Integrand size = 40, antiderivative size = 21 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=e^4+x+\log \left (3+\frac {x}{x+\frac {x^2}{e^5}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1671, 630, 31} \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x-\log \left (x+e^5\right )+\log \left (3 x+4 e^5\right ) \]
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Rule 31
Rule 630
Rule 1671
Rubi steps \begin{align*} \text {integral}& = \int \left (1-\frac {e^5}{4 e^{10}+7 e^5 x+3 x^2}\right ) \, dx \\ & = x-e^5 \int \frac {1}{4 e^{10}+7 e^5 x+3 x^2} \, dx \\ & = x-3 \int \frac {1}{3 e^5+3 x} \, dx+3 \int \frac {1}{4 e^5+3 x} \, dx \\ & = x-\log \left (e^5+x\right )+\log \left (4 e^5+3 x\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x-\log \left (e^5+x\right )+\log \left (4 e^5+3 x\right ) \]
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Time = 0.87 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(x -\ln \left ({\mathrm e}^{5}+x \right )+\ln \left (x +\frac {4 \,{\mathrm e}^{5}}{3}\right )\) | \(17\) |
norman | \(x -\ln \left ({\mathrm e}^{5}+x \right )+\ln \left (4 \,{\mathrm e}^{5}+3 x \right )\) | \(19\) |
risch | \(x -\ln \left ({\mathrm e}^{5}+x \right )+\ln \left (4 \,{\mathrm e}^{5}+3 x \right )\) | \(19\) |
default | \(x +\frac {2 \,{\mathrm e}^{5} \operatorname {arctanh}\left (\frac {7 \,{\mathrm e}^{5}+6 x}{\sqrt {{\mathrm e}^{10}}}\right )}{\sqrt {{\mathrm e}^{10}}}\) | \(43\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x + \log \left (3 \, x + 4 \, e^{5}\right ) - \log \left (x + e^{5}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x - \log {\left (x + e^{5} \right )} + \log {\left (x + \frac {4 e^{5}}{3} \right )} \]
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Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x + \log \left (3 \, x + 4 \, e^{5}\right ) - \log \left (x + e^{5}\right ) \]
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Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x + \log \left ({\left | 3 \, x + 4 \, e^{5} \right |}\right ) - \log \left ({\left | x + e^{5} \right |}\right ) \]
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Time = 12.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {4 e^{10}+3 x^2+e^5 (-1+7 x)}{4 e^{10}+7 e^5 x+3 x^2} \, dx=x+2\,\mathrm {atanh}\left (6\,x\,{\mathrm {e}}^{-5}+7\right ) \]
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