Optimal. Leaf size=24 \[ e^5+5 (-2+x)^4 x+2 \log ^2\left (e^{-x} x\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 23, normalized size of antiderivative = 0.96, number of steps
used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {14, 34, 45,
6818} \begin {gather*} 5 x (2-x)^4+2 \log ^2\left (e^{-x} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 34
Rule 45
Rule 6818
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5 (-2+x)^3 (-2+5 x)-\frac {4 (-1+x) \log \left (e^{-x} x\right )}{x}\right ) \, dx\\ &=-\left (4 \int \frac {(-1+x) \log \left (e^{-x} x\right )}{x} \, dx\right )+5 \int (-2+x)^3 (-2+5 x) \, dx\\ &=5 (2-x)^4 x+2 \log ^2\left (e^{-x} x\right )\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(24)=48\).
time = 0.04, size = 71, normalized size = 2.96 \begin {gather*} 76 x-158 x^2+120 x^3-40 x^4+5 x^5-4 x \log (x)+2 \log ^2(x)-4 x \left (-1+x-\log (x)+\log \left (e^{-x} x\right )\right )+4 \log (x) \left (x-\log (x)+\log \left (e^{-x} x\right )\right ) \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. \(56\) vs.
\(2(22)=44\).
time = 0.73, size = 57, normalized size = 2.38
method | result | size |
default | \(5 x^{5}-40 x^{4}+120 x^{3}-162 x^{2}+80 x +4 \ln \left (x \,{\mathrm e}^{-x}\right ) \ln \left (x \right )-4 x \ln \left (x \,{\mathrm e}^{-x}\right )-2 \ln \left (x \right )^{2}+4 x \ln \left (x \right )\) | \(57\) |
risch | \(\left (4 x -4 \ln \left (x \right )\right ) \ln \left ({\mathrm e}^{x}\right )+2 \ln \left (x \right )^{2}+5 x^{5}+2 i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )-2 i x \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}-2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}+2 i x \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}-40 x^{4}+120 x^{3}-162 x^{2}+80 x -2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )+2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}+2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2}-2 i \pi \ln \left (x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3}\) | \(223\) |
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. 61 vs.
\(2 (22) = 44\).
time = 0.29, size = 61, normalized size = 2.54 \begin {gather*} 5 \, x^{5} - 40 \, x^{4} + 120 \, x^{3} - 162 \, x^{2} - 4 \, x \log \left (x e^{\left (-x\right )}\right ) + 4 \, {\left (x - \log \left (x\right )\right )} \log \left (x\right ) + 4 \, \log \left (x e^{\left (-x\right )}\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2} + 80 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 35, normalized size = 1.46 \begin {gather*} 5 \, x^{5} - 40 \, x^{4} + 120 \, x^{3} - 160 \, x^{2} + 2 \, \log \left (x e^{\left (-x\right )}\right )^{2} + 80 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 32, normalized size = 1.33 \begin {gather*} 5 x^{5} - 40 x^{4} + 120 x^{3} - 160 x^{2} + 80 x + 2 \log {\left (x e^{- x} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 35, normalized size = 1.46 \begin {gather*} 5 \, x^{5} - 40 \, x^{4} + 120 \, x^{3} - 158 \, x^{2} - 4 \, x \log \left (x\right ) + 2 \, \log \left (x\right )^{2} + 80 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.53, size = 35, normalized size = 1.46 \begin {gather*} 5\,x^5-40\,x^4+120\,x^3-158\,x^2-4\,x\,\ln \left (x\right )+80\,x+2\,{\ln \left (x\right )}^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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