Optimal. Leaf size=33 \[ \frac {5}{1+x^2 \left (\frac {1+e^{2+x+e^4 x^2}-x}{x}+x\right )} \]
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Rubi [A] time = 0.34, antiderivative size = 29, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, integrand size = 113, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6688, 12, 6686} \begin {gather*} \frac {5}{x^3-x^2+e^{e^4 x^2+x+2} x+x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 \left (-1+2 x-3 x^2-2 e^{6+x+e^4 x^2} x^2-e^{2+x+e^4 x^2} (1+x)\right )}{\left (1+x+e^{2+x+e^4 x^2} x-x^2+x^3\right )^2} \, dx\\ &=5 \int \frac {-1+2 x-3 x^2-2 e^{6+x+e^4 x^2} x^2-e^{2+x+e^4 x^2} (1+x)}{\left (1+x+e^{2+x+e^4 x^2} x-x^2+x^3\right )^2} \, dx\\ &=\frac {5}{1+x+e^{2+x+e^4 x^2} x-x^2+x^3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.04, size = 29, normalized size = 0.88 \begin {gather*} \frac {5}{1+x+e^{2+x+e^4 x^2} x-x^2+x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 27, normalized size = 0.82 \begin {gather*} \frac {5}{x^{3} - x^{2} + x e^{\left (x^{2} e^{4} + x + 2\right )} + x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.32, size = 253, normalized size = 7.67 \begin {gather*} \frac {5 \, {\left (2 \, x^{5} e^{4} - 2 \, x^{4} e^{4} + x^{4} + 2 \, x^{3} e^{4} - 3 \, x^{3} + 2 \, x^{2} e^{4} + 2 \, x^{2} + x + 1\right )}}{2 \, x^{8} e^{4} - 4 \, x^{7} e^{4} + x^{7} + 6 \, x^{6} e^{4} + 2 \, x^{6} e^{\left (x^{2} e^{4} + x + 6\right )} - 4 \, x^{6} - 2 \, x^{5} e^{\left (x^{2} e^{4} + x + 6\right )} + x^{5} e^{\left (x^{2} e^{4} + x + 2\right )} + 6 \, x^{5} - 2 \, x^{4} e^{4} + 2 \, x^{4} e^{\left (x^{2} e^{4} + x + 6\right )} - 3 \, x^{4} e^{\left (x^{2} e^{4} + x + 2\right )} - 3 \, x^{4} + 4 \, x^{3} e^{4} + 2 \, x^{3} e^{\left (x^{2} e^{4} + x + 6\right )} + 2 \, x^{3} e^{\left (x^{2} e^{4} + x + 2\right )} - x^{3} + 2 \, x^{2} e^{4} + x^{2} e^{\left (x^{2} e^{4} + x + 2\right )} + 2 \, x^{2} + x e^{\left (x^{2} e^{4} + x + 2\right )} + 2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 28, normalized size = 0.85
method | result | size |
norman | \(\frac {5}{x^{3}+{\mathrm e}^{x^{2} {\mathrm e}^{4}+2+x} x -x^{2}+x +1}\) | \(28\) |
risch | \(\frac {5}{x^{3}+{\mathrm e}^{x^{2} {\mathrm e}^{4}+2+x} x -x^{2}+x +1}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 27, normalized size = 0.82 \begin {gather*} \frac {5}{x^{3} - x^{2} + x e^{\left (x^{2} e^{4} + x + 2\right )} + x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.75, size = 28, normalized size = 0.85 \begin {gather*} \frac {5}{x-x^2+x^3+x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^4}\,{\mathrm {e}}^2\,{\mathrm {e}}^x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 24, normalized size = 0.73 \begin {gather*} \frac {5}{x^{3} - x^{2} + x e^{x^{2} e^{4} + x + 2} + x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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