Optimal. Leaf size=24 \[ 5+\frac {x}{5-2 x+e^4 x^2}+\frac {\log (x)}{e} \]
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Rubi [A]
time = 0.10, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps
used = 8, number of rules used = 5, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.060, Rules used = {6, 2099, 632,
210, 652} \begin {gather*} \frac {x}{e^4 x^2-2 x+5}+\frac {\log (x)}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 210
Rule 632
Rule 652
Rule 2099
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25+(-20+5 e) x+4 x^2+e^8 x^4+e^4 \left (10 x^2-4 x^3-e x^3\right )}{e^9 x^5+e \left (25 x-20 x^2+4 x^3\right )+e^5 \left (10 x^3-4 x^4\right )} \, dx\\ &=\int \left (\frac {1}{e x}+\frac {1}{-5+2 x-e^4 x^2}-\frac {2 (-5+x)}{\left (5-2 x+e^4 x^2\right )^2}\right ) \, dx\\ &=\frac {\log (x)}{e}-2 \int \frac {-5+x}{\left (5-2 x+e^4 x^2\right )^2} \, dx+\int \frac {1}{-5+2 x-e^4 x^2} \, dx\\ &=\frac {x}{5-2 x+e^4 x^2}+\frac {\log (x)}{e}-2 \text {Subst}\left (\int \frac {1}{4 \left (1-5 e^4\right )-x^2} \, dx,x,2-2 e^4 x\right )+\int \frac {1}{5-2 x+e^4 x^2} \, dx\\ &=\frac {x}{5-2 x+e^4 x^2}+\frac {\tan ^{-1}\left (\frac {1-e^4 x}{\sqrt {-1+5 e^4}}\right )}{\sqrt {-1+5 e^4}}+\frac {\log (x)}{e}-2 \text {Subst}\left (\int \frac {1}{4 \left (1-5 e^4\right )-x^2} \, dx,x,-2+2 e^4 x\right )\\ &=\frac {x}{5-2 x+e^4 x^2}+\frac {\log (x)}{e}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 1.00 \begin {gather*} \frac {\frac {e x}{5-2 x+e^4 x^2}+\log (x)}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.26, size = 89, normalized size = 3.71
method | result | size |
risch | \(\frac {x}{x^{2} {\mathrm e}^{4}-2 x +5}+\ln \left (x \right ) {\mathrm e}^{-1}\) | \(22\) |
norman | \(\frac {\frac {x^{2} {\mathrm e}^{4}}{2}+\frac {5}{2}}{x^{2} {\mathrm e}^{4}-2 x +5}+\ln \left (x \right ) {\mathrm e}^{-1}\) | \(36\) |
default | \({\mathrm e}^{-1} \left (-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (25+\textit {\_Z}^{4} {\mathrm e}^{8}-4 \textit {\_Z}^{3} {\mathrm e}^{4}-\left (-10 \,{\mathrm e}^{4}-4\right ) \textit {\_Z}^{2}-20 \textit {\_Z} \right )}{\sum }\frac {\left (-\textit {\_R}^{2} {\mathrm e}^{5}+5 \,{\mathrm e}\right ) \ln \left (x -\textit {\_R} \right )}{5-\textit {\_R}^{3} {\mathrm e}^{8}+3 \textit {\_R}^{2} {\mathrm e}^{4}-5 \textit {\_R} \,{\mathrm e}^{4}-2 \textit {\_R}}\right )}{4}+\ln \left (x \right )\right )\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 21, normalized size = 0.88 \begin {gather*} e^{\left (-1\right )} \log \left (x\right ) + \frac {x}{x^{2} e^{4} - 2 \, x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 38, normalized size = 1.58 \begin {gather*} \frac {x e + {\left (x^{2} e^{4} - 2 \, x + 5\right )} \log \left (x\right )}{x^{2} e^{5} - {\left (2 \, x - 5\right )} e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.69, size = 19, normalized size = 0.79 \begin {gather*} \frac {x}{x^{2} e^{4} - 2 x + 5} + \frac {\log {\left (x \right )}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 22, normalized size = 0.92 \begin {gather*} e^{\left (-1\right )} \log \left ({\left | x \right |}\right ) + \frac {x}{x^{2} e^{4} - 2 \, x + 5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 28, normalized size = 1.17 \begin {gather*} {\mathrm {e}}^{-1}\,\ln \left (x\right )+\frac {x\,\mathrm {e}}{{\mathrm {e}}^5\,x^2-2\,\mathrm {e}\,x+5\,\mathrm {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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