Optimal. Leaf size=23 \[ 3+\frac {12 e^{1+e}}{5+x^2-x \log ^2(4)} \]
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Rubi [A]
time = 0.03, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps
used = 5, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6, 12, 1694,
267} \begin {gather*} \frac {12 e^{1+e}}{x^2-x \log ^2(4)+5} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 267
Rule 1694
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{1+e} \left (-24 x+12 \log ^2(4)\right )}{25+x^4+\left (-10 x-2 x^3\right ) \log ^2(4)+x^2 \left (10+\log ^4(4)\right )} \, dx\\ &=e^{1+e} \int \frac {-24 x+12 \log ^2(4)}{25+x^4+\left (-10 x-2 x^3\right ) \log ^2(4)+x^2 \left (10+\log ^4(4)\right )} \, dx\\ &=e^{1+e} \text {Subst}\left (\int -\frac {384 x}{\left (20+4 x^2-\log ^4(4)\right )^2} \, dx,x,x-\frac {\log ^2(4)}{2}\right )\\ &=-\left (\left (384 e^{1+e}\right ) \text {Subst}\left (\int \frac {x}{\left (20+4 x^2-\log ^4(4)\right )^2} \, dx,x,x-\frac {\log ^2(4)}{2}\right )\right )\\ &=\frac {12 e^{1+e}}{5+x^2-x \log ^2(4)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} \frac {12 e^{1+e}}{5+x^2-x \log ^2(4)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 40, normalized size = 1.74
method | result | size |
risch | \(-\frac {3 \,{\mathrm e}^{1+{\mathrm e}}}{x \ln \left (2\right )^{2}-\frac {x^{2}}{4}-\frac {5}{4}}\) | \(23\) |
gosper | \(-\frac {12 \,{\mathrm e}^{1+{\mathrm e}}}{4 x \ln \left (2\right )^{2}-x^{2}-5}\) | \(24\) |
norman | \(-\frac {12 \,{\mathrm e} \,{\mathrm e}^{{\mathrm e}}}{4 x \ln \left (2\right )^{2}-x^{2}-5}\) | \(24\) |
default | \(\frac {24 \,{\mathrm e}^{1+{\mathrm e}} \left (10-8 \ln \left (2\right )^{4}\right )}{\left (20-16 \ln \left (2\right )^{4}\right ) \left (x^{2}-4 x \ln \left (2\right )^{2}+5\right )}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 23, normalized size = 1.00 \begin {gather*} -\frac {12 \, e^{\left (e + 1\right )}}{4 \, x \log \left (2\right )^{2} - x^{2} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 23, normalized size = 1.00 \begin {gather*} -\frac {12 \, e^{\left (e + 1\right )}}{4 \, x \log \left (2\right )^{2} - x^{2} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 22, normalized size = 0.96 \begin {gather*} \frac {12 e e^{e}}{x^{2} - 4 x \log {\left (2 \right )}^{2} + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 23, normalized size = 1.00 \begin {gather*} -\frac {12 \, e^{\left (e + 1\right )}}{4 \, x \log \left (2\right )^{2} - x^{2} - 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.32, size = 21, normalized size = 0.91 \begin {gather*} \frac {12\,{\mathrm {e}}^{\mathrm {e}+1}}{x^2-4\,{\ln \left (2\right )}^2\,x+5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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