Optimal. Leaf size=24 \[ -25+5 e^{-5+\frac {1}{\log (x (2+\log (x)))}}+x \left (x+x^2\right ) \]
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Rubi [A]
time = 3.07, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps
used = 8, number of rules used = 5, integrand size = 120, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2641, 6874,
6820, 45, 6838} \begin {gather*} x^3+x^2+5 e^{\frac {1}{\log (x (\log (x)+2))}-5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2641
Rule 6820
Rule 6838
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {-1+5 \log (2 x+x \log (x))}{\log (2 x+x \log (x))}\right ) \left (-15-5 \log (x)+\exp \left (\frac {-1+5 \log (2 x+x \log (x))}{\log (2 x+x \log (x))}\right ) \left (4 x^2+6 x^3+\left (2 x^2+3 x^3\right ) \log (x)\right ) \log ^2(2 x+x \log (x))\right )}{x (2+\log (x)) \log ^2(2 x+x \log (x))} \, dx\\ &=\int \left (\exp \left (5-\frac {1}{\log (x (2+\log (x)))}-\frac {-1+5 \log (2 x+x \log (x))}{\log (2 x+x \log (x))}\right ) x (2+3 x)+\frac {5 \exp \left (-\frac {-1+5 \log (2 x+x \log (x))}{\log (2 x+x \log (x))}\right ) (-3-\log (x))}{x (2+\log (x)) \log ^2(2 x+x \log (x))}\right ) \, dx\\ &=5 \int \frac {\exp \left (-\frac {-1+5 \log (2 x+x \log (x))}{\log (2 x+x \log (x))}\right ) (-3-\log (x))}{x (2+\log (x)) \log ^2(2 x+x \log (x))} \, dx+\int \exp \left (5-\frac {1}{\log (x (2+\log (x)))}-\frac {-1+5 \log (2 x+x \log (x))}{\log (2 x+x \log (x))}\right ) x (2+3 x) \, dx\\ &=5 \int \frac {e^{-5+\frac {1}{\log (x (2+\log (x)))}} (-3-\log (x))}{x (2+\log (x)) \log ^2(2 x+x \log (x))} \, dx+\int x (2+3 x) \, dx\\ &=5 e^{-5+\frac {1}{\log (x (2+\log (x)))}}+\int \left (2 x+3 x^2\right ) \, dx\\ &=5 e^{-5+\frac {1}{\log (x (2+\log (x)))}}+x^2+x^3\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.06, size = 28, normalized size = 1.17 \begin {gather*} \frac {5 e^{\frac {1}{\log (x (2+\log (x)))}}+e^5 x^2 (1+x)}{e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.85, size = 208, normalized size = 8.67
method | result | size |
risch | \(x^{3}+x^{2}+5 \,{\mathrm e}^{-\frac {-5 i \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right )^{3}+5 i \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right )^{2} \mathrm {csgn}\left (i x \right )+5 i \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \left (x \right )+2\right )\right )-5 i \pi \,\mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )+2\right )\right )+10 \ln \left (x \right )+10 \ln \left (\ln \left (x \right )+2\right )-2}{-i \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right )^{3}+i \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right )^{2} \mathrm {csgn}\left (i \left (\ln \left (x \right )+2\right )\right )-i \pi \,\mathrm {csgn}\left (i x \left (\ln \left (x \right )+2\right )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\ln \left (x \right )+2\right )\right )+2 \ln \left (x \right )+2 \ln \left (\ln \left (x \right )+2\right )}}\) | \(208\) |
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. 59 vs.
\(2 (23) = 46\).
time = 0.44, size = 59, normalized size = 2.46 \begin {gather*} x^{3} + x^{2} + \frac {5 \, e^{\left (\frac {1}{\log \left (x\right ) + \log \left (\log \left (x\right ) + 2\right )}\right )} \log \left (x\right )}{e^{5} \log \left (x\right ) + 3 \, e^{5}} + \frac {15 \, e^{\left (\frac {1}{\log \left (x\right ) + \log \left (\log \left (x\right ) + 2\right )}\right )}}{e^{5} \log \left (x\right ) + 3 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (23) = 46\).
time = 0.35, size = 64, normalized size = 2.67 \begin {gather*} {\left ({\left (x^{3} + x^{2}\right )} e^{\left (\frac {5 \, \log \left (x \log \left (x\right ) + 2 \, x\right ) - 1}{\log \left (x \log \left (x\right ) + 2 \, x\right )}\right )} + 5\right )} e^{\left (-\frac {5 \, \log \left (x \log \left (x\right ) + 2 \, x\right ) - 1}{\log \left (x \log \left (x\right ) + 2 \, x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 32, normalized size = 1.33 \begin {gather*} x^{3} + x^{2} + 5 e^{- \frac {5 \log {\left (x \log {\left (x \right )} + 2 x \right )} - 1}{\log {\left (x \log {\left (x \right )} + 2 x \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.44, size = 23, normalized size = 0.96 \begin {gather*} x^2+x^3+5\,{\mathrm {e}}^{\frac {1}{\ln \left (2\,x+x\,\ln \left (x\right )\right )}}\,{\mathrm {e}}^{-5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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