Integrand size = 38, antiderivative size = 20 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=\frac {9 x}{-5+\frac {1}{16} \left (-5+e^{2 x/25}\right )} \]
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Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(20)=40\).
Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 4.90, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.342, Rules used = {6873, 12, 6874, 2216, 2215, 2221, 2317, 2438, 2222, 2320, 36, 31, 29} \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=-\frac {144 x^2}{2125}+\frac {36 (25-2 x)^2}{2125}-\frac {144 x}{85-e^{2 x/25}}+\frac {144 x}{85}+\frac {72}{85} (25-2 x) \log \left (1-\frac {1}{85} e^{2 x/25}\right )-\frac {360}{17} \log \left (85-e^{2 x/25}\right )+\frac {144}{85} x \log \left (1-\frac {1}{85} e^{2 x/25}\right ) \]
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Rule 12
Rule 29
Rule 31
Rule 36
Rule 2215
Rule 2216
Rule 2221
Rule 2222
Rule 2317
Rule 2320
Rule 2438
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-306000+e^{2 x/25} (3600-288 x)}{25 \left (85-e^{2 x/25}\right )^2} \, dx \\ & = \frac {1}{25} \int \frac {-306000+e^{2 x/25} (3600-288 x)}{\left (85-e^{2 x/25}\right )^2} \, dx \\ & = \frac {1}{25} \int \left (-\frac {24480 x}{\left (-85+e^{2 x/25}\right )^2}-\frac {144 (-25+2 x)}{-85+e^{2 x/25}}\right ) \, dx \\ & = -\left (\frac {144}{25} \int \frac {-25+2 x}{-85+e^{2 x/25}} \, dx\right )-\frac {4896}{5} \int \frac {x}{\left (-85+e^{2 x/25}\right )^2} \, dx \\ & = \frac {36 (25-2 x)^2}{2125}-\frac {144 \int \frac {e^{2 x/25} (-25+2 x)}{-85+e^{2 x/25}} \, dx}{2125}-\frac {288}{25} \int \frac {e^{2 x/25} x}{\left (-85+e^{2 x/25}\right )^2} \, dx+\frac {288}{25} \int \frac {x}{-85+e^{2 x/25}} \, dx \\ & = \frac {36 (25-2 x)^2}{2125}-\frac {144 x}{85-e^{2 x/25}}-\frac {144 x^2}{2125}+\frac {72}{85} (25-2 x) \log \left (1-\frac {1}{85} e^{2 x/25}\right )+\frac {288 \int \frac {e^{2 x/25} x}{-85+e^{2 x/25}} \, dx}{2125}+\frac {144}{85} \int \log \left (1-\frac {1}{85} e^{2 x/25}\right ) \, dx-144 \int \frac {1}{-85+e^{2 x/25}} \, dx \\ & = \frac {36 (25-2 x)^2}{2125}-\frac {144 x}{85-e^{2 x/25}}-\frac {144 x^2}{2125}+\frac {72}{85} (25-2 x) \log \left (1-\frac {1}{85} e^{2 x/25}\right )+\frac {144}{85} x \log \left (1-\frac {1}{85} e^{2 x/25}\right )-\frac {144}{85} \int \log \left (1-\frac {1}{85} e^{2 x/25}\right ) \, dx+\frac {360}{17} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{85}\right )}{x} \, dx,x,e^{2 x/25}\right )-1800 \text {Subst}\left (\int \frac {1}{(-85+x) x} \, dx,x,e^{2 x/25}\right ) \\ & = \frac {36 (25-2 x)^2}{2125}-\frac {144 x}{85-e^{2 x/25}}-\frac {144 x^2}{2125}+\frac {72}{85} (25-2 x) \log \left (1-\frac {1}{85} e^{2 x/25}\right )+\frac {144}{85} x \log \left (1-\frac {1}{85} e^{2 x/25}\right )-\frac {360}{17} \operatorname {PolyLog}\left (2,\frac {1}{85} e^{2 x/25}\right )-\frac {360}{17} \text {Subst}\left (\int \frac {1}{-85+x} \, dx,x,e^{2 x/25}\right )+\frac {360}{17} \text {Subst}\left (\int \frac {1}{x} \, dx,x,e^{2 x/25}\right )-\frac {360}{17} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{85}\right )}{x} \, dx,x,e^{2 x/25}\right ) \\ & = \frac {36 (25-2 x)^2}{2125}+\frac {144 x}{85}-\frac {144 x}{85-e^{2 x/25}}-\frac {144 x^2}{2125}-\frac {360}{17} \log \left (85-e^{2 x/25}\right )+\frac {72}{85} (25-2 x) \log \left (1-\frac {1}{85} e^{2 x/25}\right )+\frac {144}{85} x \log \left (1-\frac {1}{85} e^{2 x/25}\right ) \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=\frac {144 x}{-85+e^{2 x/25}} \]
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Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60
method | result | size |
risch | \(\frac {144 x}{{\mathrm e}^{\frac {2 x}{25}}-85}\) | \(12\) |
norman | \(\frac {144 x}{{\mathrm e}^{\frac {2 x}{25}}-85}\) | \(14\) |
parallelrisch | \(\frac {144 x}{{\mathrm e}^{\frac {2 x}{25}}-85}\) | \(14\) |
derivativedivides | \(-\frac {720 \ln \left ({\mathrm e}^{\frac {x}{25}}\right )}{17}+\frac {144 x \,{\mathrm e}^{\frac {2 x}{25}}}{85 \left ({\mathrm e}^{\frac {2 x}{25}}-85\right )}\) | \(28\) |
default | \(-\frac {720 \ln \left ({\mathrm e}^{\frac {x}{25}}\right )}{17}+\frac {144 x \,{\mathrm e}^{\frac {2 x}{25}}}{85 \left ({\mathrm e}^{\frac {2 x}{25}}-85\right )}\) | \(28\) |
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Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=\frac {144 \, x}{e^{\left (\frac {2}{25} \, x\right )} - 85} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.50 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=\frac {144 x}{e^{\frac {2 x}{25}} - 85} \]
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Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=-\frac {144}{85} \, x + \frac {144 \, x e^{\left (\frac {2}{25} \, x\right )}}{85 \, {\left (e^{\left (\frac {2}{25} \, x\right )} - 85\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=\frac {144 \, x}{e^{\left (\frac {2}{25} \, x\right )} - 85} \]
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Time = 9.26 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.55 \[ \int \frac {-306000+e^{2 x/25} (3600-288 x)}{180625-4250 e^{2 x/25}+25 e^{4 x/25}} \, dx=\frac {144\,x}{{\mathrm {e}}^{\frac {2\,x}{25}}-85} \]
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