Integrand size = 66, antiderivative size = 20 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx=\left (-2 x+e^{3+x} (x+\log (3)) \log ^8(5)\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(153\) vs. \(2(20)=40\).
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 7.65, number of steps used = 18, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2227, 2207, 2225} \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx=4 x^2+e^{2 x+6} x^2 \log ^{16}(5)-4 e^{x+3} x^2 \log ^8(5)+e^{2 x+6} x \log ^{16}(5) (1+\log (9))-e^{2 x+6} x \log ^{16}(5)-\frac {1}{2} e^{2 x+6} \log ^{16}(5) (1+\log (9))+\frac {1}{2} e^{2 x+6} \log ^{16}(5)+4 e^{x+3} \log (3) \log ^8(5)-4 e^{x+3} (x+1) \log (3) \log ^8(5)+\frac {1}{2} e^{2 x+6} \log ^{16}(5) \left (2 \log ^2(3)+\log (9)\right ) \]
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Rule 2207
Rule 2225
Rule 2227
Rubi steps \begin{align*} \text {integral}& = 4 x^2+\log ^8(5) \int e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \, dx+\log ^{16}(5) \int e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \, dx \\ & = 4 x^2+\log ^8(5) \int \left (-8 e^{3+x} x-4 e^{3+x} x^2-4 e^{3+x} (1+x) \log (3)\right ) \, dx+\log ^{16}(5) \int \left (2 e^{6+2 x} x^2+2 e^{6+2 x} x (1+\log (9))+2 e^{6+2 x} \log ^2(3) \left (1+\frac {\log (9)}{2 \log ^2(3)}\right )\right ) \, dx \\ & = 4 x^2-\left (4 \log ^8(5)\right ) \int e^{3+x} x^2 \, dx-\left (8 \log ^8(5)\right ) \int e^{3+x} x \, dx-\left (4 \log (3) \log ^8(5)\right ) \int e^{3+x} (1+x) \, dx+\left (2 \log ^{16}(5)\right ) \int e^{6+2 x} x^2 \, dx+\left (2 \log ^{16}(5) (1+\log (9))\right ) \int e^{6+2 x} x \, dx+\left (\log ^{16}(5) \left (2 \log ^2(3)+\log (9)\right )\right ) \int e^{6+2 x} \, dx \\ & = 4 x^2-8 e^{3+x} x \log ^8(5)-4 e^{3+x} x^2 \log ^8(5)-4 e^{3+x} (1+x) \log (3) \log ^8(5)+e^{6+2 x} x^2 \log ^{16}(5)+e^{6+2 x} x \log ^{16}(5) (1+\log (9))+\frac {1}{2} e^{6+2 x} \log ^{16}(5) \left (2 \log ^2(3)+\log (9)\right )+\left (8 \log ^8(5)\right ) \int e^{3+x} \, dx+\left (8 \log ^8(5)\right ) \int e^{3+x} x \, dx+\left (4 \log (3) \log ^8(5)\right ) \int e^{3+x} \, dx-\left (2 \log ^{16}(5)\right ) \int e^{6+2 x} x \, dx-\left (\log ^{16}(5) (1+\log (9))\right ) \int e^{6+2 x} \, dx \\ & = 4 x^2+8 e^{3+x} \log ^8(5)-4 e^{3+x} x^2 \log ^8(5)+4 e^{3+x} \log (3) \log ^8(5)-4 e^{3+x} (1+x) \log (3) \log ^8(5)-e^{6+2 x} x \log ^{16}(5)+e^{6+2 x} x^2 \log ^{16}(5)-\frac {1}{2} e^{6+2 x} \log ^{16}(5) (1+\log (9))+e^{6+2 x} x \log ^{16}(5) (1+\log (9))+\frac {1}{2} e^{6+2 x} \log ^{16}(5) \left (2 \log ^2(3)+\log (9)\right )-\left (8 \log ^8(5)\right ) \int e^{3+x} \, dx+\log ^{16}(5) \int e^{6+2 x} \, dx \\ & = 4 x^2-4 e^{3+x} x^2 \log ^8(5)+4 e^{3+x} \log (3) \log ^8(5)-4 e^{3+x} (1+x) \log (3) \log ^8(5)+\frac {1}{2} e^{6+2 x} \log ^{16}(5)-e^{6+2 x} x \log ^{16}(5)+e^{6+2 x} x^2 \log ^{16}(5)-\frac {1}{2} e^{6+2 x} \log ^{16}(5) (1+\log (9))+e^{6+2 x} x \log ^{16}(5) (1+\log (9))+\frac {1}{2} e^{6+2 x} \log ^{16}(5) \left (2 \log ^2(3)+\log (9)\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(20)=40\).
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx=4 x^2-4 e^{3+x} \left (x^2+x \log (3)\right ) \log ^8(5)+e^{6+2 x} \log ^{16}(5) \left (x^2+\log ^2(3)+x \log (9)\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(19)=38\).
