Optimal. Leaf size=33 \[ \frac {e^{2-x} \left (\frac {1}{3} e^{-5+x} x-\log (3)\right )^4}{\log ^4\left (\frac {2}{x^2}\right )} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(33)=66\).
time = 4.51, antiderivative size = 111, normalized size of antiderivative = 3.36, number of steps
used = 20, number of rules used = 7, integrand size = 91, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 6874,
2407, 2334, 2337, 2209, 2326} \begin {gather*} \frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{x-8} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{3 x-18} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{2 x-13} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2326
Rule 2334
Rule 2337
Rule 2407
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{81} \int \frac {e^{2-x} \left (e^{-5+x} x-3 \log (3)\right )^4 \left (8 e^{-5+x} x-24 \log (3)+\left (e^{-5+x} \left (4 x+3 x^2\right )+3 x \log (3)\right ) \log \left (\frac {2}{x^2}\right )\right )}{\left (e^{-5+x} x^2-3 x \log (3)\right ) \log ^5\left (\frac {2}{x^2}\right )} \, dx\\ &=\frac {1}{81} \int \left (-\frac {108 \log ^3(3) \left (8+\log \left (\frac {2}{x^2}\right )\right )}{e^3 \log ^5\left (\frac {2}{x^2}\right )}-\frac {81 e^{2-x} \log ^4(3) \left (-8+x \log \left (\frac {2}{x^2}\right )\right )}{x \log ^5\left (\frac {2}{x^2}\right )}+\frac {54 e^{-8+x} x \log ^2(3) \left (8+2 \log \left (\frac {2}{x^2}\right )+x \log \left (\frac {2}{x^2}\right )\right )}{\log ^5\left (\frac {2}{x^2}\right )}-\frac {12 e^{-13+2 x} x^2 \log (3) \left (8+3 \log \left (\frac {2}{x^2}\right )+2 x \log \left (\frac {2}{x^2}\right )\right )}{\log ^5\left (\frac {2}{x^2}\right )}+\frac {e^{-18+3 x} x^3 \left (8+4 \log \left (\frac {2}{x^2}\right )+3 x \log \left (\frac {2}{x^2}\right )\right )}{\log ^5\left (\frac {2}{x^2}\right )}\right ) \, dx\\ &=\frac {1}{81} \int \frac {e^{-18+3 x} x^3 \left (8+4 \log \left (\frac {2}{x^2}\right )+3 x \log \left (\frac {2}{x^2}\right )\right )}{\log ^5\left (\frac {2}{x^2}\right )} \, dx-\frac {1}{27} (4 \log (3)) \int \frac {e^{-13+2 x} x^2 \left (8+3 \log \left (\frac {2}{x^2}\right )+2 x \log \left (\frac {2}{x^2}\right )\right )}{\log ^5\left (\frac {2}{x^2}\right )} \, dx+\frac {1}{3} \left (2 \log ^2(3)\right ) \int \frac {e^{-8+x} x \left (8+2 \log \left (\frac {2}{x^2}\right )+x \log \left (\frac {2}{x^2}\right )\right )}{\log ^5\left (\frac {2}{x^2}\right )} \, dx-\frac {\left (4 \log ^3(3)\right ) \int \frac {8+\log \left (\frac {2}{x^2}\right )}{\log ^5\left (\frac {2}{x^2}\right )} \, dx}{3 e^3}-\log ^4(3) \int \frac {e^{2-x} \left (-8+x \log \left (\frac {2}{x^2}\right )\right )}{x \log ^5\left (\frac {2}{x^2}\right )} \, dx\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}-\frac {\left (4 \log ^3(3)\right ) \int \left (\frac {8}{\log ^5\left (\frac {2}{x^2}\right )}+\frac {1}{\log ^4\left (\frac {2}{x^2}\right )}\right ) \, dx}{3 e^3}\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}-\frac {\left (4 \log ^3(3)\right ) \int \frac {1}{\log ^4\left (\frac {2}{x^2}\right )} \, dx}{3 e^3}-\frac {\left (32 \log ^3(3)\right ) \int \frac {1}{\log ^5\left (\frac {2}{x^2}\right )} \, dx}{3 e^3}\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}-\frac {2 x \log ^3(3)}{9 e^3 \log ^3\left (\frac {2}{x^2}\right )}+\frac {\left (2 \log ^3(3)\right ) \int \frac {1}{\log ^3\left (\frac {2}{x^2}\right )} \, dx}{9 e^3}+\frac {\left (4 \log ^3(3)\right ) \int \frac {1}{\log ^4\left (\frac {2}{x^2}\right )} \, dx}{3 e^3}\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}+\frac {x \log ^3(3)}{18 e^3 \log ^2\left (\frac {2}{x^2}\right )}-\frac {\log ^3(3) \int \frac {1}{\log ^2\left (\frac {2}{x^2}\right )} \, dx}{18 e^3}-\frac {\left (2 \log ^3(3)\right ) \int \frac {1}{\log ^3\left (\frac {2}{x^2}\right )} \, dx}{9 e^3}\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}-\frac {x \log ^3(3)}{36 e^3 \log \left (\frac {2}{x^2}\right )}+\frac {\log ^3(3) \int \frac {1}{\log \left (\frac {2}{x^2}\right )} \, dx}{36 e^3}+\frac {\log ^3(3) \int \frac {1}{\log ^2\left (\frac {2}{x^2}\right )} \, dx}{18 e^3}\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}-\frac {\log ^3(3) \int \frac {1}{\log \left (\frac {2}{x^2}\right )} \, dx}{36 e^3}-\frac {\left (\sqrt {\frac {1}{x^2}} x \log ^3(3)\right ) \text {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (\frac {2}{x^2}\right )\right )}{36 \sqrt {2} e^3}\\ &=-\frac {\sqrt {\frac {1}{x^2}} x \text {Ei}\left (-\frac {1}{2} \log \left (\frac {2}{x^2}\right )\right ) \log ^3(3)}{36 \sqrt {2} e^3}+\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}+\frac {\left (\sqrt {\frac {1}{x^2}} x \log ^3(3)\right ) \text {Subst}\left (\int \frac {e^{-x/2}}{x} \, dx,x,\log \left (\frac {2}{x^2}\right )\right )}{36 \sqrt {2} e^3}\\ &=\frac {e^{-18+3 x} x^4}{81 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 e^{-13+2 x} x^3 \log (3)}{27 \log ^4\left (\frac {2}{x^2}\right )}+\frac {2 e^{-8+x} x^2 \log ^2(3)}{3 \log ^4\left (\frac {2}{x^2}\right )}-\frac {4 x \log ^3(3)}{3 e^3 \log ^4\left (\frac {2}{x^2}\right )}+\frac {e^{2-x} \log ^4(3)}{\log ^4\left (\frac {2}{x^2}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.54, size = 34, normalized size = 1.03 \begin {gather*} \frac {e^{-18-x} \left (e^x x-3 e^5 \log (3)\right )^4}{81 \log ^4\left (\frac {2}{x^2}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.95, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (3 x^{2}+4 x \right ) {\mathrm e}^{x -5}+3 x \ln \left (3\right )\right ) \ln \left (\frac {2}{x^{2}}\right )+8 x \,{\mathrm e}^{x -5}-24 \ln \left (3\right )\right ) {\mathrm e}^{4 \ln \left (\frac {x \,{\mathrm e}^{x -5}-3 \ln \left (3\right )}{3 \ln \left (\frac {2}{x^{2}}\right )}\right )+2-x}}{\left (x^{2} {\mathrm e}^{x -5}-3 x \ln \left (3\right )\right ) \ln \left (\frac {2}{x^{2}}\right )}\, dx\]
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. 108 vs.
\(2 (30) = 60\).
time = 0.59, size = 108, normalized size = 3.27 \begin {gather*} \frac {x^{4} e^{\left (3 \, x\right )} - 12 \, x^{3} e^{\left (2 \, x + 5\right )} \log \left (3\right ) + 54 \, x^{2} e^{\left (x + 10\right )} \log \left (3\right )^{2} - 108 \, x e^{15} \log \left (3\right )^{3} + 81 \, e^{\left (-x + 20\right )} \log \left (3\right )^{4}}{81 \, {\left (e^{18} \log \left (2\right )^{4} - 8 \, e^{18} \log \left (2\right )^{3} \log \left (x\right ) + 24 \, e^{18} \log \left (2\right )^{2} \log \left (x\right )^{2} - 32 \, e^{18} \log \left (2\right ) \log \left (x\right )^{3} + 16 \, e^{18} \log \left (x\right )^{4}\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. 72 vs.
