Integrand size = 174, antiderivative size = 34 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=-4+x-\frac {-4+(4-x)^2 x^4}{1-\log \left (4-e^{x^2}\right )} \]
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\[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4-256 x^3+160 x^4-24 x^5-e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )-\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )-\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{\left (4-e^{x^2}\right ) \left (1-\log \left (4-e^{x^2}\right )\right )^2} \, dx \\ & = \int \left (-\frac {8 x \left (-4+16 x^4-8 x^5+x^6\right )}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2}+\frac {1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7-2 \log \left (4-e^{x^2}\right )+64 x^3 \log \left (4-e^{x^2}\right )-40 x^4 \log \left (4-e^{x^2}\right )+6 x^5 \log \left (4-e^{x^2}\right )+\log ^2\left (4-e^{x^2}\right )}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2}\right ) \, dx \\ & = -\left (8 \int \frac {x \left (-4+16 x^4-8 x^5+x^6\right )}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx\right )+\int \frac {1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7-2 \log \left (4-e^{x^2}\right )+64 x^3 \log \left (4-e^{x^2}\right )-40 x^4 \log \left (4-e^{x^2}\right )+6 x^5 \log \left (4-e^{x^2}\right )+\log ^2\left (4-e^{x^2}\right )}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx \\ & = -\left (8 \int \left (-\frac {4 x}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2}+\frac {16 x^5}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2}-\frac {8 x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2}+\frac {x^7}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2}\right ) \, dx\right )+\int \left (1-\frac {2 x \left (-4+16 x^4-8 x^5+x^6\right )}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2}+\frac {2 x^3 \left (32-20 x+3 x^2\right )}{-1+\log \left (4-e^{x^2}\right )}\right ) \, dx \\ & = x-2 \int \frac {x \left (-4+16 x^4-8 x^5+x^6\right )}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx+2 \int \frac {x^3 \left (32-20 x+3 x^2\right )}{-1+\log \left (4-e^{x^2}\right )} \, dx-8 \int \frac {x^7}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx+32 \int \frac {x}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-128 \int \frac {x^5}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx \\ & = x-2 \int \left (-\frac {4 x}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2}+\frac {16 x^5}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2}-\frac {8 x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2}+\frac {x^7}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2}\right ) \, dx+2 \int \left (\frac {32 x^3}{-1+\log \left (4-e^{x^2}\right )}-\frac {20 x^4}{-1+\log \left (4-e^{x^2}\right )}+\frac {3 x^5}{-1+\log \left (4-e^{x^2}\right )}\right ) \, dx-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+16 \text {Subst}\left (\int \frac {1}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = x-2 \int \frac {x^7}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+6 \int \frac {x^5}{-1+\log \left (4-e^{x^2}\right )} \, dx+8 \int \frac {x}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx+16 \int \frac {x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx+16 \text {Subst}\left (\int \frac {1}{(-4+x) x (-1+\log (4-x))^2} \, dx,x,e^{x^2}\right )-32 \int \frac {x^5}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-40 \int \frac {x^4}{-1+\log \left (4-e^{x^2}\right )} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx+64 \int \frac {x^3}{-1+\log \left (4-e^{x^2}\right )} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = x+3 \text {Subst}\left (\int \frac {x^2}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )+4 \text {Subst}\left (\int \frac {1}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+16 \int \frac {x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-16 \text {Subst}\left (\int \frac {x^2}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+16 \text {Subst}\left (\int \left (\frac {1}{4 (-4+x) (-1+\log (4-x))^2}-\frac {1}{4 x (-1+\log (4-x))^2}\right ) \, dx,x,e^{x^2}\right )+32 \text {Subst}\left (\int \frac {x}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-40 \int \frac {x^4}{-1+\log \left (4-e^{x^2}\right )} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {x^3}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = x+3 \text {Subst}\left (\int \frac {x^2}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+4 \text {Subst}\left (\int \frac {1}{(-4+x) (-1+\log (4-x))^2} \, dx,x,e^{x^2}\right )+16 \int \frac {x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-16 \text {Subst}\left (\int \frac {x^2}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+32 \text {Subst}\left (\int \frac {x}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-40 \int \frac {x^4}{-1+\log \left (4-e^{x^2}\right )} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {x^3}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = x+3 \text {Subst}\left (\int \frac {x^2}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+4 \text {Subst}\left (\int \frac {1}{x (-1+\log (x))^2} \, dx,x,4-e^{x^2}\right )+16 \int \frac {x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-16 \text {Subst}\left (\int \frac {x^2}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+32 \text {Subst}\left (\int \frac {x}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-40 \int \frac {x^4}{-1+\log \left (4-e^{x^2}\right )} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {x^3}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = x+3 \text {Subst}\left (\int \frac {x^2}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )+4 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,-1+\log \left (4-e^{x^2}\right )\right )-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+16 \int \frac {x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-16 \text {Subst}\left (\int \frac {x^2}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+32 \text {Subst}\left (\int \frac {x}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-40 \int \frac {x^4}{-1+\log \left (4-e^{x^2}\right )} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {x^3}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ & = x+\frac {4}{1-\log \left (4-e^{x^2}\right )}+3 \text {Subst}\left (\int \frac {x^2}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-4 \text {Subst}\left (\int \frac {x^3}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+16 \int \frac {x^6}{\left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-16 \text {Subst}\left (\int \frac {x^2}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )+32 \text {Subst}\left (\int \frac {x}{-1+\log \left (4-e^x\right )} \, dx,x,x^2\right )-40 \int \frac {x^4}{-1+\log \left (4-e^{x^2}\right )} \, dx+64 \int \frac {x^6}{\left (-4+e^{x^2}\right ) \left (-1+\log \left (4-e^{x^2}\right )\right )^2} \, dx-64 \text {Subst}\left (\int \frac {x^2}{\left (-4+e^x\right ) \left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right )-\text {Subst}\left (\int \frac {x^3}{\left (-1+\log \left (4-e^x\right )\right )^2} \, dx,x,x^2\right ) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=x+\frac {-4+16 x^4-8 x^5+x^6}{-1+\log \left (4-e^{x^2}\right )} \]
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Time = 0.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94
method | result | size |
risch | \(x +\frac {x^{6}-8 x^{5}+16 x^{4}-4}{\ln \left (4-{\mathrm e}^{x^{2}}\right )-1}\) | \(32\) |
parallelrisch | \(\frac {8 x^{6}-64 x^{5}+128 x^{4}+8 x \ln \left (4-{\mathrm e}^{x^{2}}\right )-8 x -32 \ln \left (4-{\mathrm e}^{x^{2}}\right )}{8 \ln \left (4-{\mathrm e}^{x^{2}}\right )-8}\) | \(58\) |
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Time = 0.35 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {x^{6} - 8 \, x^{5} + 16 \, x^{4} + x \log \left (-e^{\left (x^{2}\right )} + 4\right ) - x - 4}{\log \left (-e^{\left (x^{2}\right )} + 4\right ) - 1} \]
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Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=x + \frac {x^{6} - 8 x^{5} + 16 x^{4} - 4}{\log {\left (4 - e^{x^{2}} \right )} - 1} \]
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Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {x^{6} - 8 \, x^{5} + 16 \, x^{4} + x \log \left (-e^{\left (x^{2}\right )} + 4\right ) - x - 4}{\log \left (-e^{\left (x^{2}\right )} + 4\right ) - 1} \]
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Time = 0.33 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=\frac {x^{6} - 8 \, x^{5} + 16 \, x^{4} + x \log \left (-e^{\left (x^{2}\right )} + 4\right ) - x - 4}{\log \left (-e^{\left (x^{2}\right )} + 4\right ) - 1} \]
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Time = 0.32 (sec) , antiderivative size = 169, normalized size of antiderivative = 4.97 \[ \int \frac {-4+256 x^3-160 x^4+24 x^5+e^{x^2} \left (1+8 x-64 x^3+40 x^4-38 x^5+16 x^6-2 x^7\right )+\left (8-256 x^3+160 x^4-24 x^5+e^{x^2} \left (-2+64 x^3-40 x^4+6 x^5\right )\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )}{-4+e^{x^2}+\left (8-2 e^{x^2}\right ) \log \left (4-e^{x^2}\right )+\left (-4+e^{x^2}\right ) \log ^2\left (4-e^{x^2}\right )} \, dx=x-\frac {{\mathrm {e}}^{-x^2}\,\left (4\,{\mathrm {e}}^{x^2}-32\,x^2\,{\mathrm {e}}^{x^2}+20\,x^3\,{\mathrm {e}}^{x^2}-19\,x^4\,{\mathrm {e}}^{x^2}+8\,x^5\,{\mathrm {e}}^{x^2}-x^6\,{\mathrm {e}}^{x^2}+128\,x^2-80\,x^3+12\,x^4\right )+{\mathrm {e}}^{-x^2}\,\ln \left (4-{\mathrm {e}}^{x^2}\right )\,\left ({\mathrm {e}}^{x^2}-4\right )\,\left (3\,x^4-20\,x^3+32\,x^2\right )}{\ln \left (4-{\mathrm {e}}^{x^2}\right )-1}+32\,x^2-20\,x^3+3\,x^4-{\mathrm {e}}^{-x^2}\,\left (12\,x^4-80\,x^3+128\,x^2\right ) \]
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