Integrand size = 96, antiderivative size = 28 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-e^{x+\frac {3}{4+x^2}} x^2+\frac {4+x}{1+x} \]
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\[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=\int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {-3 \left (4+x^2\right )^2-e^{x+\frac {3}{4+x^2}} x (1+x)^2 \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+4 x+x^2+x^3\right )^2} \, dx \\ & = \int \left (-\frac {3}{(1+x)^2}-\frac {e^{x+\frac {3}{4+x^2}} x \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+x^2\right )^2}\right ) \, dx \\ & = \frac {3}{1+x}-\int \frac {e^{x+\frac {3}{4+x^2}} x \left (32+16 x+10 x^2+8 x^3+2 x^4+x^5\right )}{\left (4+x^2\right )^2} \, dx \\ & = \frac {3}{1+x}-\int \left (2 e^{x+\frac {3}{4+x^2}} x+e^{x+\frac {3}{4+x^2}} x^2+\frac {24 e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2}-\frac {6 e^{x+\frac {3}{4+x^2}} x}{4+x^2}\right ) \, dx \\ & = \frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx+6 \int \frac {e^{x+\frac {3}{4+x^2}} x}{4+x^2} \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx \\ & = \frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx+6 \int \left (-\frac {e^{x+\frac {3}{4+x^2}}}{2 (2 i-x)}+\frac {e^{x+\frac {3}{4+x^2}}}{2 (2 i+x)}\right ) \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx \\ & = \frac {3}{1+x}-2 \int e^{x+\frac {3}{4+x^2}} x \, dx-3 \int \frac {e^{x+\frac {3}{4+x^2}}}{2 i-x} \, dx+3 \int \frac {e^{x+\frac {3}{4+x^2}}}{2 i+x} \, dx-24 \int \frac {e^{x+\frac {3}{4+x^2}} x}{\left (4+x^2\right )^2} \, dx-\int e^{x+\frac {3}{4+x^2}} x^2 \, dx \\ \end{align*}
Time = 0.88 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-e^{x+\frac {3}{4+x^2}} x^2+\frac {3}{1+x} \]
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Time = 1.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {3}{1+x}-{\mathrm e}^{\frac {x^{3}+4 x +3}{x^{2}+4}} x^{2}\) | \(31\) |
parallelrisch | \(-\frac {{\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}} x^{3}+x^{2} {\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}}+3 x}{1+x}\) | \(44\) |
parts | \(\frac {3}{1+x}+\frac {-4 x^{2} {\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}}-{\mathrm e}^{x} {\mathrm e}^{\frac {3}{x^{2}+4}} x^{4}}{x^{2}+4}\) | \(52\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-\frac {{\left (x^{3} + x^{2}\right )} e^{\left (\frac {x^{3} + 4 \, x + 3}{x^{2} + 4}\right )} - 3}{x + 1} \]
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Time = 11.75 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.68 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=- x^{2} e^{x} e^{\frac {3}{x^{2} + 4}} + \frac {3}{x + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (27) = 54\).
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.25 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-x^{2} e^{\left (x + \frac {3}{x^{2} + 4}\right )} + \frac {6 \, {\left (11 \, x^{2} - 5 \, x + 24\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} + \frac {3 \, {\left (7 \, x^{2} - 10 \, x - 12\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} - \frac {12 \, {\left (x^{2} - 5 \, x - 16\right )}}{25 \, {\left (x^{3} + x^{2} + 4 \, x + 4\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (27) = 54\).
Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.46 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=-\frac {x^{3} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} + x^{2} e^{\left (\frac {4 \, x^{3} - 3 \, x^{2} + 16 \, x}{4 \, {\left (x^{2} + 4\right )}} + \frac {3}{4}\right )} - 3}{x + 1} \]
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Time = 12.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {-48-24 x^2-3 x^4+e^{x+\frac {3}{4+x^2}} \left (-32 x-80 x^2-74 x^3-44 x^4-28 x^5-13 x^6-4 x^7-x^8\right )}{16+32 x+24 x^2+16 x^3+9 x^4+2 x^5+x^6} \, dx=\frac {3}{x+1}-x^2\,{\mathrm {e}}^{\frac {3}{x^2+4}}\,{\mathrm {e}}^x \]
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