Optimal. Leaf size=28 \[ \log \left (\frac {-81+x}{4+\frac {\frac {3}{5}-e^4}{2 e}-x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 24, normalized size of antiderivative = 0.86, number of steps
used = 5, number of rules used = 4, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {12, 2006, 630,
31} \begin {gather*} \log (81-x)-\log \left (-10 e x-5 e^4+40 e+3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 31
Rule 630
Rule 2006
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\left (-3+770 e+5 e^4\right ) \int \frac {1}{243-3 x+e^4 (-405+5 x)+e \left (3240-850 x+10 x^2\right )} \, dx\\ &=\left (-3+770 e+5 e^4\right ) \int \frac {1}{81 \left (3+40 e-5 e^4\right )-\left (3+850 e-5 e^4\right ) x+10 e x^2} \, dx\\ &=(10 e) \int \frac {1}{-810 e+10 e x} \, dx-(10 e) \int \frac {1}{-3-40 e+5 e^4+10 e x} \, dx\\ &=\log (81-x)-\log \left (3+40 e-5 e^4-10 e x\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 0.82 \begin {gather*} -\log \left (3-5 e^4-10 e (-4+x)\right )+\log (81-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs.
\(2(24)=48\).
time = 0.41, size = 60, normalized size = 2.14
method | result | size |
norman | \(-\ln \left (10 x \,{\mathrm e}-40 \,{\mathrm e}+5 \,{\mathrm e}^{4}-3\right )+\ln \left (x -81\right )\) | \(24\) |
risch | \(-\ln \left (10 x \,{\mathrm e}-40 \,{\mathrm e}+5 \,{\mathrm e}^{4}-3\right )+\ln \left (81-x \right )\) | \(26\) |
default | \(\left (5 \,{\mathrm e}^{4}+770 \,{\mathrm e}-3\right ) \left (-\frac {\ln \left (10 x \,{\mathrm e}-40 \,{\mathrm e}+5 \,{\mathrm e}^{4}-3\right )}{5 \,{\mathrm e}^{4}+770 \,{\mathrm e}-3}+\frac {\ln \left (x -81\right )}{5 \,{\mathrm e}^{4}+770 \,{\mathrm e}-3}\right )\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 60 vs.
\(2 (22) = 44\).
time = 0.26, size = 60, normalized size = 2.14 \begin {gather*} -{\left (\frac {\log \left (10 \, x e + 5 \, e^{4} - 40 \, e - 3\right )}{5 \, e^{4} + 770 \, e - 3} - \frac {\log \left (x - 81\right )}{5 \, e^{4} + 770 \, e - 3}\right )} {\left (5 \, e^{4} + 770 \, e - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 21, normalized size = 0.75 \begin {gather*} -\log \left (10 \, {\left (x - 4\right )} e + 5 \, e^{4} - 3\right ) + \log \left (x - 81\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 272 vs.
\(2 (19) = 38\).
time = 0.45, size = 272, normalized size = 9.71 \begin {gather*} \left (- \frac {\log {\left (x + \frac {- 850 e - \frac {4620 e}{-3 + 5 e^{4} + 770 e} - 3 - \frac {30 e^{4}}{-3 + 5 e^{4} + 770 e} + \frac {9}{-3 + 5 e^{4} + 770 e} + \frac {25 e^{8}}{-3 + 5 e^{4} + 770 e} + 5 e^{4} + \frac {7700 e^{5}}{-3 + 5 e^{4} + 770 e} + \frac {592900 e^{2}}{-3 + 5 e^{4} + 770 e}}{20 e} \right )}}{-3 + 5 e^{4} + 770 e} + \frac {\log {\left (x + \frac {- 850 e - \frac {592900 e^{2}}{-3 + 5 e^{4} + 770 e} - \frac {7700 e^{5}}{-3 + 5 e^{4} + 770 e} - \frac {25 e^{8}}{-3 + 5 e^{4} + 770 e} - 3 - \frac {9}{-3 + 5 e^{4} + 770 e} + \frac {30 e^{4}}{-3 + 5 e^{4} + 770 e} + \frac {4620 e}{-3 + 5 e^{4} + 770 e} + 5 e^{4}}{20 e} \right )}}{-3 + 5 e^{4} + 770 e}\right ) \left (-3 + 5 e^{4} + 770 e\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 67 vs.
\(2 (22) = 44\).
time = 0.39, size = 67, normalized size = 2.39 \begin {gather*} -{\left (\frac {e \log \left ({\left | 10 \, x e + 5 \, e^{4} - 40 \, e - 3 \right |}\right )}{5 \, e^{5} + 770 \, e^{2} - 3 \, e} - \frac {\log \left ({\left | x - 81 \right |}\right )}{5 \, e^{4} + 770 \, e - 3}\right )} {\left (5 \, e^{4} + 770 \, e - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 29, normalized size = 1.04 \begin {gather*} \mathrm {atan}\left (\frac {\mathrm {e}\,\left (x\,40{}\mathrm {i}-3240{}\mathrm {i}\right )}{1540\,\mathrm {e}+10\,{\mathrm {e}}^4-6}+1{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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