Integrand size = 109, antiderivative size = 28 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=x^2 \left (-2+x+\frac {x+\frac {e^x}{\frac {1}{6}+x^2}}{x}\right )^2 \]
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\[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \left (x \left (1-3 x+2 x^2\right ) \left (1+6 x^2\right )^3+36 e^{2 x} \left (1-12 x+6 x^2\right )+6 e^x \left (-1+x+x^2+48 x^4-36 x^5+36 x^6\right )\right )}{\left (1+6 x^2\right )^3} \, dx \\ & = 2 \int \frac {x \left (1-3 x+2 x^2\right ) \left (1+6 x^2\right )^3+36 e^{2 x} \left (1-12 x+6 x^2\right )+6 e^x \left (-1+x+x^2+48 x^4-36 x^5+36 x^6\right )}{\left (1+6 x^2\right )^3} \, dx \\ & = 2 \int \left ((-1+x) x (-1+2 x)+\frac {36 e^{2 x} \left (1-12 x+6 x^2\right )}{\left (1+6 x^2\right )^3}+\frac {6 e^x \left (-1+x+7 x^2-6 x^3+6 x^4\right )}{\left (1+6 x^2\right )^2}\right ) \, dx \\ & = 2 \int (-1+x) x (-1+2 x) \, dx+12 \int \frac {e^x \left (-1+x+7 x^2-6 x^3+6 x^4\right )}{\left (1+6 x^2\right )^2} \, dx+72 \int \frac {e^{2 x} \left (1-12 x+6 x^2\right )}{\left (1+6 x^2\right )^3} \, dx \\ & = (1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+12 \int \left (\frac {e^x}{6}+\frac {2 e^x (-1+x)}{\left (1+6 x^2\right )^2}+\frac {e^x (5-6 x)}{6 \left (1+6 x^2\right )}\right ) \, dx \\ & = (1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+2 \int e^x \, dx+2 \int \frac {e^x (5-6 x)}{1+6 x^2} \, dx+24 \int \frac {e^x (-1+x)}{\left (1+6 x^2\right )^2} \, dx \\ & = 2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+2 \int \left (\frac {\left (5 i+\sqrt {6}\right ) e^x}{2 \left (i-\sqrt {6} x\right )}+\frac {\left (5 i-\sqrt {6}\right ) e^x}{2 \left (i+\sqrt {6} x\right )}\right ) \, dx+24 \int \left (-\frac {e^x}{\left (1+6 x^2\right )^2}+\frac {e^x x}{\left (1+6 x^2\right )^2}\right ) \, dx \\ & = 2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-24 \int \frac {e^x}{\left (1+6 x^2\right )^2} \, dx+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx+\left (5 i-\sqrt {6}\right ) \int \frac {e^x}{i+\sqrt {6} x} \, dx+\left (5 i+\sqrt {6}\right ) \int \frac {e^x}{i-\sqrt {6} x} \, dx \\ & = 2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx-24 \int \left (-\frac {3 e^x}{2 \left (i \sqrt {6}-6 x\right )^2}-\frac {3 e^x}{2 \left (i \sqrt {6}+6 x\right )^2}-\frac {3 e^x}{-6-36 x^2}\right ) \, dx \\ & = 2 e^x+(1-x)^2 x^2+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx+36 \int \frac {e^x}{\left (i \sqrt {6}-6 x\right )^2} \, dx+36 \int \frac {e^x}{\left (i \sqrt {6}+6 x\right )^2} \, dx+72 \int \frac {e^x}{-6-36 x^2} \, dx \\ & = 2 e^x+\frac {6 e^x}{i \sqrt {6}-6 x}+(1-x)^2 x^2-\frac {\sqrt {6} e^x}{i+\sqrt {6} x}+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-6 \int \frac {e^x}{i \sqrt {6}-6 x} \, dx+6 \int \frac {e^x}{i \sqrt {6}+6 x} \, dx+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx+72 \int \left (-\frac {i e^x}{12 \left (i-\sqrt {6} x\right )}-\frac {i e^x}{12 \left (i+\sqrt {6} x\right )}\right ) \, dx \\ & = 2 e^x+\frac {6 e^x}{i \sqrt {6}-6 x}+(1-x)^2 x^2-\frac {\sqrt {6} e^x}{i+\sqrt {6} x}+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-6 