Optimal. Leaf size=18 \[ -x-\frac {4 e^4}{-x+x^2} \]
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Rubi [A]
time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.33, number of steps
used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {1608, 27, 1634}
\begin {gather*} -x+\frac {4 e^4}{1-x}+\frac {4 e^4}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 1608
Rule 1634
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x^2+2 x^3-x^4+e^4 (-4+8 x)}{x^2 \left (1-2 x+x^2\right )} \, dx\\ &=\int \frac {-x^2+2 x^3-x^4+e^4 (-4+8 x)}{(-1+x)^2 x^2} \, dx\\ &=\int \left (-1+\frac {4 e^4}{(-1+x)^2}-\frac {4 e^4}{x^2}\right ) \, dx\\ &=\frac {4 e^4}{1-x}+\frac {4 e^4}{x}-x\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.22 \begin {gather*} -\frac {4 e^4}{-1+x}+\frac {4 e^4}{x}-x \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 21, normalized size = 1.17
method | result | size |
risch | \(-x -\frac {4 \,{\mathrm e}^{4}}{x \left (x -1\right )}\) | \(17\) |
default | \(-x +\frac {4 \,{\mathrm e}^{4}}{x}-\frac {4 \,{\mathrm e}^{4}}{x -1}\) | \(21\) |
gosper | \(-\frac {x^{3}+4 \,{\mathrm e}^{4}-x}{x \left (x -1\right )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 0.94 \begin {gather*} -x - \frac {4 \, e^{4}}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 24, normalized size = 1.33 \begin {gather*} -\frac {x^{3} - x^{2} + 4 \, e^{4}}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 12, normalized size = 0.67 \begin {gather*} - x - \frac {4 e^{4}}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 17, normalized size = 0.94 \begin {gather*} -x - \frac {4 \, e^{4}}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 16, normalized size = 0.89 \begin {gather*} -x-\frac {4\,{\mathrm {e}}^4}{x\,\left (x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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