Optimal. Leaf size=24 \[ \frac {2 \left (-x+\frac {e^{2 x}}{(81+x)^2}-\log (x)\right )}{x^2} \]
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Rubi [B] time = 0.64, antiderivative size = 76, normalized size of antiderivative = 3.17, number of steps used = 22, number of rules used = 8, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6688, 12, 14, 6742, 2177, 2178, 37, 2304} \begin {gather*} \frac {(1-x)^2}{x^2}+\frac {2 e^{2 x}}{6561 x^2}-\frac {1}{x^2}-\frac {2 \log (x)}{x^2}-\frac {4 e^{2 x}}{531441 x}+\frac {4 e^{2 x}}{531441 (x+81)}+\frac {2 e^{2 x}}{6561 (x+81)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 37
Rule 2177
Rule 2178
Rule 2304
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-1+x+\frac {2 e^{2 x} \left (-81+79 x+x^2\right )}{(81+x)^3}+2 \log (x)\right )}{x^3} \, dx\\ &=2 \int \frac {-1+x+\frac {2 e^{2 x} \left (-81+79 x+x^2\right )}{(81+x)^3}+2 \log (x)}{x^3} \, dx\\ &=2 \int \left (\frac {2 e^{2 x} \left (-81+79 x+x^2\right )}{x^3 (81+x)^3}+\frac {-1+x+2 \log (x)}{x^3}\right ) \, dx\\ &=2 \int \frac {-1+x+2 \log (x)}{x^3} \, dx+4 \int \frac {e^{2 x} \left (-81+79 x+x^2\right )}{x^3 (81+x)^3} \, dx\\ &=2 \int \left (\frac {-1+x}{x^3}+\frac {2 \log (x)}{x^3}\right ) \, dx+4 \int \left (-\frac {e^{2 x}}{6561 x^3}+\frac {82 e^{2 x}}{531441 x^2}-\frac {2 e^{2 x}}{531441 x}-\frac {e^{2 x}}{6561 (81+x)^3}+\frac {80 e^{2 x}}{531441 (81+x)^2}+\frac {2 e^{2 x}}{531441 (81+x)}\right ) \, dx\\ &=-\frac {8 \int \frac {e^{2 x}}{x} \, dx}{531441}+\frac {8 \int \frac {e^{2 x}}{81+x} \, dx}{531441}+\frac {320 \int \frac {e^{2 x}}{(81+x)^2} \, dx}{531441}-\frac {4 \int \frac {e^{2 x}}{x^3} \, dx}{6561}-\frac {4 \int \frac {e^{2 x}}{(81+x)^3} \, dx}{6561}+\frac {328 \int \frac {e^{2 x}}{x^2} \, dx}{531441}+2 \int \frac {-1+x}{x^3} \, dx+4 \int \frac {\log (x)}{x^3} \, dx\\ &=-\frac {1}{x^2}+\frac {2 e^{2 x}}{6561 x^2}+\frac {(1-x)^2}{x^2}-\frac {328 e^{2 x}}{531441 x}+\frac {2 e^{2 x}}{6561 (81+x)^2}-\frac {320 e^{2 x}}{531441 (81+x)}-\frac {8 \text {Ei}(2 x)}{531441}+\frac {8 \text {Ei}(2 (81+x))}{531441 e^{162}}-\frac {2 \log (x)}{x^2}-\frac {4 \int \frac {e^{2 x}}{x^2} \, dx}{6561}-\frac {4 \int \frac {e^{2 x}}{(81+x)^2} \, dx}{6561}+\frac {640 \int \frac {e^{2 x}}{81+x} \, dx}{531441}+\frac {656 \int \frac {e^{2 x}}{x} \, dx}{531441}\\ &=-\frac {1}{x^2}+\frac {2 e^{2 x}}{6561 x^2}+\frac {(1-x)^2}{x^2}-\frac {4 e^{2 x}}{531441 x}+\frac {2 e^{2 x}}{6561 (81+x)^2}+\frac {4 e^{2 x}}{531441 (81+x)}+\frac {8 \text {Ei}(2 x)}{6561}+\frac {8 \text {Ei}(2 (81+x))}{6561 e^{162}}-\frac {2 \log (x)}{x^2}-\frac {8 \int \frac {e^{2 x}}{x} \, dx}{6561}-\frac {8 \int \frac {e^{2 x}}{81+x} \, dx}{6561}\\ &=-\frac {1}{x^2}+\frac {2 e^{2 x}}{6561 x^2}+\frac {(1-x)^2}{x^2}-\frac {4 e^{2 x}}{531441 x}+\frac {2 e^{2 x}}{6561 (81+x)^2}+\frac {4 e^{2 x}}{531441 (81+x)}-\frac {2 \log (x)}{x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.14, size = 33, normalized size = 1.38 \begin {gather*} \frac {2 \left (e^{2 x}-x (81+x)^2-(81+x)^2 \log (x)\right )}{x^2 (81+x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.54, size = 47, normalized size = 1.96 \begin {gather*} -\frac {2 \, {\left (x^{3} + 162 \, x^{2} + {\left (x^{2} + 162 \, x + 6561\right )} \log \relax (x) + 6561 \, x - e^{\left (2 \, x\right )}\right )}}{x^{4} + 162 \, x^{3} + 6561 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 51, normalized size = 2.12 \begin {gather*} -\frac {2 \, {\left (x^{3} + x^{2} \log \relax (x) + 162 \, x^{2} + 162 \, x \log \relax (x) + 6561 \, x - e^{\left (2 \, x\right )} + 6561 \, \log \relax (x)\right )}}{x^{4} + 162 \, x^{3} + 6561 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 37, normalized size = 1.54
method | result | size |
risch | \(-\frac {2 \ln \relax (x )}{x^{2}}-\frac {2 \left (x^{3}+162 x^{2}+6561 x -{\mathrm e}^{2 x}\right )}{x^{2} \left (81+x \right )^{2}}\) | \(37\) |
default | \(-\frac {2}{x}-\frac {2 \ln \relax (x )}{x^{2}}+\frac {8 \,{\mathrm e}^{2 x}}{6561 \left (2 x +162\right )^{2}}+\frac {8 \,{\mathrm e}^{2 x}}{531441 \left (2 x +162\right )}-\frac {4 \,{\mathrm e}^{2 x}}{531441 x}+\frac {2 \,{\mathrm e}^{2 x}}{6561 x^{2}}\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 150, normalized size = 6.25 \begin {gather*} -\frac {4 \, x^{3} + 486 \, x^{2} + 8748 \, x - 177147}{27 \, {\left (x^{4} + 162 \, x^{3} + 6561 \, x^{2}\right )}} - \frac {26 \, {\left (2 \, x^{2} + 243 \, x + 4374\right )}}{9 \, {\left (x^{3} + 162 \, x^{2} + 6561 \, x\right )}} - \frac {x^{2} + 2 \, {\left (x^{2} + 162 \, x + 6561\right )} \log \relax (x) + 162 \, x - 2 \, e^{\left (2 \, x\right )} + 6561}{x^{4} + 162 \, x^{3} + 6561 \, x^{2}} + \frac {80 \, {\left (2 \, x + 243\right )}}{27 \, {\left (x^{2} + 162 \, x + 6561\right )}} - \frac {2 \, x + 81}{x^{2} + 162 \, x + 6561} - \frac {242}{x^{2} + 162 \, x + 6561} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.50, size = 35, normalized size = 1.46 \begin {gather*} \frac {2\,{\mathrm {e}}^{2\,x}}{x^4+162\,x^3+6561\,x^2}-\frac {2\,\ln \relax (x)}{x^2}-\frac {2}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 31, normalized size = 1.29 \begin {gather*} \frac {2 e^{2 x}}{x^{4} + 162 x^{3} + 6561 x^{2}} - \frac {2}{x} - \frac {2 \log {\relax (x )}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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