\(2 (30) = 60\).
time = 0.41, size = 72, normalized size = 2.18 \begin {gather*} \frac {{\left (x^{4} e^{\left (4 \, x - 20\right )} - 12 \, x^{3} e^{\left (3 \, x - 15\right )} \log \left (3\right ) + 54 \, x^{2} e^{\left (2 \, x - 10\right )} \log \left (3\right )^{2} - 108 \, x e^{\left (x - 5\right )} \log \left (3\right )^{3} + 81 \, \log \left (3\right )^{4}\right )} e^{\left (-x + 2\right )}}{81 \, \log \left (\frac {2}{x^{2}}\right )^{4}} \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. 136 vs.
\(2 (26) = 52\).
time = 0.35, size = 136, normalized size = 4.12 \begin {gather*} - \frac {4 x \log {\left (3 \right )}^{3}}{3 e^{3} \log {\left (\frac {2}{x^{2}} \right )}^{4}} + \frac {81 x^{4} e^{15} e^{3 x - 6} \log {\left (\frac {2}{x^{2}} \right )}^{12} - 972 x^{3} e^{18} e^{2 x - 4} \log {\left (3 \right )} \log {\left (\frac {2}{x^{2}} \right )}^{12} + 4374 x^{2} e^{21} e^{x - 2} \log {\left (3 \right )}^{2} \log {\left (\frac {2}{x^{2}} \right )}^{12} + 6561 e^{27} e^{2 - x} \log {\left (3 \right )}^{4} \log {\left (\frac {2}{x^{2}} \right )}^{12}}{6561 e^{27} \log {\left (\frac {2}{x^{2}} \right )}^{16}} \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. 256 vs.
\(2 (30) = 60\).
time = 1.77, size = 256, normalized size = 7.76 \begin {gather*} \frac {x^{4} e^{\left (4 \, x\right )} - 12 \, x^{3} e^{\left (3 \, x + 5\right )} \log \left (3\right ) + 54 \, x^{2} e^{\left (2 \, x + 10\right )} \log \left (3\right )^{2} - 108 \, x e^{\left (x + 15\right )} \log \left (3\right )^{3} + 81 \, e^{20} \log \left (3\right )^{4}}{81 \, {\left (e^{\left (x + 18\right )} \log \left (2\right )^{4} - 8 \, e^{\left (x + 18\right )} \log \left (2\right )^{3} \log \left (x\right ) + 24 \, e^{\left (x + 18\right )} \log \left (2\right )^{2} \log \left (x\right )^{2} - 32 \, e^{\left (x + 18\right )} \log \left (2\right ) \log \left (x\right )^{3} + 16 \, e^{\left (x + 18\right )} \log \left (x\right )^{4} - 8 \, e^{\left (x + 18\right )} \log \left (2\right )^{3} \log \left (\mathrm {sgn}\left (x\right )\right ) + 48 \, e^{\left (x + 18\right )} \log \left (2\right )^{2} \log \left (x\right ) \log \left (\mathrm {sgn}\left (x\right )\right ) - 96 \, e^{\left (x + 18\right )} \log \left (2\right ) \log \left (x\right )^{2} \log \left (\mathrm {sgn}\left (x\right )\right ) + 64 \, e^{\left (x + 18\right )} \log \left (x\right )^{3} \log \left (\mathrm {sgn}\left (x\right )\right ) + 24 \, e^{\left (x + 18\right )} \log \left (2\right )^{2} \log \left (\mathrm {sgn}\left (x\right )\right )^{2} - 96 \, e^{\left (x + 18\right )} \log \left (2\right ) \log \left (x\right ) \log \left (\mathrm {sgn}\left (x\right )\right )^{2} + 96 \, e^{\left (x + 18\right )} \log \left (x\right )^{2} \log \left (\mathrm {sgn}\left (x\right )\right )^{2} - 32 \, e^{\left (x + 18\right )} \log \left (2\right ) \log \left (\mathrm {sgn}\left (x\right )\right )^{3} + 64 \, e^{\left (x + 18\right )} \log \left (x\right ) \log \left (\mathrm {sgn}\left (x\right )\right )^{3} + 16 \, e^{\left (x + 18\right )} \log \left (\mathrm {sgn}\left (x\right )\right )^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.42, size = 1729, normalized size = 52.39 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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