Optimal. Leaf size=31 \[ \frac {1-\frac {e^2}{x}-x^4}{1+2 e^4 x-x^2} \]
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Rubi [B] time = 0.23, antiderivative size = 65, normalized size of antiderivative = 2.10, number of steps used = 8, number of rules used = 5, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {6, 2074, 638, 618, 206} \begin {gather*} x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{-x^2+2 e^4 x+1}+2 e^4 x-\frac {e^2}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 206
Rule 618
Rule 638
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^6 x+2 x^3-4 x^5+2 x^7+e^2 \left (1-3 x^2\right )+e^4 \left (-2 x^2-6 x^6\right )}{x^2+\left (-2+4 e^8\right ) x^4+x^6+e^4 \left (4 x^3-4 x^5\right )} \, dx\\ &=\int \left (2 e^4+\frac {e^2}{x^2}+2 x+\frac {2 \left (-e^2 \left (1+4 e^2+2 e^8+4 e^{10}\right )+e^6 \left (1-8 e^2-8 e^{10}\right ) x\right )}{\left (1+2 e^4 x-x^2\right )^2}+\frac {e^2+4 e^4+8 e^{12}}{1+2 e^4 x-x^2}\right ) \, dx\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+2 \int \frac {-e^2 \left (1+4 e^2+2 e^8+4 e^{10}\right )+e^6 \left (1-8 e^2-8 e^{10}\right ) x}{\left (1+2 e^4 x-x^2\right )^2} \, dx+\left (e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \int \frac {1}{1+2 e^4 x-x^2} \, dx\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{1+2 e^4 x-x^2}-\left (e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \int \frac {1}{1+2 e^4 x-x^2} \, dx-\left (2 e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+e^8\right )-x^2} \, dx,x,2 e^4-2 x\right )\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{1+2 e^4 x-x^2}-\frac {e^2 \left (1+4 e^2+8 e^{10}\right ) \tanh ^{-1}\left (\frac {e^4-x}{\sqrt {1+e^8}}\right )}{\sqrt {1+e^8}}+\left (2 e^2 \left (1+4 e^2+8 e^{10}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (1+e^8\right )-x^2} \, dx,x,2 e^4-2 x\right )\\ &=-\frac {e^2}{x}+2 e^4 x+x^2+\frac {e^2 \left (2 e^4 \left (1-2 e^2\right )-\left (1+4 e^2+8 e^{10}\right ) x\right )}{1+2 e^4 x-x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 57, normalized size = 1.84 \begin {gather*} \frac {e^2+2 e^4 x^2+8 e^{12} x^2-4 e^8 x \left (-1+x^2\right )+x^3 \left (-1+x^2\right )}{x \left (-1-2 e^4 x+x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 53, normalized size = 1.71 \begin {gather*} \frac {x^{5} - x^{3} + 8 \, x^{2} e^{12} + 2 \, x^{2} e^{4} - 4 \, {\left (x^{3} - x\right )} e^{8} + e^{2}}{x^{3} - 2 \, x^{2} e^{4} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 31, normalized size = 1.00
method | result | size |
gosper | \(-\frac {x^{5}+{\mathrm e}^{2}-x}{x \left (2 x \,{\mathrm e}^{4}-x^{2}+1\right )}\) | \(31\) |
norman | \(\frac {-x^{5}+x -{\mathrm e}^{2}}{x \left (2 x \,{\mathrm e}^{4}-x^{2}+1\right )}\) | \(32\) |
risch | \(2 x \,{\mathrm e}^{4}+x^{2}+\frac {-4 \left (2 \,{\mathrm e}^{8}+1\right ) {\mathrm e}^{4} x^{2}-4 x \,{\mathrm e}^{8}-{\mathrm e}^{2}}{x \left (2 x \,{\mathrm e}^{4}-x^{2}+1\right )}\) | \(52\) |
default | \(2 x \,{\mathrm e}^{4}+x^{2}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (1+\textit {\_Z}^{4}-4 \textit {\_Z}^{3} {\mathrm e}^{4}+\left (4 \,{\mathrm e}^{8}-2\right ) \textit {\_Z}^{2}+4 \textit {\_Z} \,{\mathrm e}^{4}\right )}{\sum }\frac {\left (\left (-8 \,{\mathrm e}^{12}-{\mathrm e}^{2}-4 \,{\mathrm e}^{4}\right ) \textit {\_R}^{2}+4 \left (-2 \,{\mathrm e}^{8}+{\mathrm e}^{6}\right ) \textit {\_R} -{\mathrm e}^{2}-4 \,{\mathrm e}^{4}-4 \,{\mathrm e}^{10}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R} \,{\mathrm e}^{8}-3 \textit {\_R}^{2} {\mathrm e}^{4}+\textit {\_R}^{3}+{\mathrm e}^{4}-\textit {\_R}}\right )}{4}-\frac {{\mathrm e}^{2}}{x}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 46, normalized size = 1.48 \begin {gather*} x^{2} + 2 \, x e^{4} + \frac {4 \, x^{2} {\left (2 \, e^{12} + e^{4}\right )} + 4 \, x e^{8} + e^{2}}{x^{3} - 2 \, x^{2} e^{4} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 48, normalized size = 1.55 \begin {gather*} 2\,x\,{\mathrm {e}}^4-\frac {\left (4\,{\mathrm {e}}^4+8\,{\mathrm {e}}^{12}\right )\,x^2+4\,{\mathrm {e}}^8\,x+{\mathrm {e}}^2}{-x^3+2\,{\mathrm {e}}^4\,x^2+x}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.86, size = 46, normalized size = 1.48 \begin {gather*} x^{2} + 2 x e^{4} + \frac {x^{2} \left (4 e^{4} + 8 e^{12}\right ) + 4 x e^{8} + e^{2}}{x^{3} - 2 x^{2} e^{4} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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