Integrand size = 127, antiderivative size = 25 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=e^{\left (-2+\frac {2+x}{16+5 \left (-5-4 x+\log ^2(2)\right )}\right )^2} \]
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Time = 1.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6820, 12, 6838} \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=\exp \left (\frac {\left (41 x+10 \left (2-\log ^2(2)\right )\right )^2}{\left (20 x+9-5 \log ^2(2)\right )^2}\right ) \]
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Rule 12
Rule 6820
Rule 6838
Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \exp \left (\frac {\left (20+41 x-10 \log ^2(2)\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}\right ) \left (20+41 x-10 \log ^2(2)\right ) \left (-31-5 \log ^2(2)\right )}{\left (9+20 x-5 \log ^2(2)\right )^3} \, dx \\ & = -\left (\left (2 \left (31+5 \log ^2(2)\right )\right ) \int \frac {\exp \left (\frac {\left (20+41 x-10 \log ^2(2)\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}\right ) \left (20+41 x-10 \log ^2(2)\right )}{\left (9+20 x-5 \log ^2(2)\right )^3} \, dx\right ) \\ & = \exp \left (\frac {\left (41 x+10 \left (2-\log ^2(2)\right )\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}\right ) \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=\frac {2 e^{\frac {\left (20+41 x-10 \log ^2(2)\right )^2}{\left (9+20 x-5 \log ^2(2)\right )^2}} \left (31+5 \log ^2(2)\right )}{62+10 \log ^2(2)} \]
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Time = 0.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
method | result | size |
risch | \({\mathrm e}^{\frac {\left (10 \ln \left (2\right )^{2}-41 x -20\right )^{2}}{\left (5 \ln \left (2\right )^{2}-20 x -9\right )^{2}}}\) | \(29\) |
parallelrisch | \({\mathrm e}^{\frac {100 \ln \left (2\right )^{4}+\left (-820 x -400\right ) \ln \left (2\right )^{2}+1681 x^{2}+1640 x +400}{25 \ln \left (2\right )^{4}-200 x \ln \left (2\right )^{2}-90 \ln \left (2\right )^{2}+400 x^{2}+360 x +81}}\) | \(60\) |
gosper | \({\mathrm e}^{\frac {100 \ln \left (2\right )^{4}-820 x \ln \left (2\right )^{2}-400 \ln \left (2\right )^{2}+1681 x^{2}+1640 x +400}{25 \ln \left (2\right )^{4}-200 x \ln \left (2\right )^{2}-90 \ln \left (2\right )^{2}+400 x^{2}+360 x +81}}\) | \(63\) |
norman | \(\frac {\left (25 \ln \left (2\right )^{4}-90 \ln \left (2\right )^{2}+81\right ) {\mathrm e}^{\frac {100 \ln \left (2\right )^{4}+\left (-820 x -400\right ) \ln \left (2\right )^{2}+1681 x^{2}+1640 x +400}{25 \ln \left (2\right )^{4}+\left (-200 x -90\right ) \ln \left (2\right )^{2}+400 x^{2}+360 x +81}}+\left (-200 \ln \left (2\right )^{2}+360\right ) x \,{\mathrm e}^{\frac {100 \ln \left (2\right )^{4}+\left (-820 x -400\right ) \ln \left (2\right )^{2}+1681 x^{2}+1640 x +400}{25 \ln \left (2\right )^{4}+\left (-200 x -90\right ) \ln \left (2\right )^{2}+400 x^{2}+360 x +81}}+400 x^{2} {\mathrm e}^{\frac {100 \ln \left (2\right )^{4}+\left (-820 x -400\right ) \ln \left (2\right )^{2}+1681 x^{2}+1640 x +400}{25 \ln \left (2\right )^{4}+\left (-200 x -90\right ) \ln \left (2\right )^{2}+400 x^{2}+360 x +81}}}{\left (5 \ln \left (2\right )^{2}-20 x -9\right )^{2}}\) | \(214\) |
derivativedivides | \(\text {Expression too large to display}\) | \(875\) |
default | \(\text {Expression too large to display}\) | \(875\) |
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=e^{\left (\frac {100 \, \log \left (2\right )^{4} - 20 \, {\left (41 \, x + 20\right )} \log \left (2\right )^{2} + 1681 \, x^{2} + 1640 \, x + 400}{25 \, \log \left (2\right )^{4} - 10 \, {\left (20 \, x + 9\right )} \log \left (2\right )^{2} + 400 \, x^{2} + 360 \, x + 81}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (19) = 38\).
