Optimal. Leaf size=33 \[ 5-\frac {x^2}{e^{\frac {23 x+\log (4)}{e^2-x}}-x^2} \]
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Rubi [A] time = 3.81, antiderivative size = 34, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {6, 1594, 6688, 6711, 32} \begin {gather*} \frac {1}{1-\frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}}{x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 32
Rule 1594
Rule 6688
Rule 6711
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x-2 x^3+x^2 \left (27 e^2+\log (4)\right )\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx\\ &=\int \frac {e^{\frac {23 x+\log (4)}{e^2-x}} x \left (-2 e^4-2 x^2+x \left (27 e^2+\log (4)\right )\right )}{e^4 x^4-2 e^2 x^5+x^6+e^{\frac {2 (23 x+\log (4))}{e^2-x}} \left (e^4-2 e^2 x+x^2\right )+e^{\frac {23 x+\log (4)}{e^2-x}} \left (-2 e^4 x^2+4 e^2 x^3-2 x^4\right )} \, dx\\ &=\int \frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}} x \left (-2 e^4-2 x^2+x \left (27 e^2+\log (4)\right )\right )}{\left (e^2-x\right )^2 \left (4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}-x^2\right )^2} \, dx\\ &=\operatorname {Subst}\left (\int \frac {1}{(-1+x)^2} \, dx,x,\frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}}{x^2}\right )\\ &=\frac {1}{1-\frac {4^{\frac {1}{e^2-x}} e^{\frac {23 x}{e^2-x}}}{x^2}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 49, normalized size = 1.48 \begin {gather*} -\frac {e^{23} x^2}{4^{-\frac {1}{-e^2+x}} e^{-\frac {23 e^2}{-e^2+x}}-e^{23} x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 31, normalized size = 0.94 \begin {gather*} \frac {x^{2}}{x^{2} - e^{\left (-\frac {23 \, x + 2 \, \log \relax (2)}{x - e^{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x^{3} - 27 \, x^{2} e^{2} - 2 \, x^{2} \log \relax (2) + 2 \, x e^{4}\right )} e^{\left (-\frac {23 \, x + 2 \, \log \relax (2)}{x - e^{2}}\right )}}{x^{6} - 2 \, x^{5} e^{2} + x^{4} e^{4} - 2 \, {\left (x^{4} - 2 \, x^{3} e^{2} + x^{2} e^{4}\right )} e^{\left (-\frac {23 \, x + 2 \, \log \relax (2)}{x - e^{2}}\right )} + {\left (x^{2} - 2 \, x e^{2} + e^{4}\right )} e^{\left (-\frac {2 \, {\left (23 \, x + 2 \, \log \relax (2)\right )}}{x - e^{2}}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 5.14, size = 31, normalized size = 0.94
method | result | size |
risch | \(\frac {x^{2}}{x^{2}-{\mathrm e}^{\frac {2 \ln \relax (2)+23 x}{{\mathrm e}^{2}-x}}}\) | \(31\) |
norman | \(\frac {-x \,{\mathrm e}^{\frac {2 \ln \relax (2)+23 x}{{\mathrm e}^{2}-x}}+{\mathrm e}^{2} {\mathrm e}^{\frac {2 \ln \relax (2)+23 x}{{\mathrm e}^{2}-x}}}{\left (x^{2}-{\mathrm e}^{\frac {2 \ln \relax (2)+23 x}{{\mathrm e}^{2}-x}}\right ) \left ({\mathrm e}^{2}-x \right )}\) | \(79\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{x^{2} e^{\left (\frac {23 \, e^{2}}{x - e^{2}} + \frac {2 \, \log \relax (2)}{x - e^{2}} + 23\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 164, normalized size = 4.97 \begin {gather*} -\frac {x^3\,{\left (x^2-2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4\right )}^2\,\left (2\,{\mathrm {e}}^4-27\,x\,{\mathrm {e}}^2-2\,x\,\ln \relax (2)+2\,x^2\right )}{\left (\frac {{\mathrm {e}}^{-\frac {23\,x}{x-{\mathrm {e}}^2}}}{2^{\frac {2}{x-{\mathrm {e}}^2}}}-x^2\right )\,\left (2\,x\,{\mathrm {e}}^{12}-35\,x^6\,{\mathrm {e}}^2+122\,x^5\,{\mathrm {e}}^4-178\,x^4\,{\mathrm {e}}^6+122\,x^3\,{\mathrm {e}}^8-35\,x^2\,{\mathrm {e}}^{10}-x^6\,\ln \relax (4)+2\,x^7+4\,x^5\,{\mathrm {e}}^2\,\ln \relax (4)-6\,x^4\,{\mathrm {e}}^4\,\ln \relax (4)+4\,x^3\,{\mathrm {e}}^6\,\ln \relax (4)-x^2\,{\mathrm {e}}^8\,\ln \relax (4)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 22, normalized size = 0.67 \begin {gather*} - \frac {x^{2}}{- x^{2} + e^{\frac {23 x + 2 \log {\relax (2 )}}{- x + e^{2}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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