Optimal. Leaf size=28 \[ 3+x+\frac {3}{1+x+20 e^{-x-x \log (4)} \left (1+x^2\right )} \]
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Rubi [F] time = 37.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (-2+2 x+x^2\right )+e^{x+x \log (4)} \left (100-80 x+100 x^2+40 x^3+\left (60+60 x^2\right ) \log (4)\right )}{400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (1+2 x+x^2\right )+e^{x+x \log (4)} \left (40+40 x+40 x^2+40 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {400+2^{1+4 x} e^{2 x} (-1+x)+\left (800+(4 e)^{2 x}\right ) x^2+400 x^4+5\ 4^{1+x} e^x \left (5-4 x+2 x^3+\log (64)+x^2 (5+\log (64))\right )}{\left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx\\ &=\int \left (\frac {x^2}{(1+x)^2}-\frac {40 x^2 \left (1+x^2\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )}+\frac {2 \left (200-2^{4 x} e^{2 x}+400 x-2^{4 x} e^{2 x} x+800 x^2+2^{4 x} e^{2 x} x^2+800 x^3+2^{4 x} e^{2 x} x^3+1000 x^4+400 x^5+5\ 2^{2+2 x} e^x x^5+400 x^6+45\ 2^{1+2 x} e^x x^4 \left (1+\frac {2 \log (2)}{3}\right )+15\ 2^{2+2 x} e^x x (1+\log (4))+25\ 2^{1+2 x} e^x \left (1+\frac {\log (64)}{5}\right )+5\ 2^{4+2 x} e^x x^3 \left (1+\frac {\log (64)}{4}\right )+5\ 2^{2+2 x} e^x x^2 (1+\log (64))\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {200-2^{4 x} e^{2 x}+400 x-2^{4 x} e^{2 x} x+800 x^2+2^{4 x} e^{2 x} x^2+800 x^3+2^{4 x} e^{2 x} x^3+1000 x^4+400 x^5+5\ 2^{2+2 x} e^x x^5+400 x^6+45\ 2^{1+2 x} e^x x^4 \left (1+\frac {2 \log (2)}{3}\right )+15\ 2^{2+2 x} e^x x (1+\log (4))+25\ 2^{1+2 x} e^x \left (1+\frac {\log (64)}{5}\right )+5\ 2^{4+2 x} e^x x^3 \left (1+\frac {\log (64)}{4}\right )+5\ 2^{2+2 x} e^x x^2 (1+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-40 \int \frac {x^2 \left (1+x^2\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx+\int \frac {x^2}{(1+x)^2} \, dx\\ &=2 \int \frac {16^x e^{2 x} (-1+x) (1+x)^2+200 \left (1+x^2\right )^2 \left (1+2 x+2 x^2\right )+5\ 2^{1+2 x} e^x \left (5+2 x^5+6 x (1+\log (4))+\log (64)+2 x^2 (1+\log (64))+2 x^3 (4+\log (64))+x^4 (9+\log (64))\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-40 \int \left (\frac {4}{20+(4 e)^x+(4 e)^x x+20 x^2}-\frac {2 x}{20+(4 e)^x+(4 e)^x x+20 x^2}+\frac {x^2}{20+(4 e)^x+(4 e)^x x+20 x^2}+\frac {2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )}-\frac {6}{(1+x) \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )}\right ) \, dx+\int \left (1+\frac {1}{(1+x)^2}-\frac {2}{1+x}\right ) \, dx\\ &=x-\frac {1}{1+x}-2 \log (1+x)+2 \int \left (\frac {200}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}-\frac {16^x e^{2 x}}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {400 x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}-\frac {2^{4 x} e^{2 x} x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {800 x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {2^{4 x} e^{2 x} x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {800 x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {16^x e^{2 x} x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {1000 x^4}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {400 x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 4^{1+x} e^x x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {400 x^6}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {15\ 4^{1+x} e^x x (1+\log (4))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {25\ 2^{1+2 x} e^x \left (1+\frac {\log (64)}{5}\right )}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 4^{1+x} e^x x^2 (1+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 4^{1+x} e^x x^3 (4+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}+\frac {5\ 2^{1+2 x} e^x x^4 (9+\log (64))}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2}\right ) \, dx-40 \int \frac {x^2}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+80 \int \frac {x}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx-80 \int \frac {1}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx-160 \int \frac {1}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+240 \int \frac {1}{(1+x) \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx\\ &=x-\frac {1}{1+x}-2 \log (1+x)-2 \int \frac {16^x e^{2 x}}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-2 \int \frac {2^{4 x} e^{2 x} x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+2 \int \frac {2^{4 x} e^{2 x} x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+2 \int \frac {16^x e^{2 x} x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+10 \int \frac {4^{1+x} e^x x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx-40 \int \frac {x^2}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+80 \int \frac {x}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx-80 \int \frac {1}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx-160 \int \frac {1}{20+(4 e)^x+(4 e)^x x+20 x^2} \, dx+240 \int \frac {1}{(1+x) \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )} \, dx+400 \int \frac {1}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+800 \int \frac {x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+800 \int \frac {x^5}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+800 \int \frac {x^6}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+1600 \int \frac {x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+1600 \int \frac {x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+2000 \int \frac {x^4}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(30 (1+\log (4))) \int \frac {4^{1+x} e^x x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (1+\log (64))) \int \frac {4^{1+x} e^x x^2}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (4+\log (64))) \int \frac {4^{1+x} e^x x^3}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (5+\log (64))) \int \frac {2^{1+2 x} e^x}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx+(10 (9+\log (64))) \int \frac {2^{1+2 x} e^x x^4}{(1+x)^2 \left (20+(4 e)^x+(4 e)^x x+20 x^2\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F] time = 7.71, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (-2+2 x+x^2\right )+e^{x+x \log (4)} \left (100-80 x+100 x^2+40 x^3+\left (60+60 x^2\right ) \log (4)\right )}{400+800 x^2+400 x^4+e^{2 x+2 x \log (4)} \left (1+2 x+x^2\right )+e^{x+x \log (4)} \left (40+40 x+40 x^2+40 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.55, size = 46, normalized size = 1.64 \begin {gather*} \frac {20 \, x^{3} + {\left (x^{2} + x + 3\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20 \, x}{20 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.85, size = 69, normalized size = 2.46 \begin {gather*} \frac {20 \, x^{3} + x^{2} e^{\left (2 \, x \log \relax (2) + x\right )} + x e^{\left (2 \, x \log \relax (2) + x\right )} + 20 \, x + 3 \, e^{\left (2 \, x \log \relax (2) + x\right )}}{20 \, x^{2} + x e^{\left (2 \, x \log \relax (2) + x\right )} + e^{\left (2 \, x \log \relax (2) + x\right )} + 20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 44, normalized size = 1.57
method | result | size |
risch | \(x +\frac {3}{x +1}-\frac {60 \left (x^{2}+1\right )}{\left (x +1\right ) \left ({\mathrm e}^{x} 4^{x} x +20 x^{2}+{\mathrm e}^{x} 4^{x}+20\right )}\) | \(44\) |
norman | \(\frac {-20 x^{2}+2 \,{\mathrm e}^{x +2 x \ln \relax (2)}+{\mathrm e}^{x +2 x \ln \relax (2)} x^{2}+20 x +20 x^{3}-20}{{\mathrm e}^{x +2 x \ln \relax (2)} x +20 x^{2}+{\mathrm e}^{x +2 x \ln \relax (2)}+20}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 46, normalized size = 1.64 \begin {gather*} \frac {20 \, x^{3} + {\left (x^{2} + x + 3\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20 \, x}{20 \, x^{2} + {\left (x + 1\right )} e^{\left (2 \, x \log \relax (2) + x\right )} + 20} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{2\,x+4\,x\,\ln \relax (2)}\,\left (x^2+2\,x-2\right )+800\,x^2+400\,x^4+{\mathrm {e}}^{x+2\,x\,\ln \relax (2)}\,\left (2\,\ln \relax (2)\,\left (60\,x^2+60\right )-80\,x+100\,x^2+40\,x^3+100\right )+400}{{\mathrm {e}}^{2\,x+4\,x\,\ln \relax (2)}\,\left (x^2+2\,x+1\right )+{\mathrm {e}}^{x+2\,x\,\ln \relax (2)}\,\left (40\,x^3+40\,x^2+40\,x+40\right )+800\,x^2+400\,x^4+400} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 48, normalized size = 1.71 \begin {gather*} x + \frac {- 60 x^{2} - 60}{20 x^{3} + 20 x^{2} + 20 x + \left (x^{2} + 2 x + 1\right ) e^{x + 2 x \log {\relax (2 )}} + 20} + \frac {3}{x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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