Integrand size = 31, antiderivative size = 19 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=10-\frac {x}{1+e^x}+5 (-9+3 x) \]
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Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(19)=38\).
Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 3.16, number of steps used = 17, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {6820, 6874, 2215, 2221, 2317, 2438, 2216, 2222, 2320, 36, 29, 31} \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=-\frac {x^2}{2}+\frac {1}{2} (1-x)^2-\frac {x}{e^x+1}+16 x+(1-x) \log \left (e^x+1\right )+x \log \left (e^x+1\right )-\log \left (e^x+1\right ) \]
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Rule 29
Rule 31
Rule 36
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {14+15 e^{2 x}+e^x (29+x)}{\left (1+e^x\right )^2} \, dx \\ & = \int \left (15+\frac {-1+x}{1+e^x}-\frac {x}{\left (1+e^x\right )^2}\right ) \, dx \\ & = 15 x+\int \frac {-1+x}{1+e^x} \, dx-\int \frac {x}{\left (1+e^x\right )^2} \, dx \\ & = \frac {1}{2} (1-x)^2+15 x-\int \frac {e^x (-1+x)}{1+e^x} \, dx+\int \frac {e^x x}{\left (1+e^x\right )^2} \, dx-\int \frac {x}{1+e^x} \, dx \\ & = \frac {1}{2} (1-x)^2+15 x-\frac {x}{1+e^x}-\frac {x^2}{2}+(1-x) \log \left (1+e^x\right )+\int \frac {1}{1+e^x} \, dx+\int \frac {e^x x}{1+e^x} \, dx+\int \log \left (1+e^x\right ) \, dx \\ & = \frac {1}{2} (1-x)^2+15 x-\frac {x}{1+e^x}-\frac {x^2}{2}+(1-x) \log \left (1+e^x\right )+x \log \left (1+e^x\right )-\int \log \left (1+e^x\right ) \, dx+\text {Subst}\left (\int \frac {1}{x (1+x)} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right ) \\ & = \frac {1}{2} (1-x)^2+15 x-\frac {x}{1+e^x}-\frac {x^2}{2}+(1-x) \log \left (1+e^x\right )+x \log \left (1+e^x\right )-\text {Li}_2\left (-e^x\right )+\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {1}{1+x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^x\right ) \\ & = \frac {1}{2} (1-x)^2+16 x-\frac {x}{1+e^x}-\frac {x^2}{2}-\log \left (1+e^x\right )+(1-x) \log \left (1+e^x\right )+x \log \left (1+e^x\right ) \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=15 x-\frac {x}{1+e^x} \]
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Time = 0.32 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74
method | result | size |
risch | \(15 x -\frac {x}{{\mathrm e}^{x}+1}\) | \(14\) |
default | \(\frac {x \,{\mathrm e}^{x}}{{\mathrm e}^{x}+1}+14 \ln \left ({\mathrm e}^{x}\right )\) | \(17\) |
norman | \(\frac {14 x +15 \,{\mathrm e}^{x} x}{{\mathrm e}^{x}+1}\) | \(17\) |
parallelrisch | \(\frac {14 x +15 \,{\mathrm e}^{x} x}{{\mathrm e}^{x}+1}\) | \(17\) |
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Time = 0.24 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=\frac {15 \, x e^{x} + 14 \, x}{e^{x} + 1} \]
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Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.42 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=15 x - \frac {x}{e^{x} + 1} \]
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Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=14 \, x + \frac {x e^{x}}{e^{x} + 1} \]
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Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=\frac {15 \, x e^{x} + 14 \, x}{e^{x} + 1} \]
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Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {14+15 e^{2 x}+e^x (29+x)}{1+2 e^x+e^{2 x}} \, dx=15\,x-\frac {x}{{\mathrm {e}}^x+1} \]
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