Optimal. Leaf size=71 \[ \frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (-x^7+\sqrt {2} x^2+\sqrt {2} x+x+1\right )-\left (\sqrt {2}-1\right ) \log \left (x^7+\sqrt {2} x^2+\left (\sqrt {2}-1\right ) x-1\right )\right ) \]
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Rubi [F] time = 0.75, 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 = {} \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \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} \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*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 1.00 \[ \frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (-x^7+\sqrt {2} x^2+\sqrt {2} x+x+1\right )-\left (\sqrt {2}-1\right ) \log \left (x^7+\sqrt {2} x^2+\left (\sqrt {2}-1\right ) x-1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 137, normalized size = 1.93 \[ \frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{14} - 2 \, x^{8} - 2 \, x^{7} + 2 \, x^{4} + 4 \, x^{3} + 3 \, x^{2} - 2 \, \sqrt {2} {\left (x^{9} + x^{8} - x^{3} - 2 \, x^{2} - x\right )} + 2 \, x + 1}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\right ) + \frac {1}{2} \, \log \left (x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.29, size = 94, normalized size = 1.32 \[ -\frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} + \sqrt {2} x^{2} + \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} - \sqrt {2} x^{2} - \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 102, normalized size = 1.44 \[ \frac {\ln \left (x^{7}+\sqrt {2}\, x^{2}+\left (\sqrt {2}-1\right ) x -1\right )}{2}-\frac {\sqrt {2}\, \ln \left (x^{7}+\sqrt {2}\, x^{2}+\left (\sqrt {2}-1\right ) x -1\right )}{2}+\frac {\ln \left (x^{7}-\sqrt {2}\, x^{2}+\left (-1-\sqrt {2}\right ) x -1\right )}{2}+\frac {\sqrt {2}\, \ln \left (x^{7}-\sqrt {2}\, x^{2}+\left (-1-\sqrt {2}\right ) x -1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {7 \, x^{13} + 10 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 4 \, x^{3} - 4 \, x^{2} + 3 \, x + 3}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 103, normalized size = 1.45 \[ \frac {\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2}-\frac {\sqrt {2}\,\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\sqrt {2}\,\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 76, normalized size = 1.07 \[ \left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} - \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) - 1 \right )} + \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} + \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) - 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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