Integrand size = 65, antiderivative size = 19 \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=\left (1+e^{-\frac {1}{2} x \left (\frac {625}{x^8}+\log (x)\right )}\right ) x \]
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Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(19)=38\).
Time = 0.93 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.53, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 6874, 6820, 8, 2326} \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=x-\frac {e^{-\frac {625}{2 x^7}} x^{-\frac {x}{2}-7} \left (-x^8+x^8 (-\log (x))+4375\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (x^8 \log (x)+625\right )}{x^8}} \]
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Rule 8
Rule 12
Rule 2326
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{x^7} \, dx \\ & = \frac {1}{2} \int \left (2 e^{\frac {625}{2 x^7}-\frac {625+x^8 \log (x)}{2 x^7}} x^{x/2}+\frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7-x^8-x^8 \log (x)\right )}{x^7}\right ) \, dx \\ & = \frac {1}{2} \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7-x^8-x^8 \log (x)\right )}{x^7} \, dx+\int e^{\frac {625}{2 x^7}-\frac {625+x^8 \log (x)}{2 x^7}} x^{x/2} \, dx \\ & = -\frac {e^{-\frac {625}{2 x^7}} x^{-7-\frac {x}{2}} \left (4375-x^8-x^8 \log (x)\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (625+x^8 \log (x)\right )}{x^8}}+\int 1 \, dx \\ & = x-\frac {e^{-\frac {625}{2 x^7}} x^{-7-\frac {x}{2}} \left (4375-x^8-x^8 \log (x)\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (625+x^8 \log (x)\right )}{x^8}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=\frac {1}{2} \left (2 x+2 e^{-\frac {625}{2 x^7}} x^{1-\frac {x}{2}}\right ) \]
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Time = 1.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
risch | \(x +x \,x^{-\frac {x}{2}} {\mathrm e}^{-\frac {625}{2 x^{7}}}\) | \(18\) |
parallelrisch | \(-\frac {\left (-2 \,{\mathrm e}^{\frac {x^{8} \ln \left (x \right )+625}{2 x^{7}}} x^{8}-2 x^{8}\right ) {\mathrm e}^{-\frac {x^{8} \ln \left (x \right )+625}{2 x^{7}}}}{2 x^{7}}\) | \(47\) |
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Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx={\left (x e^{\left (\frac {x^{8} \log \left (x\right ) + 625}{2 \, x^{7}}\right )} + x\right )} e^{\left (-\frac {x^{8} \log \left (x\right ) + 625}{2 \, x^{7}}\right )} \]
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Time = 0.14 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=x + x e^{- \frac {\frac {x^{8} \log {\left (x \right )}}{2} + \frac {625}{2}}{x^{7}}} \]
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none
Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=x e^{\left (-\frac {1}{2} \, x \log \left (x\right ) - \frac {625}{2 \, x^{7}}\right )} + x \]
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\[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=\int { -\frac {{\left (x^{8} \log \left (x\right ) + x^{8} - 2 \, x^{7} e^{\left (\frac {x^{8} \log \left (x\right ) + 625}{2 \, x^{7}}\right )} - 2 \, x^{7} - 4375\right )} e^{\left (-\frac {x^{8} \log \left (x\right ) + 625}{2 \, x^{7}}\right )}}{2 \, x^{7}} \,d x } \]
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Time = 12.97 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{2 x^7} \, dx=x\,\left ({\mathrm {e}}^{-\frac {x\,\ln \left (x\right )}{2}-\frac {625}{2\,x^7}}+1\right ) \]
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