Integrand size = 71, antiderivative size = 71 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (1+x+\sqrt {2} x+\sqrt {2} x^2-x^7\right )-\left (-1+\sqrt {2}\right ) \log \left (-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right )\right ) \]
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\[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+\frac {1}{14} \int \frac {28+56 x+28 x^2+168 x^7+140 x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \\ & = \frac {1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+\frac {1}{14} \int \frac {28 \left (1+2 x+x^2+6 x^7+5 x^8\right )}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \\ & = \frac {1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+2 \int \frac {1+2 x+x^2+6 x^7+5 x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \\ & = \frac {1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+2 \int \left (\frac {1}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac {2 x}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac {x^2}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac {6 x^7}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}+\frac {5 x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}}\right ) \, dx \\ & = \frac {1}{2} \log \left (1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}\right )+2 \int \frac {1}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+2 \int \frac {x^2}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+4 \int \frac {x}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+10 \int \frac {x^8}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx+12 \int \frac {x^7}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (1+x+\sqrt {2} x+\sqrt {2} x^2-x^7\right )-\left (-1+\sqrt {2}\right ) \log \left (-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right )\right ) \]
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Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86
method | result | size |
default | \(\left (\frac {1}{2}+\frac {\sqrt {2}}{2}\right ) \ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )+\left (-\frac {\sqrt {2}}{2}+\frac {1}{2}\right ) \ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )\) | \(61\) |
risch | \(\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )}{2}+\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right ) \sqrt {2}}{2}-\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right ) \sqrt {2}}{2}+\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )}{2}\) | \(102\) |
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Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (56) = 112\).
Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.93 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{14} - 2 \, x^{8} - 2 \, x^{7} + 2 \, x^{4} + 4 \, x^{3} + 3 \, x^{2} - 2 \, \sqrt {2} {\left (x^{9} + x^{8} - x^{3} - 2 \, x^{2} - x\right )} + 2 \, x + 1}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\right ) + \frac {1}{2} \, \log \left (x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right ) \]
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Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} - \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) - 1 \right )} + \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} + \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) - 1 \right )} \]
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\[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\int { \frac {7 \, x^{13} + 10 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 4 \, x^{3} - 4 \, x^{2} + 3 \, x + 3}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} + \sqrt {2} x^{2} + \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} - \sqrt {2} x^{2} - \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1 \right |}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.45 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2}-\frac {\sqrt {2}\,\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\sqrt {2}\,\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2} \]
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