Optimal. Leaf size=23 \[ \frac {\frac {19683}{e-2 x}+\frac {1}{2} \left (4+x^2\right )}{x^2} \]
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Rubi [A]
time = 0.08, antiderivative size = 33, normalized size of antiderivative = 1.43, number of steps
used = 4, number of rules used = 3, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {1608, 27, 1834}
\begin {gather*} \frac {19683+2 e}{e x^2}+\frac {39366}{e^2 x}+\frac {78732}{e^2 (e-2 x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 1834
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 e^2+118098 x-16 x^2+e (-39366+16 x)}{x^3 \left (e^2-4 e x+4 x^2\right )} \, dx\\ &=\int \frac {-4 e^2+118098 x-16 x^2+e (-39366+16 x)}{(e-2 x)^2 x^3} \, dx\\ &=\int \left (\frac {157464}{e^2 (e-2 x)^2}-\frac {2 (19683+2 e)}{e x^3}-\frac {39366}{e^2 x^2}\right ) \, dx\\ &=\frac {78732}{e^2 (e-2 x)}+\frac {19683+2 e}{e x^2}+\frac {39366}{e^2 x}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.87 \begin {gather*} -\frac {-19683-2 e+4 x}{(e-2 x) x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 22, normalized size = 0.96
method | result | size |
gosper | \(\frac {19683-4 x +2 \,{\mathrm e}}{x^{2} \left ({\mathrm e}-2 x \right )}\) | \(22\) |
norman | \(\frac {19683-4 x +2 \,{\mathrm e}}{x^{2} \left ({\mathrm e}-2 x \right )}\) | \(22\) |
risch | \(\frac {19683-4 x +2 \,{\mathrm e}}{x^{2} \left ({\mathrm e}-2 x \right )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 1.09 \begin {gather*} \frac {4 \, x - 2 \, e - 19683}{2 \, x^{3} - x^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 25, normalized size = 1.09 \begin {gather*} \frac {4 \, x - 2 \, e - 19683}{2 \, x^{3} - x^{2} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 22, normalized size = 0.96 \begin {gather*} - \frac {- 4 x + 2 e + 19683}{2 x^{3} - e x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 33, normalized size = 1.43 \begin {gather*} -\frac {78732 \, e^{\left (-2\right )}}{2 \, x - e} + \frac {{\left (39366 \, x + 2 \, e^{2} + 19683 \, e\right )} e^{\left (-2\right )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 24, normalized size = 1.04 \begin {gather*} \frac {2\,\mathrm {e}-4\,x+19683}{x^2\,\mathrm {e}-2\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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