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55
method | result | size |
risch | \(\ln \left (5\right )^{16} \left (\ln \left (3\right )^{2}+2 x \ln \left (3\right )+x^{2}\right ) {\mathrm e}^{2 x +6}+\ln \left (5\right )^{8} \left (-4 x \ln \left (3\right )-4 x^{2}\right ) {\mathrm e}^{3+x}+4 x^{2}\) | \(51\) |
default | \(\ln \left (5\right )^{16} \ln \left (3\right )^{2} {\mathrm e}^{2 x +6}+2 \ln \left (5\right )^{16} \ln \left (3\right ) {\mathrm e}^{2 x +6} x +\ln \left (5\right )^{16} {\mathrm e}^{2 x +6} x^{2}-4 \ln \left (5\right )^{8} \ln \left (3\right ) {\mathrm e}^{3+x} x -4 \ln \left (5\right )^{8} {\mathrm e}^{3+x} x^{2}+4 x^{2}\) | \(77\) |
norman | \(\ln \left (5\right )^{16} \ln \left (3\right )^{2} {\mathrm e}^{2 x +6}+2 \ln \left (5\right )^{16} \ln \left (3\right ) {\mathrm e}^{2 x +6} x +\ln \left (5\right )^{16} {\mathrm e}^{2 x +6} x^{2}-4 \ln \left (5\right )^{8} \ln \left (3\right ) {\mathrm e}^{3+x} x -4 \ln \left (5\right )^{8} {\mathrm e}^{3+x} x^{2}+4 x^{2}\) | \(77\) |
parallelrisch | \(\ln \left (5\right )^{16} \ln \left (3\right )^{2} {\mathrm e}^{2 x +6}+2 \ln \left (5\right )^{16} \ln \left (3\right ) {\mathrm e}^{2 x +6} x +\ln \left (5\right )^{16} {\mathrm e}^{2 x +6} x^{2}-4 \ln \left (5\right )^{8} \ln \left (3\right ) {\mathrm e}^{3+x} x -4 \ln \left (5\right )^{8} {\mathrm e}^{3+x} x^{2}+4 x^{2}\) | \(77\) |
parts | \(\ln \left (5\right )^{16} \ln \left (3\right )^{2} {\mathrm e}^{2 x +6}+2 \ln \left (5\right )^{16} \ln \left (3\right ) {\mathrm e}^{2 x +6} x +\ln \left (5\right )^{16} {\mathrm e}^{2 x +6} x^{2}-4 \ln \left (5\right )^{8} \ln \left (3\right ) {\mathrm e}^{3+x} x -4 \ln \left (5\right )^{8} {\mathrm e}^{3+x} x^{2}+4 x^{2}\) | \(77\) |
derivativedivides | \(-72-24 x +\ln \left (5\right )^{8} \left (24 \,{\mathrm e}^{3+x} \left (3+x \right )-36 \,{\mathrm e}^{3+x}-4 \,{\mathrm e}^{3+x} \left (3+x \right )^{2}+8 \,{\mathrm e}^{3+x} \ln \left (3\right )-4 \ln \left (3\right ) \left ({\mathrm e}^{3+x} \left (3+x \right )-{\mathrm e}^{3+x}\right )\right )+\ln \left (5\right )^{16} \left (9 \,{\mathrm e}^{2 x +6}-6 \,{\mathrm e}^{2 x +6} \left (3+x \right )+{\mathrm e}^{2 x +6} \left (3+x \right )^{2}-6 \,{\mathrm e}^{2 x +6} \ln \left (3\right )+{\mathrm e}^{2 x +6} \ln \left (3\right )^{2}+2 \,{\mathrm e}^{2 x +6} \ln \left (3\right ) \left (3+x \right )\right )+4 \left (3+x \right )^{2}\) | \(143\) |
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx={\left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )} e^{\left (2 \, x + 6\right )} \log \left (5\right )^{16} - 4 \, {\left (x^{2} + x \log \left (3\right )\right )} e^{\left (x + 3\right )} \log \left (5\right )^{8} + 4 \, x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (19) = 38\).
Time = 0.11 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.50 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx=4 x^{2} + \left (- 4 x^{2} \log {\left (5 \right )}^{8} - 4 x \log {\left (3 \right )} \log {\left (5 \right )}^{8}\right ) e^{x + 3} + \left (x^{2} \log {\left (5 \right )}^{16} + 2 x \log {\left (3 \right )} \log {\left (5 \right )}^{16} + \log {\left (3 \right )}^{2} \log {\left (5 \right )}^{16}\right ) e^{2 x + 6} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (19) = 38\).
Time = 0.33 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.80 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx={\left (x^{2} e^{6} + 2 \, x e^{6} \log \left (3\right ) + e^{6} \log \left (3\right )^{2}\right )} e^{\left (2 \, x\right )} \log \left (5\right )^{16} - 4 \, {\left ({\left (x e^{3} - e^{3}\right )} e^{x} \log \left (3\right ) + {\left (x^{2} e^{3} - 2 \, x e^{3} + 2 \, e^{3}\right )} e^{x} + 2 \, {\left (x e^{3} - e^{3}\right )} e^{x} + e^{\left (x + 3\right )} \log \left (3\right )\right )} \log \left (5\right )^{8} + 4 \, x^{2} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (19) = 38\).
Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.40 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx={\left (x^{2} + 2 \, x \log \left (3\right ) + \log \left (3\right )^{2}\right )} e^{\left (2 \, x + 6\right )} \log \left (5\right )^{16} - 4 \, {\left (x^{2} + x \log \left (3\right )\right )} e^{\left (x + 3\right )} \log \left (5\right )^{8} + 4 \, x^{2} \]
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Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.35 \[ \int \left (8 x+e^{3+x} \left (-8 x-4 x^2+(-4-4 x) \log (3)\right ) \log ^8(5)+e^{6+2 x} \left (2 x+2 x^2+(2+4 x) \log (3)+2 \log ^2(3)\right ) \log ^{16}(5)\right ) \, dx={\left ({\mathrm {e}}^{x+3}\,\ln \left (3\right )\,{\ln \left (5\right )}^8-2\,x+x\,{\mathrm {e}}^{x+3}\,{\ln \left (5\right )}^8\right )}^2 \]
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