Optimal. Leaf size=24 \[ \log \left (2-\frac {x \left (x+\log \left (-4+\left (2+e^2\right )^2 x\right )\right )}{\log (5)}\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 3, integrand size = 158, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6, 6688, 6684} \begin {gather*} \log \left (x^2+x \log \left (\left (2+e^2\right )^2 x-4\right )-2 \log (5)\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 6684
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x+8 x^2+e^4 \left (x+2 x^2\right )+e^2 \left (4 x+8 x^2\right )+\left (-4+4 x+4 e^2 x+e^4 x\right ) \log \left (-4+4 x+4 e^2 x+e^4 x\right )}{-4 x^2+e^4 x^3+\left (4+4 e^2\right ) x^3+\left (8-8 x-8 e^2 x-2 e^4 x\right ) \log (5)+\left (-4 x+4 x^2+4 e^2 x^2+e^4 x^2\right ) \log \left (-4+4 x+4 e^2 x+e^4 x\right )} \, dx\\ &=\int \frac {-4 x+8 x^2+e^4 \left (x+2 x^2\right )+e^2 \left (4 x+8 x^2\right )+\left (-4+4 x+4 e^2 x+e^4 x\right ) \log \left (-4+4 x+4 e^2 x+e^4 x\right )}{-4 x^2+\left (4+4 e^2+e^4\right ) x^3+\left (8-8 x-8 e^2 x-2 e^4 x\right ) \log (5)+\left (-4 x+4 x^2+4 e^2 x^2+e^4 x^2\right ) \log \left (-4+4 x+4 e^2 x+e^4 x\right )} \, dx\\ &=\int \frac {-x \left (-4+8 x+e^4 (1+2 x)+e^2 (4+8 x)\right )-\left (-4+\left (2+e^2\right )^2 x\right ) \log \left (-4+\left (2+e^2\right )^2 x\right )}{\left (4-\left (2+e^2\right )^2 x\right ) \left (x^2-2 \log (5)+x \log \left (-4+\left (2+e^2\right )^2 x\right )\right )} \, dx\\ &=\log \left (x^2-2 \log (5)+x \log \left (-4+\left (2+e^2\right )^2 x\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 23, normalized size = 0.96 \begin {gather*} \log \left (x^2-2 \log (5)+x \log \left (-4+\left (2+e^2\right )^2 x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 33, normalized size = 1.38 \begin {gather*} \log \relax (x) + \log \left (\frac {x^{2} + x \log \left (x e^{4} + 4 \, x e^{2} + 4 \, x - 4\right ) - 2 \, \log \relax (5)}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 29, normalized size = 1.21 \begin {gather*} \log \left (-x^{2} - x \log \left (x e^{4} + 4 \, x e^{2} + 4 \, x - 4\right ) + 2 \, \log \relax (5)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 32, normalized size = 1.33
method | result | size |
norman | \(\ln \left (-x^{2}-\ln \left (x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2} x +4 x -4\right ) x +2 \ln \relax (5)\right )\) | \(32\) |
risch | \(\ln \relax (x )+\ln \left (\ln \left (x \,{\mathrm e}^{4}+4 \,{\mathrm e}^{2} x +4 x -4\right )-\frac {-x^{2}+2 \ln \relax (5)}{x}\right )\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 31, normalized size = 1.29 \begin {gather*} \log \relax (x) + \log \left (\frac {x^{2} + x \log \left (x {\left (e^{4} + 4 \, e^{2} + 4\right )} - 4\right ) - 2 \, \log \relax (5)}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 33, normalized size = 1.38 \begin {gather*} \ln \left (\frac {x^2-\ln \left (25\right )+x\,\ln \left (4\,x+4\,x\,{\mathrm {e}}^2+x\,{\mathrm {e}}^4-4\right )}{x}\right )+\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.73, size = 32, normalized size = 1.33 \begin {gather*} \log {\relax (x )} + \log {\left (\log {\left (4 x + 4 x e^{2} + x e^{4} - 4 \right )} + \frac {x^{2} - 2 \log {\relax (5 )}}{x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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