Optimal. Leaf size=29 \[ -7-x+\frac {1}{3} \left (4 x+\frac {x}{\frac {e^2}{4 x}+x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.83, number of steps used = 4, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {28, 1814, 21, 8} \begin {gather*} \frac {x}{3}-\frac {e^2}{3 \left (4 x^2+e^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 21
Rule 28
Rule 1814
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=48 \int \frac {e^4+16 x^4+e^2 \left (8 x+8 x^2\right )}{\left (12 e^2+48 x^2\right )^2} \, dx\\ &=-\frac {e^2}{3 \left (e^2+4 x^2\right )}-\frac {2 \int \frac {-2 e^4-8 e^2 x^2}{12 e^2+48 x^2} \, dx}{e^2}\\ &=-\frac {e^2}{3 \left (e^2+4 x^2\right )}+\frac {\int 1 \, dx}{3}\\ &=\frac {x}{3}-\frac {e^2}{3 \left (e^2+4 x^2\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.76 \begin {gather*} \frac {1}{3} \left (x-\frac {e^2}{e^2+4 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 24, normalized size = 0.83 \begin {gather*} \frac {4 \, x^{3} + {\left (x - 1\right )} e^{2}}{3 \, {\left (4 \, x^{2} + e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 19, normalized size = 0.66
method | result | size |
risch | \(\frac {x}{3}-\frac {{\mathrm e}^{2}}{3 \left (4 x^{2}+{\mathrm e}^{2}\right )}\) | \(19\) |
gosper | \(\frac {4 x^{3}+{\mathrm e}^{2} x -{\mathrm e}^{2}}{12 x^{2}+3 \,{\mathrm e}^{2}}\) | \(27\) |
norman | \(\frac {\frac {4 x^{3}}{3}+\frac {{\mathrm e}^{2} x}{3}-\frac {{\mathrm e}^{2}}{3}}{4 x^{2}+{\mathrm e}^{2}}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 18, normalized size = 0.62 \begin {gather*} \frac {1}{3} \, x - \frac {e^{2}}{3 \, {\left (4 \, x^{2} + e^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 20, normalized size = 0.69 \begin {gather*} \frac {x}{3}-\frac {{\mathrm {e}}^2}{3\,\left (4\,x^2+{\mathrm {e}}^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 15, normalized size = 0.52 \begin {gather*} \frac {x}{3} - \frac {e^{2}}{12 x^{2} + 3 e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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