Integrand size = 126, antiderivative size = 23 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=\left (\frac {1}{2}-e^x-x\right ) \left (-x^2+\log (3)\right )^4 \]
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Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(23)=46\).
Time = 0.32 (sec) , antiderivative size = 121, normalized size of antiderivative = 5.26, number of steps used = 57, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2227, 2207, 2225} \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^9-e^x x^8+\frac {x^8}{2}+4 x^7 \log (3)+4 e^x x^6 \log (3)-2 x^6 \log (3)-6 x^5 \log ^2(3)-6 e^x x^4 \log ^2(3)+3 x^4 \log ^2(3)+4 x^3 \log ^3(3)+4 e^x x^2 \log ^3(3)-2 x^2 \log ^3(3)-x \log ^4(3)-e^x \log ^4(3) \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = \frac {x^8}{2}-x^9-x \log ^4(3)+\log (3) \int \left (-12 x^5+28 x^6\right ) \, dx+\log ^2(3) \int \left (12 x^3-30 x^4\right ) \, dx+\log ^3(3) \int \left (-4 x+12 x^2\right ) \, dx+\int e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right ) \, dx \\ & = \frac {x^8}{2}-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3)+\int \left (-8 e^x x^7-e^x x^8+4 e^x x^5 (6+x) \log (3)-6 e^x x^3 (4+x) \log ^2(3)+4 e^x x (2+x) \log ^3(3)-e^x \log ^4(3)\right ) \, dx \\ & = \frac {x^8}{2}-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3)-8 \int e^x x^7 \, dx+(4 \log (3)) \int e^x x^5 (6+x) \, dx-\left (6 \log ^2(3)\right ) \int e^x x^3 (4+x) \, dx+\left (4 \log ^3(3)\right ) \int e^x x (2+x) \, dx-\log ^4(3) \int e^x \, dx-\int e^x x^8 \, dx \\ & = -8 e^x x^7+\frac {x^8}{2}-e^x x^8-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+8 \int e^x x^7 \, dx+56 \int e^x x^6 \, dx+(4 \log (3)) \int \left (6 e^x x^5+e^x x^6\right ) \, dx-\left (6 \log ^2(3)\right ) \int \left (4 e^x x^3+e^x x^4\right ) \, dx+\left (4 \log ^3(3)\right ) \int \left (2 e^x x+e^x x^2\right ) \, dx \\ & = 56 e^x x^6+\frac {x^8}{2}-e^x x^8-x^9-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-56 \int e^x x^6 \, dx-336 \int e^x x^5 \, dx+(4 \log (3)) \int e^x x^6 \, dx+(24 \log (3)) \int e^x x^5 \, dx-\left (6 \log ^2(3)\right ) \int e^x x^4 \, dx-\left (24 \log ^2(3)\right ) \int e^x x^3 \, dx+\left (4 \log ^3(3)\right ) \int e^x x^2 \, dx+\left (8 \log ^3(3)\right ) \int e^x x \, dx \\ & = -336 e^x x^5+\frac {x^8}{2}-e^x x^8-x^9+24 e^x x^5 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)-24 e^x x^3 \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)+8 e^x x \log ^3(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+336 \int e^x x^5 \, dx+1680 \int e^x x^4 \, dx-(24 \log (3)) \int e^x x^5 \, dx-(120 \log (3)) \int e^x x^4 \, dx+\left (24 \log ^2(3)\right ) \int e^x x^3 \, dx+\left (72 \log ^2(3)\right ) \int e^x x^2 \, dx-\left (8 \log ^3(3)\right ) \int e^x \, dx-\left (8 \log ^3(3)\right ) \int e^x x \, dx \\ & = 1680 e^x x^4+\frac {x^8}{2}-e^x x^8-x^9-120 e^x x^4 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+72 e^x x^2 \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-8 e^x \log ^3(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-1680 \int e^x x^4 \, dx-6720 \int e^x x^3 \, dx+(120 \log (3)) \int e^x x^4 \, dx+(480 \log (3)) \int e^x x^3 \, dx-\left (72 \log ^2(3)\right ) \int e^x x^2 \, dx-\left (144 \log ^2(3)\right ) \int e^x x \, dx+\left (8 \log ^3(3)\right ) \int e^x \, dx \\ & = -6720 e^x x^3+\frac {x^8}{2}-e^x x^8-x^9+480 e^x x^3 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)-144 e^x x \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+6720 \int e^x x^3 \, dx+20160 \int e^x x^2 \, dx-(480 \log (3)) \int e^x x^3 \, dx-(1440 \log (3)) \int e^x x^2 \, dx+\left (144 \log ^2(3)\right ) \int e^x \, dx+\left (144 \log ^2(3)\right ) \int e^x x \, dx \\ & = 20160 e^x x^2+\frac {x^8}{2}-e^x x^8-x^9-1440 e^x x^2 \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+144 e^x \log ^2(3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-20160 \int e^x x^2 \, dx-40320 \int e^x x \, dx+(1440 \log (3)) \int e^x x^2 \, dx+(2880 \log (3)) \int e^x x \, dx-\left (144 \log ^2(3)\right ) \int e^x \, dx \\ & = -40320 e^x x+\frac {x^8}{2}-e^x x^8-x^9+2880 e^x x \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)+40320 \int e^x \, dx+40320 \int e^x x \, dx-(2880 \log (3)) \int e^x \, dx-(2880 \log (3)) \int e^x x \, dx \\ & = 40320 e^x+\frac {x^8}{2}-e^x x^8-x^9-2880 e^x \log (3)-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3)-40320 \int e^x \, dx+(2880 \log (3)) \int e^x \, dx \\ & = \frac {x^8}{2}-e^x x^8-x^9-2 x^6 \log (3)+4 e^x x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 e^x x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 e^x x^2 \log ^3(3)+4 x^3 \log ^3(3)-e^x \log ^4(3)-x \log ^4(3) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(23)=46\).
Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.70 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=\frac {x^8}{2}-x^9-e^x \left (x^2-\log (3)\right )^4-2 x^6 \log (3)+4 x^7 \log (3)+3 x^4 \log ^2(3)-6 x^5 \log ^2(3)-2 x^2 \log ^3(3)+4 x^3 \log ^3(3)-x \log ^4(3) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(20)=40\).
Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.74
method | result | size |
risch | \(\left (-x^{8}+4 x^{6} \ln \left (3\right )-6 x^{4} \ln \left (3\right )^{2}+4 \ln \left (3\right )^{3} x^{2}-\ln \left (3\right )^{4}\right ) {\mathrm e}^{x}-x \ln \left (3\right )^{4}+4 x^{3} \ln \left (3\right )^{3}-2 \ln \left (3\right )^{3} x^{2}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 \ln \left (3\right ) x^{7}-2 x^{6} \ln \left (3\right )-x^{9}+\frac {x^{8}}{2}\) | \(109\) |
default | \(4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}-{\mathrm e}^{x} \ln \left (3\right )^{4}-x^{8} {\mathrm e}^{x}+4 x^{3} \ln \left (3\right )^{3}-2 \ln \left (3\right )^{3} x^{2}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 \ln \left (3\right ) x^{7}-2 x^{6} \ln \left (3\right )+\frac {x^{8}}{2}-x^{9}-x \ln \left (3\right )^{4}\) | \(115\) |
norman | \(4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}-{\mathrm e}^{x} \ln \left (3\right )^{4}-x^{8} {\mathrm e}^{x}+4 x^{3} \ln \left (3\right )^{3}-2 \ln \left (3\right )^{3} x^{2}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 \ln \left (3\right ) x^{7}-2 x^{6} \ln \left (3\right )+\frac {x^{8}}{2}-x^{9}-x \ln \left (3\right )^{4}\) | \(115\) |
parallelrisch | \(4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}-{\mathrm e}^{x} \ln \left (3\right )^{4}-x^{8} {\mathrm e}^{x}+4 x^{3} \ln \left (3\right )^{3}-2 \ln \left (3\right )^{3} x^{2}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 \ln \left (3\right ) x^{7}-2 x^{6} \ln \left (3\right )+\frac {x^{8}}{2}-x^{9}-x \ln \left (3\right )^{4}\) | \(115\) |
parts | \(4 \,{\mathrm e}^{x} \ln \left (3\right )^{3} x^{2}-6 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{4}+4 \,{\mathrm e}^{x} \ln \left (3\right ) x^{6}-{\mathrm e}^{x} \ln \left (3\right )^{4}-x^{8} {\mathrm e}^{x}+4 x^{3} \ln \left (3\right )^{3}-2 \ln \left (3\right )^{3} x^{2}-6 x^{5} \ln \left (3\right )^{2}+3 x^{4} \ln \left (3\right )^{2}+4 \ln \left (3\right ) x^{7}-2 x^{6} \ln \left (3\right )+\frac {x^{8}}{2}-x^{9}-x \ln \left (3\right )^{4}\) | \(115\) |
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^{9} + \frac {1}{2} \, x^{8} - x \log \left (3\right )^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \left (3\right )^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \left (3\right )^{2} - {\left (x^{8} - 4 \, x^{6} \log \left (3\right ) + 6 \, x^{4} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right )^{3} + \log \left (3\right )^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \left (3\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (15) = 30\).
Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.87 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=- x^{9} + \frac {x^{8}}{2} + 4 x^{7} \log {\left (3 \right )} - 2 x^{6} \log {\left (3 \right )} - 6 x^{5} \log {\left (3 \right )}^{2} + 3 x^{4} \log {\left (3 \right )}^{2} + 4 x^{3} \log {\left (3 \right )}^{3} - 2 x^{2} \log {\left (3 \right )}^{3} - x \log {\left (3 \right )}^{4} + \left (- x^{8} + 4 x^{6} \log {\left (3 \right )} - 6 x^{4} \log {\left (3 \right )}^{2} + 4 x^{2} \log {\left (3 \right )}^{3} - \log {\left (3 \right )}^{4}\right ) e^{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^{9} + \frac {1}{2} \, x^{8} - x \log \left (3\right )^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \left (3\right )^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \left (3\right )^{2} - {\left (x^{8} - 4 \, x^{6} \log \left (3\right ) + 6 \, x^{4} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right )^{3} + \log \left (3\right )^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \left (3\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (21) = 42\).
Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=-x^{9} + \frac {1}{2} \, x^{8} - x \log \left (3\right )^{4} + 2 \, {\left (2 \, x^{3} - x^{2}\right )} \log \left (3\right )^{3} - 3 \, {\left (2 \, x^{5} - x^{4}\right )} \log \left (3\right )^{2} - {\left (x^{8} - 4 \, x^{6} \log \left (3\right ) + 6 \, x^{4} \log \left (3\right )^{2} - 4 \, x^{2} \log \left (3\right )^{3} + \log \left (3\right )^{4}\right )} e^{x} + 2 \, {\left (2 \, x^{7} - x^{6}\right )} \log \left (3\right ) \]
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Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.96 \[ \int \left (4 x^7-9 x^8+\left (-12 x^5+28 x^6\right ) \log (3)+\left (12 x^3-30 x^4\right ) \log ^2(3)+\left (-4 x+12 x^2\right ) \log ^3(3)-\log ^4(3)+e^x \left (-8 x^7-x^8+\left (24 x^5+4 x^6\right ) \log (3)+\left (-24 x^3-6 x^4\right ) \log ^2(3)+\left (8 x+4 x^2\right ) \log ^3(3)-\log ^4(3)\right )\right ) \, dx=4\,x^3\,{\ln \left (3\right )}^3-2\,x^2\,{\ln \left (3\right )}^3+3\,x^4\,{\ln \left (3\right )}^2-6\,x^5\,{\ln \left (3\right )}^2-{\mathrm {e}}^x\,{\ln \left (3\right )}^4-x^8\,{\mathrm {e}}^x-x\,{\ln \left (3\right )}^4-2\,x^6\,\ln \left (3\right )+4\,x^7\,\ln \left (3\right )+\frac {x^8}{2}-x^9+4\,x^6\,{\mathrm {e}}^x\,\ln \left (3\right )+4\,x^2\,{\mathrm {e}}^x\,{\ln \left (3\right )}^3-6\,x^4\,{\mathrm {e}}^x\,{\ln \left (3\right )}^2 \]
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