Optimal. Leaf size=23 \[ \frac {e^5}{-7-e^{2 x}+\frac {1}{-1+x}-x} \]
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Rubi [F]
time = 0.58, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^5 \left (2-2 x+x^2\right )+e^{5+2 x} \left (2-4 x+2 x^2\right )}{64-96 x+20 x^2+12 x^3+x^4+e^{4 x} \left (1-2 x+x^2\right )+e^{2 x} \left (16-28 x+10 x^2+2 x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^5 \left (2+2 e^{2 x} (-1+x)^2-2 x+x^2\right )}{\left (8-e^{2 x} (-1+x)-6 x-x^2\right )^2} \, dx\\ &=e^5 \int \frac {2+2 e^{2 x} (-1+x)^2-2 x+x^2}{\left (8-e^{2 x} (-1+x)-6 x-x^2\right )^2} \, dx\\ &=e^5 \int \left (\frac {2 (-1+x)}{-8-e^{2 x}+6 x+e^{2 x} x+x^2}-\frac {14-26 x+9 x^2+2 x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}\right ) \, dx\\ &=-\left (e^5 \int \frac {14-26 x+9 x^2+2 x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx\right )+\left (2 e^5\right ) \int \frac {-1+x}{-8-e^{2 x}+6 x+e^{2 x} x+x^2} \, dx\\ &=-\left (e^5 \int \left (\frac {14}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}-\frac {26 x}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}+\frac {9 x^2}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}+\frac {2 x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2}\right ) \, dx\right )+\left (2 e^5\right ) \int \left (-\frac {1}{-8-e^{2 x}+6 x+e^{2 x} x+x^2}+\frac {x}{-8-e^{2 x}+6 x+e^{2 x} x+x^2}\right ) \, dx\\ &=-\left (\left (2 e^5\right ) \int \frac {x^3}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx\right )-\left (2 e^5\right ) \int \frac {1}{-8-e^{2 x}+6 x+e^{2 x} x+x^2} \, dx+\left (2 e^5\right ) \int \frac {x}{-8-e^{2 x}+6 x+e^{2 x} x+x^2} \, dx-\left (9 e^5\right ) \int \frac {x^2}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx-\left (14 e^5\right ) \int \frac {1}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx+\left (26 e^5\right ) \int \frac {x}{\left (-8-e^{2 x}+6 x+e^{2 x} x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 1.56, size = 28, normalized size = 1.22 \begin {gather*} \frac {e^5 (1-x)}{-8+e^{2 x} (-1+x)+6 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 30, normalized size = 1.30
method | result | size |
risch | \(-\frac {{\mathrm e}^{5} \left (x -1\right )}{x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x}+x^{2}+6 x -8}\) | \(30\) |
norman | \(\frac {-x \,{\mathrm e}^{5}+{\mathrm e}^{5}}{x \,{\mathrm e}^{2 x}-{\mathrm e}^{2 x}+x^{2}+6 x -8}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 29, normalized size = 1.26 \begin {gather*} -\frac {x e^{5} - e^{5}}{x^{2} + {\left (x - 1\right )} e^{\left (2 \, x\right )} + 6 \, x - 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 31, normalized size = 1.35 \begin {gather*} -\frac {{\left (x - 1\right )} e^{10}}{{\left (x^{2} + 6 \, x - 8\right )} e^{5} + {\left (x - 1\right )} e^{\left (2 \, x + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 24, normalized size = 1.04 \begin {gather*} \frac {- x e^{5} + e^{5}}{x^{2} + 6 x + \left (x - 1\right ) e^{2 x} - 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 62 vs.
\(2 (20) = 40\).
time = 0.45, size = 62, normalized size = 2.70 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, x + 5\right )} e^{10} - 7 \, e^{10}\right )}}{{\left (2 \, x + 5\right )}^{2} e^{5} + 2 \, {\left (2 \, x + 5\right )} e^{5} + 2 \, {\left (2 \, x + 5\right )} e^{\left (2 \, x + 5\right )} - 67 \, e^{5} - 14 \, e^{\left (2 \, x + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 54, normalized size = 2.35 \begin {gather*} -\frac {\frac {{\mathrm {e}}^{2\,x+5}}{8}+\frac {x\,{\mathrm {e}}^5}{4}-\frac {x\,{\mathrm {e}}^{2\,x+5}}{8}-\frac {x^2\,{\mathrm {e}}^5}{8}}{6\,x-{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+x^2-8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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