}^{3/2}+2\right )+3}-\frac {4}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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