Optimal. Leaf size=26 \[ \log \left (\log \left (\left (x+x \left (3-e^{-5+2 e^{-2+x}}+2 x\right )\right )^2\right )\right ) \]
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Rubi [A]
time = 0.47, antiderivative size = 29, normalized size of antiderivative = 1.12, number of steps
used = 3, number of rules used = 3, integrand size = 126, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6820, 12,
6816} \begin {gather*} \log \left (\log \left (\frac {x^2 \left (e^{2 e^{x-2}}-2 e^5 (x+2)\right )^2}{e^{10}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6816
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{2+2 e^{-2+x}}+4 e^{2 e^{-2+x}+x} x-8 e^7 (1+x)}{e^2 x \left (e^{2 e^{-2+x}}-2 e^5 (2+x)\right ) \log \left (\frac {x^2 \left (e^{2 e^{-2+x}}-2 e^5 (2+x)\right )^2}{e^{10}}\right )} \, dx\\ &=\frac {\int \frac {2 e^{2+2 e^{-2+x}}+4 e^{2 e^{-2+x}+x} x-8 e^7 (1+x)}{x \left (e^{2 e^{-2+x}}-2 e^5 (2+x)\right ) \log \left (\frac {x^2 \left (e^{2 e^{-2+x}}-2 e^5 (2+x)\right )^2}{e^{10}}\right )} \, dx}{e^2}\\ &=\log \left (\log \left (\frac {x^2 \left (e^{2 e^{-2+x}}-2 e^5 (2+x)\right )^2}{e^{10}}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 29, normalized size = 1.12 \begin {gather*} \log \left (\log \left (\frac {x^2 \left (e^{2 e^{-2+x}}-2 e^5 (2+x)\right )^2}{e^{10}}\right )\right ) \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 = 0.44, size = 373, normalized size = 14.35
method | result | size |
risch | \(\ln \left (\ln \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )-\frac {i \left (\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )-2 \pi \,\mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )\right ) \mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right ) \mathrm {csgn}\left (i x^{2} \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )^{2}+\pi \mathrm {csgn}\left (i x^{2} \left (\left (2+x \right ) {\mathrm e}^{5}-\frac {{\mathrm e}^{2 \,{\mathrm e}^{x -2}}}{2}\right )^{2}\right )^{3}+4 i \ln \left (x \right )+4 i \ln \left (2\right )-20 i\right )}{4}\right )\) | \(373\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 25, normalized size = 0.96 \begin {gather*} \log \left (\log \left (2 \, x e^{5} + 4 \, e^{5} - e^{\left (2 \, e^{\left (x - 2\right )}\right )}\right ) + \log \left (x\right ) - 5\right ) \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. 68 vs.
\(2 (24) = 48\).
time = 0.36, size = 68, normalized size = 2.62 \begin {gather*} \log \left (\log \left ({\left (x^{2} e^{\left (2 \, {\left (5 \, e^{2} + 2 \, e^{x}\right )} e^{\left (-2\right )}\right )} + 4 \, {\left (x^{4} + 4 \, x^{3} + 4 \, x^{2}\right )} e^{20} - 4 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{\left ({\left (5 \, e^{2} + 2 \, e^{x}\right )} e^{\left (-2\right )} + 10\right )}\right )} e^{\left (-20\right )}\right )\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. 63 vs.
\(2 (26) = 52\).
time = 0.62, size = 63, normalized size = 2.42 \begin {gather*} \log {\left (\log {\left (\frac {x^{2} e^{\frac {4 e^{x}}{e^{2}}} + \left (- 4 x^{3} - 8 x^{2}\right ) e^{5} e^{\frac {2 e^{x}}{e^{2}}} + \left (4 x^{4} + 16 x^{3} + 16 x^{2}\right ) e^{10}}{e^{10}} \right )} \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. 92 vs.
\(2 (24) = 48\).
time = 0.52, size = 92, normalized size = 3.54 \begin {gather*} \log \left (-\log \left (2 \, x e^{5} \mathrm {sgn}\left (2 \, x e^{5} + 4 \, e^{5} - e^{\left (2 \, e^{\left (x - 2\right )}\right )}\right ) + 4 \, e^{5} \mathrm {sgn}\left (2 \, x e^{5} + 4 \, e^{5} - e^{\left (2 \, e^{\left (x - 2\right )}\right )}\right ) - e^{\left (2 \, e^{\left (x - 2\right )}\right )} \mathrm {sgn}\left (2 \, x e^{5} + 4 \, e^{5} - e^{\left (2 \, e^{\left (x - 2\right )}\right )}\right )\right ) - \log \left (x \mathrm {sgn}\left (x\right )\right ) + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.22, size = 57, normalized size = 2.19 \begin {gather*} \ln \left (\ln \left (x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^x}+{\mathrm {e}}^{10}\,\left (4\,x^4+16\,x^3+16\,x^2\right )-{\mathrm {e}}^{2\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^x}\,{\mathrm {e}}^5\,\left (4\,x^3+8\,x^2\right )\right )-10\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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