Optimal. Leaf size=35 \[ \frac {x^2}{\left (\frac {x}{e^5}+x \log \left (\frac {-x+x^2}{-1+\log (5)+\log (5-x)}\right )\right )^2} \]
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Rubi [A]
time = 1.96, antiderivative size = 34, normalized size of antiderivative = 0.97, number of steps
used = 4, number of rules used = 4, integrand size = 357, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6820, 12,
6840, 32} \begin {gather*} \frac {e^{10}}{\left (e^5 \log \left (\frac {(1-x) x}{1-\log (5 (5-x))}\right )+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 6820
Rule 6840
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{15} \left (x (12-11 \log (5))+5 (-1+\log (5))+x^2 (-3+\log (25))+\left (5-11 x+2 x^2\right ) \log (5-x)\right )}{x \left (5-6 x+x^2\right ) (1-\log (-5 (-5+x))) \left (1+e^5 \log \left (\frac {(-1+x) x}{-1+\log (-5 (-5+x))}\right )\right )^3} \, dx\\ &=\left (2 e^{15}\right ) \int \frac {x (12-11 \log (5))+5 (-1+\log (5))+x^2 (-3+\log (25))+\left (5-11 x+2 x^2\right ) \log (5-x)}{x \left (5-6 x+x^2\right ) (1-\log (-5 (-5+x))) \left (1+e^5 \log \left (\frac {(-1+x) x}{-1+\log (-5 (-5+x))}\right )\right )^3} \, dx\\ &=-\left (\left (2 e^{10}\right ) \text {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,e^5 \log \left (\frac {(-1+x) x}{-1+\log (-5 (-5+x))}\right )\right )\right )\\ &=\frac {e^{10}}{\left (1+e^5 \log \left (\frac {(1-x) x}{1-\log (5 (5-x))}\right )\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 1.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{15} \left (10-24 x+6 x^2\right )+e^{15} \left (-10+22 x-4 x^2\right ) \log (5)+e^{15} \left (-10+22 x-4 x^2\right ) \log (5-x)}{-5 x+6 x^2-x^3+\left (5 x-6 x^2+x^3\right ) \log (5)+\left (5 x-6 x^2+x^3\right ) \log (5-x)+\left (e^5 \left (-15 x+18 x^2-3 x^3\right )+e^5 \left (15 x-18 x^2+3 x^3\right ) \log (5)+e^5 \left (15 x-18 x^2+3 x^3\right ) \log (5-x)\right ) \log \left (\frac {-x+x^2}{-1+\log (5)+\log (5-x)}\right )+\left (e^{10} \left (-15 x+18 x^2-3 x^3\right )+e^{10} \left (15 x-18 x^2+3 x^3\right ) \log (5)+e^{10} \left (15 x-18 x^2+3 x^3\right ) \log (5-x)\right ) \log ^2\left (\frac {-x+x^2}{-1+\log (5)+\log (5-x)}\right )+\left (e^{15} \left (-5 x+6 x^2-x^3\right )+e^{15} \left (5 x-6 x^2+x^3\right ) \log (5)+e^{15} \left (5 x-6 x^2+x^3\right ) \log (5-x)\right ) \log ^3\left (\frac {-x+x^2}{-1+\log (5)+\log (5-x)}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 9.48, size = 346, normalized size = 9.89
method | result | size |
risch | \(-\frac {4 \,{\mathrm e}^{10}}{\left ({\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )-{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}+{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )-{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}-{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}+{\mathrm e}^{5} \pi \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{3}-{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}+{\mathrm e}^{5} \pi \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{3}+2 i {\mathrm e}^{5} \ln \left (x \right )+2 i {\mathrm e}^{5} \ln \left (x -1\right )-2 i {\mathrm e}^{5} \ln \left (\ln \left (5-x \right )+\ln \left (5\right )-1\right )+2 i\right )^{2}}\) | \(340\) |
default | \(-\frac {4 \,{\mathrm e}^{15} {\mathrm e}^{-5}}{\left ({\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )+{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}+{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )+{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (i \left (x -1\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}-{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}-{\mathrm e}^{5} \pi \mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{3}+{\mathrm e}^{5} \pi \,\mathrm {csgn}\left (\frac {i \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right ) \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{2}+{\mathrm e}^{5} \pi \mathrm {csgn}\left (\frac {i x \left (x -1\right )}{\ln \left (5-x \right )+\ln \left (5\right )-1}\right )^{3}+2 i {\mathrm e}^{5} \ln \left (-x \right )+2 i {\mathrm e}^{5} \ln \left (1-x \right )-2 i {\mathrm e}^{5} \ln \left (\ln \left (5-x \right )+\ln \left (5\right )-1\right )+2 i\right )^{2}}\) | \(346\) |
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. 87 vs.
\(2 (34) = 68\).
time = 0.73, size = 87, normalized size = 2.49 \begin {gather*} \frac {e^{10}}{e^{10} \log \left (x - 1\right )^{2} + e^{10} \log \left (x\right )^{2} + e^{10} \log \left (\log \left (5\right ) + \log \left (-x + 5\right ) - 1\right )^{2} + 2 \, {\left (e^{10} \log \left (x\right ) + e^{5}\right )} \log \left (x - 1\right ) + 2 \, e^{5} \log \left (x\right ) - 2 \, {\left (e^{10} \log \left (x - 1\right ) + e^{10} \log \left (x\right ) + e^{5}\right )} \log \left (\log \left (5\right ) + \log \left (-x + 5\right ) - 1\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 58, normalized size = 1.66 \begin {gather*} \frac {e^{10}}{e^{10} \log \left (\frac {x^{2} - x}{\log \left (5\right ) + \log \left (-x + 5\right ) - 1}\right )^{2} + 2 \, e^{5} \log \left (\frac {x^{2} - x}{\log \left (5\right ) + \log \left (-x + 5\right ) - 1}\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 48, normalized size = 1.37 \begin {gather*} \frac {e^{10}}{e^{10} \log {\left (\frac {x^{2} - x}{\log {\left (5 - x \right )} - 1 + \log {\left (5 \right )}} \right )}^{2} + 2 e^{5} \log {\left (\frac {x^{2} - x}{\log {\left (5 - x \right )} - 1 + \log {\left (5 \right )}} \right )} + 1} \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. 86 vs.
\(2 (34) = 68\).
time = 9.26, size = 86, normalized size = 2.46 \begin {gather*} \frac {e^{10}}{e^{10} \log \left (x^{2} - x\right )^{2} - 2 \, e^{10} \log \left (x^{2} - x\right ) \log \left (\log \left (5\right ) + \log \left (-x + 5\right ) - 1\right ) + e^{10} \log \left (\log \left (5\right ) + \log \left (-x + 5\right ) - 1\right )^{2} + 2 \, e^{5} \log \left (x^{2} - x\right ) - 2 \, e^{5} \log \left (\log \left (5\right ) + \log \left (-x + 5\right ) - 1\right ) + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.37, size = 32, normalized size = 0.91 \begin {gather*} \frac {{\mathrm {e}}^{10}}{{\left ({\mathrm {e}}^5\,\ln \left (-\frac {x-x^2}{\ln \left (5\right )+\ln \left (5-x\right )-1}\right )+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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