Optimal. Leaf size=38 \[ \frac {x^2-\frac {\log (5)}{x}}{-x-e x-x^2+e^{2 x^2} x^2} \]
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Rubi [F] time = 2.99, 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 {-x^3-e x^3+(-2-2 e-3 x) \log (5)+e^{2 x^2} \left (-4 x^6+\left (3 x+4 x^3\right ) \log (5)\right )}{x^3+e^2 x^3+2 x^4+x^5+e^{4 x^2} x^5+e \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (-2 x^4-2 e x^4-2 x^5\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 {-x^3-e x^3+(-2-2 e-3 x) \log (5)+e^{2 x^2} \left (-4 x^6+\left (3 x+4 x^3\right ) \log (5)\right )}{\left (1+e^2\right ) x^3+2 x^4+x^5+e^{4 x^2} x^5+e \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (-2 x^4-2 e x^4-2 x^5\right )} \, dx\\ &=\int \frac {(-1-e) x^3+(-2-2 e-3 x) \log (5)+e^{2 x^2} \left (-4 x^6+\left (3 x+4 x^3\right ) \log (5)\right )}{\left (1+e^2\right ) x^3+2 x^4+x^5+e^{4 x^2} x^5+e \left (2 x^3+2 x^4\right )+e^{2 x^2} \left (-2 x^4-2 e x^4-2 x^5\right )} \, dx\\ &=\int \frac {-4 e^{2 x^2} x^6-2 (1+e) \log (5)+3 \left (-1+e^{2 x^2}\right ) x \log (5)-x^3 \left (1+e-4 e^{2 x^2} \log (5)\right )}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2} \, dx\\ &=\int \left (\frac {\left (1+e+4 (1+e) x^2+4 x^3\right ) \left (-x^3+\log (5)\right )}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2}-\frac {4 x^5-4 x^2 \log (5)-\log (125)}{x^3 \left (-1-e-x+e^{2 x^2} x\right )}\right ) \, dx\\ &=\int \frac {\left (1+e+4 (1+e) x^2+4 x^3\right ) \left (-x^3+\log (5)\right )}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2} \, dx-\int \frac {4 x^5-4 x^2 \log (5)-\log (125)}{x^3 \left (-1-e-x+e^{2 x^2} x\right )} \, dx\\ &=\int \left (-\frac {4 (1+e) x^2}{\left (1+e+x-e^{2 x^2} x\right )^2}-\frac {4 x^3}{\left (-1-e-x+e^{2 x^2} x\right )^2}-\frac {1+e-4 \log (5)}{\left (1+e+x-e^{2 x^2} x\right )^2}+\frac {(1+e) \log (5)}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2}+\frac {4 (1+e) \log (5)}{x \left (1+e+x-e^{2 x^2} x\right )^2}\right ) \, dx-\int \left (\frac {4 x^2}{-1-e-x+e^{2 x^2} x}-\frac {4 \log (5)}{x \left (-1-e-x+e^{2 x^2} x\right )}-\frac {\log (125)}{x^3 \left (-1-e-x+e^{2 x^2} x\right )}\right ) \, dx\\ &=-\left (4 \int \frac {x^3}{\left (-1-e-x+e^{2 x^2} x\right )^2} \, dx\right )-4 \int \frac {x^2}{-1-e-x+e^{2 x^2} x} \, dx-(4 (1+e)) \int \frac {x^2}{\left (1+e+x-e^{2 x^2} x\right )^2} \, dx+(4 \log (5)) \int \frac {1}{x \left (-1-e-x+e^{2 x^2} x\right )} \, dx+((1+e) \log (5)) \int \frac {1}{x^3 \left (1+e+x-e^{2 x^2} x\right )^2} \, dx+(4 (1+e) \log (5)) \int \frac {1}{x \left (1+e+x-e^{2 x^2} x\right )^2} \, dx+(-1-e+4 \log (5)) \int \frac {1}{\left (1+e+x-e^{2 x^2} x\right )^2} \, dx+\log (125) \int \frac {1}{x^3 \left (-1-e-x+e^{2 x^2} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.72, size = 32, normalized size = 0.84 \begin {gather*} -\frac {-x^3+\log (5)}{x^2 \left (-1-e-x+e^{2 x^2} x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 39, normalized size = 1.03 \begin {gather*} \frac {x^{3} - \log \relax (5)}{x^{3} e^{\left (2 \, x^{2}\right )} - x^{3} - x^{2} e - x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 29, normalized size = 0.76
method | result | size |
risch | \(\frac {-x^{3}+\ln \relax (5)}{x^{2} \left (-x \,{\mathrm e}^{2 x^{2}}+{\mathrm e}+x +1\right )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 36, normalized size = 0.95 \begin {gather*} \frac {x^{3} - \log \relax (5)}{x^{3} e^{\left (2 \, x^{2}\right )} - x^{3} - x^{2} {\left (e + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.72, size = 35, normalized size = 0.92 \begin {gather*} \frac {\ln \relax (5)-x^3}{x^2\,\mathrm {e}-x^3\,{\mathrm {e}}^{2\,x^2}+x^2+x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 29, normalized size = 0.76 \begin {gather*} \frac {x^{3} - \log {\relax (5 )}}{x^{3} e^{2 x^{2}} - x^{3} - e x^{2} - x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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