i \int \frac {e^x}{i-\sqrt {6} x} \, dx-6 i \int \frac {e^x}{i+\sqrt {6} x} \, dx+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx \\ & = 2 e^x+\frac {6 e^x}{i \sqrt {6}-6 x}+(1-x)^2 x^2-\frac {\sqrt {6} e^x}{i+\sqrt {6} x}+\frac {36 e^{2 x}}{\left (1+6 x^2\right )^2}+e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+i \sqrt {6} e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6+5 i \sqrt {6}\right ) e^{\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (-\frac {i-\sqrt {6} x}{\sqrt {6}}\right )+e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-i \sqrt {6} e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )-\frac {1}{6} \left (6-5 i \sqrt {6}\right ) e^{-\frac {i}{\sqrt {6}}} \operatorname {ExpIntegralEi}\left (\frac {i+\sqrt {6} x}{\sqrt {6}}\right )+24 \int \frac {e^x x}{\left (1+6 x^2\right )^2} \, dx \\ \end{align*}
Time = 6.46 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {\left (6 e^x+x \left (-1+x-6 x^2+6 x^3\right )\right )^2}{\left (1+6 x^2\right )^2} \]
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Time = 0.23 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61
method | result | size |
risch | \(x^{4}-2 x^{3}+x^{2}+\frac {36 \,{\mathrm e}^{2 x}}{\left (6 x^{2}+1\right )^{2}}+\frac {12 x \left (-1+x \right ) {\mathrm e}^{x}}{6 x^{2}+1}\) | \(45\) |
norman | \(\frac {-\frac {10 x^{2}}{3}-2 x^{3}-24 x^{5}+48 x^{6}-72 x^{7}+36 x^{8}+36 \,{\mathrm e}^{2 x}-12 \,{\mathrm e}^{x} x +12 \,{\mathrm e}^{x} x^{2}-72 \,{\mathrm e}^{x} x^{3}+72 \,{\mathrm e}^{x} x^{4}-\frac {13}{36}}{\left (6 x^{2}+1\right )^{2}}\) | \(75\) |
parallelrisch | \(\frac {1296 x^{8}-2592 x^{7}+1728 x^{6}-864 x^{5}+2592 \,{\mathrm e}^{x} x^{4}-13-2592 \,{\mathrm e}^{x} x^{3}-72 x^{3}+432 \,{\mathrm e}^{x} x^{2}-120 x^{2}-432 \,{\mathrm e}^{x} x +1296 \,{\mathrm e}^{2 x}}{1296 x^{4}+432 x^{2}+36}\) | \(81\) |
parts | \(x^{4}-2 x^{3}+x^{2}+\frac {3 \,{\mathrm e}^{2 x} \left (54 x^{3}+6 x^{2}+15 x +1\right )}{36 x^{4}+12 x^{2}+1}-\frac {36 \,{\mathrm e}^{2 x} \left (6 x^{3}+x -1\right )}{36 x^{4}+12 x^{2}+1}+\frac {3 \,{\mathrm e}^{2 x} \left (18 x^{3}-6 x^{2}-3 x -1\right )}{36 x^{4}+12 x^{2}+1}-\frac {2 \,{\mathrm e}^{x}}{6 x^{2}+1}-\frac {12 \,{\mathrm e}^{x} x}{6 x^{2}+1}+2 \,{\mathrm e}^{x}\) | \(142\) |
default | \(x^{4}-2 x^{3}+x^{2}+2 \,{\mathrm e}^{x}+\frac {3 \,{\mathrm e}^{2 x} \left (54 x^{3}+6 x^{2}+15 x +1\right )}{36 x^{4}+12 x^{2}+1}+\frac {3 \,{\mathrm e}^{2 x} \left (18 x^{3}-6 x^{2}-3 x -1\right )}{36 x^{4}+12 x^{2}+1}+\frac {{\mathrm e}^{x} \left (324 x^{3}-6 x^{2}+42 x -1\right )}{864 x^{4}+288 x^{2}+24}-\frac {{\mathrm e}^{x} \left (6 x^{3}+48 x^{2}+x +6\right )}{4 \left (36 x^{4}+12 x^{2}+1\right )}-\frac {36 \,{\mathrm e}^{2 x} \left (6 x^{3}+x -1\right )}{36 x^{4}+12 x^{2}+1}-\frac {{\mathrm e}^{x} \left (108 x^{3}+6 x^{2}+30 x +1\right )}{4 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {{\mathrm e}^{x} \left (6 x^{3}+x -2\right )}{144 x^{4}+48 x^{2}+4}-\frac {108 \left (-\frac {13}{144} x^{3}-\frac {11}{864} x \right )}{\left (6 x^{2}+1\right )^{2}}-\frac {216 \left (\frac {1}{288} x^{3}-\frac {1}{1728} x \right )}{\left (6 x^{2}+1\right )^{2}}+\frac {-\frac {81}{4} x^{3}-\frac {21}{8} x}{\left (6 x^{2}+1\right )^{2}}-\frac {{\mathrm e}^{x} \left (180 x^{3}-6 x^{2}+18 x -1\right )}{3 \left (36 x^{4}+12 x^{2}+1\right )}+\frac {{\mathrm e}^{x} \left (36 x^{3}-6 x^{2}-6 x -1\right )}{864 x^{4}+288 x^{2}+24}-\frac {3888 \left (-\frac {5}{1728} x^{3}-\frac {1}{3456} x \right )}{\left (6 x^{2}+1\right )^{2}}\) | \(384\) |