Time = 0.46 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=e^{\frac {1681 x^{2} + 1640 x + \left (- 820 x - 400\right ) \log {\left (2 \right )}^{2} + 100 \log {\left (2 \right )}^{4} + 400}{400 x^{2} + 360 x + \left (- 200 x - 90\right ) \log {\left (2 \right )}^{2} + 25 \log {\left (2 \right )}^{4} + 81}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (22) = 44\).
Time = 46.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 5.88 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=e^{\left (\frac {\log \left (2\right )^{4}}{16 \, {\left (25 \, \log \left (2\right )^{4} - 40 \, {\left (5 \, \log \left (2\right )^{2} - 9\right )} x + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 81\right )}} + \frac {31 \, \log \left (2\right )^{2}}{40 \, {\left (25 \, \log \left (2\right )^{4} - 40 \, {\left (5 \, \log \left (2\right )^{2} - 9\right )} x + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 81\right )}} - \frac {41 \, \log \left (2\right )^{2}}{40 \, {\left (5 \, \log \left (2\right )^{2} - 20 \, x - 9\right )}} + \frac {961}{400 \, {\left (25 \, \log \left (2\right )^{4} - 40 \, {\left (5 \, \log \left (2\right )^{2} - 9\right )} x + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 81\right )}} - \frac {1271}{200 \, {\left (5 \, \log \left (2\right )^{2} - 20 \, x - 9\right )}} + \frac {1681}{400}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 217, normalized size of antiderivative = 8.68 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx=e^{\left (\frac {100 \, \log \left (2\right )^{4}}{25 \, \log \left (2\right )^{4} - 200 \, x \log \left (2\right )^{2} + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 360 \, x + 81} - \frac {820 \, x \log \left (2\right )^{2}}{25 \, \log \left (2\right )^{4} - 200 \, x \log \left (2\right )^{2} + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 360 \, x + 81} + \frac {1681 \, x^{2}}{25 \, \log \left (2\right )^{4} - 200 \, x \log \left (2\right )^{2} + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 360 \, x + 81} - \frac {400 \, \log \left (2\right )^{2}}{25 \, \log \left (2\right )^{4} - 200 \, x \log \left (2\right )^{2} + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 360 \, x + 81} + \frac {1640 \, x}{25 \, \log \left (2\right )^{4} - 200 \, x \log \left (2\right )^{2} + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 360 \, x + 81} + \frac {400}{25 \, \log \left (2\right )^{4} - 200 \, x \log \left (2\right )^{2} + 400 \, x^{2} - 90 \, \log \left (2\right )^{2} + 360 \, x + 81}\right )} \]
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Time = 13.34 (sec) , antiderivative size = 222, normalized size of antiderivative = 8.88 \[ \int \frac {e^{\frac {400+1640 x+1681 x^2+(-400-820 x) \log ^2(2)+100 \log ^4(2)}{81+360 x+400 x^2+(-90-200 x) \log ^2(2)+25 \log ^4(2)}} \left (1240+2542 x+(-420+410 x) \log ^2(2)-100 \log ^4(2)\right )}{-729-4860 x-10800 x^2-8000 x^3+\left (1215+5400 x+6000 x^2\right ) \log ^2(2)+(-675-1500 x) \log ^4(2)+125 \log ^6(2)} \, dx={\mathrm {e}}^{\frac {1640\,x}{360\,x-200\,x\,{\ln \left (2\right )}^2-90\,{\ln \left (2\right )}^2+25\,{\ln \left (2\right )}^4+400\,x^2+81}}\,{\mathrm {e}}^{\frac {100\,{\ln \left (2\right )}^4}{360\,x-200\,x\,{\ln \left (2\right )}^2-90\,{\ln \left (2\right )}^2+25\,{\ln \left (2\right )}^4+400\,x^2+81}}\,{\mathrm {e}}^{-\frac {400\,{\ln \left (2\right )}^2}{360\,x-200\,x\,{\ln \left (2\right )}^2-90\,{\ln \left (2\right )}^2+25\,{\ln \left (2\right )}^4+400\,x^2+81}}\,{\mathrm {e}}^{\frac {1681\,x^2}{360\,x-200\,x\,{\ln \left (2\right )}^2-90\,{\ln \left (2\right )}^2+25\,{\ln \left (2\right )}^4+400\,x^2+81}}\,{\mathrm {e}}^{\frac {400}{360\,x-200\,x\,{\ln \left (2\right )}^2-90\,{\ln \left (2\right )}^2+25\,{\ln \left (2\right )}^4+400\,x^2+81}}\,{\mathrm {e}}^{-\frac {820\,x\,{\ln \left (2\right )}^2}{360\,x-200\,x\,{\ln \left (2\right )}^2-90\,{\ln \left (2\right )}^2+25\,{\ln \left (2\right )}^4+400\,x^2+81}} \]
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