Integrand size = 84, antiderivative size = 26 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=e^x-2 \left (e^4+\frac {3}{x \left (e^{x/21}+x\right )}\right ) \]
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\[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=\int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 x^2 \left (e^{x/21}+x\right )^2} \, dx \\ & = \frac {1}{7} \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{x^2 \left (e^{x/21}+x\right )^2} \, dx \\ & = \frac {1}{7} \int \left (7 e^x-\frac {2 (-21+x)}{x \left (e^{x/21}+x\right )^2}+\frac {2 (21+x)}{x^2 \left (e^{x/21}+x\right )}\right ) \, dx \\ & = -\left (\frac {2}{7} \int \frac {-21+x}{x \left (e^{x/21}+x\right )^2} \, dx\right )+\frac {2}{7} \int \frac {21+x}{x^2 \left (e^{x/21}+x\right )} \, dx+\int e^x \, dx \\ & = e^x-\frac {2}{7} \int \left (\frac {1}{\left (e^{x/21}+x\right )^2}-\frac {21}{x \left (e^{x/21}+x\right )^2}\right ) \, dx+\frac {2}{7} \int \left (\frac {21}{x^2 \left (e^{x/21}+x\right )}+\frac {1}{x \left (e^{x/21}+x\right )}\right ) \, dx \\ & = e^x-\frac {2}{7} \int \frac {1}{\left (e^{x/21}+x\right )^2} \, dx+\frac {2}{7} \int \frac {1}{x \left (e^{x/21}+x\right )} \, dx+6 \int \frac {1}{x \left (e^{x/21}+x\right )^2} \, dx+6 \int \frac {1}{x^2 \left (e^{x/21}+x\right )} \, dx \\ & = e^x+\frac {2}{7} \int \frac {1}{x \left (e^{x/21}+x\right )} \, dx+6 \int \frac {1}{x \left (e^{x/21}+x\right )^2} \, dx+6 \int \frac {1}{x^2 \left (e^{x/21}+x\right )} \, dx-6 \text {Subst}\left (\int \frac {1}{\left (e^x+21 x\right )^2} \, dx,x,\frac {x}{21}\right ) \\ \end{align*}
Time = 1.43 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=\frac {1}{7} \left (7 e^x-\frac {42}{x \left (e^{x/21}+x\right )}\right ) \]
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Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65
method | result | size |
risch | \({\mathrm e}^{x}-\frac {6}{x \left ({\mathrm e}^{\frac {x}{21}}+x \right )}\) | \(17\) |
parts | \({\mathrm e}^{x}-\frac {6}{x \left ({\mathrm e}^{\frac {x}{21}}+x \right )}\) | \(17\) |
parallelrisch | \(\frac {7 \,{\mathrm e}^{x} x^{2}-42+7 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {x}{21}}}{7 x \left ({\mathrm e}^{\frac {x}{21}}+x \right )}\) | \(32\) |
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Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=\frac {x^{2} e^{x} + x e^{\left (\frac {22}{21} \, x\right )} - 6}{x^{2} + x e^{\left (\frac {1}{21} \, x\right )}} \]
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Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=e^{x} - \frac {6}{x^{2} + x e^{\frac {x}{21}}} \]
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Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=\frac {x^{2} e^{x} + x e^{\left (\frac {22}{21} \, x\right )} - 6}{x^{2} + x e^{\left (\frac {1}{21} \, x\right )}} \]
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Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx=\frac {x^{2} e^{x} + x e^{\left (\frac {22}{21} \, x\right )} - 6}{x^{2} + x e^{\left (\frac {1}{21} \, x\right )}} \]
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Time = 8.16 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.62 \[ \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx={\mathrm {e}}^x-\frac {6}{x\,\left (x+{\mathrm {e}}^{x/21}\right )} \]
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