Optimal. Leaf size=27 \[ \frac {2}{e^6}-\frac {30 (3+x)}{x \left (5+x-e^3 x^2\right )} \]
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Rubi [A] time = 0.18, antiderivative size = 29, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 4, integrand size = 64, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2074, 638, 618, 206} \begin {gather*} -\frac {6 \left (3 e^3 x+2\right )}{-e^3 x^2+x+5}-\frac {18}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 638
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {18}{x^2}-\frac {6 \left (-2+30 e^3+7 e^3 x\right )}{\left (-5-x+e^3 x^2\right )^2}-\frac {18 e^3}{-5-x+e^3 x^2}\right ) \, dx\\ &=-\frac {18}{x}-6 \int \frac {-2+30 e^3+7 e^3 x}{\left (-5-x+e^3 x^2\right )^2} \, dx-\left (18 e^3\right ) \int \frac {1}{-5-x+e^3 x^2} \, dx\\ &=-\frac {18}{x}-\frac {6 \left (2+3 e^3 x\right )}{5+x-e^3 x^2}+\left (18 e^3\right ) \int \frac {1}{-5-x+e^3 x^2} \, dx+\left (36 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+20 e^3-x^2} \, dx,x,-1+2 e^3 x\right )\\ &=-\frac {18}{x}-\frac {6 \left (2+3 e^3 x\right )}{5+x-e^3 x^2}-\frac {36 e^3 \tanh ^{-1}\left (\frac {1-2 e^3 x}{\sqrt {1+20 e^3}}\right )}{\sqrt {1+20 e^3}}-\left (36 e^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+20 e^3-x^2} \, dx,x,-1+2 e^3 x\right )\\ &=-\frac {18}{x}-\frac {6 \left (2+3 e^3 x\right )}{5+x-e^3 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} -\frac {30 (3+x)}{5 x+x^2-e^3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 22, normalized size = 0.81 \begin {gather*} \frac {30 \, {\left (x + 3\right )}}{x^{3} e^{3} - x^{2} - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 30, normalized size = 1.11 \begin {gather*} -\frac {18}{x} + 2.79551404258800 \times 10^{9} \, \log \left (x + 0.474661298947000\right ) - 2.79550903378200 \times 10^{9} \, \log \left (x + 0.474661296102000\right ) + 2.89251442174200 \times 10^{9} \, \log \left (x - 0.524448364156000\right ) - 2.89252337572800 \times 10^{9} \, \log \left (x - 0.524448367629000\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 22, normalized size = 0.81
method | result | size |
gosper | \(\frac {30 x +90}{x \left (x^{2} {\mathrm e}^{3}-x -5\right )}\) | \(22\) |
norman | \(\frac {30 x +90}{x \left (x^{2} {\mathrm e}^{3}-x -5\right )}\) | \(23\) |
risch | \(\frac {30 x +90}{x \left (x^{2} {\mathrm e}^{3}-x -5\right )}\) | \(23\) |
default | \(-\frac {18}{x}+3 \left (\munderset {\textit {\_R} =\RootOf \left (25+\textit {\_Z}^{4} {\mathrm e}^{6}-2 \textit {\_Z}^{3} {\mathrm e}^{3}+\left (-10 \,{\mathrm e}^{3}+1\right ) \textit {\_Z}^{2}+10 \textit {\_Z} \right )}{\sum }\frac {\left (-3 \textit {\_R}^{2} {\mathrm e}^{6}-4 \textit {\_R} \,{\mathrm e}^{3}-15 \,{\mathrm e}^{3}+2\right ) \ln \left (x -\textit {\_R} \right )}{5+2 \textit {\_R}^{3} {\mathrm e}^{6}-3 \textit {\_R}^{2} {\mathrm e}^{3}-10 \textit {\_R} \,{\mathrm e}^{3}+\textit {\_R}}\right )\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 22, normalized size = 0.81 \begin {gather*} \frac {30 \, {\left (x + 3\right )}}{x^{3} e^{3} - x^{2} - 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 20, normalized size = 0.74 \begin {gather*} -\frac {30\,\left (x+3\right )}{x\,\left (-{\mathrm {e}}^3\,x^2+x+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.94, size = 20, normalized size = 0.74 \begin {gather*} - \frac {- 30 x - 90}{x^{3} e^{3} - x^{2} - 5 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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