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (28) = 56\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {36 \, x^{8} - 72 \, x^{7} + 48 \, x^{6} - 24 \, x^{5} + 13 \, x^{4} - 2 \, x^{3} + x^{2} + 12 \, {\left (6 \, x^{4} - 6 \, x^{3} + x^{2} - x\right )} e^{x} + 36 \, e^{\left (2 \, x\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.54 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=x^{4} - 2 x^{3} + x^{2} + \frac {\left (216 x^{2} + 36\right ) e^{2 x} + \left (432 x^{6} - 432 x^{5} + 144 x^{4} - 144 x^{3} + 12 x^{2} - 12 x\right ) e^{x}}{216 x^{6} + 108 x^{4} + 18 x^{2} + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 260 vs. \(2 (28) = 56\).
Time = 0.32 (sec) , antiderivative size = 260, normalized size of antiderivative = 9.29 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=x^{4} - 2 \, x^{3} + x^{2} + \frac {78 \, x^{3} + 11 \, x}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {3 \, {\left (54 \, x^{3} + 7 \, x\right )}}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {9 \, {\left (10 \, x^{3} + x\right )}}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {6 \, x^{3} - x}{8 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {48 \, x^{2} + 7}{36 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {36 \, x^{2} + 5}{6 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} - \frac {5 \, {\left (12 \, x^{2} + 1\right )}}{18 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} + \frac {8 \, x^{2} + 1}{36 \, x^{4} + 12 \, x^{2} + 1} + \frac {12 \, {\left ({\left (6 \, x^{4} - 6 \, x^{3} + x^{2} - x\right )} e^{x} + 3 \, e^{\left (2 \, x\right )}\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} - \frac {1}{12 \, {\left (36 \, x^{4} + 12 \, x^{2} + 1\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
Time = 0.26 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.89 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {36 \, x^{8} - 72 \, x^{7} + 48 \, x^{6} - 24 \, x^{5} + 72 \, x^{4} e^{x} + 13 \, x^{4} - 72 \, x^{3} e^{x} - 2 \, x^{3} + 12 \, x^{2} e^{x} + x^{2} - 12 \, x e^{x} + 36 \, e^{\left (2 \, x\right )}}{36 \, x^{4} + 12 \, x^{2} + 1} \]
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Time = 8.55 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {2 x-6 x^2+40 x^3-108 x^4+288 x^5-648 x^6+864 x^7-1296 x^8+864 x^9+e^{2 x} \left (72-864 x+432 x^2\right )+e^x \left (-12+12 x+12 x^2+576 x^4-432 x^5+432 x^6\right )}{1+18 x^2+108 x^4+216 x^6} \, dx=\frac {{\mathrm {e}}^{2\,x}}{x^4+\frac {x^2}{3}+\frac {1}{36}}+x^2-2\,x^3+x^4-\frac {{\mathrm {e}}^x\,\left (2\,x-2\,x^2\right )}{x^2+\frac {1}{6}} \]
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