Optimal. Leaf size=25 \[ -1+\left (e^{-4+x}-\frac {4+\frac {x^3}{\log (x)}}{x}\right )^2 \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(56\) vs. \(2(25)=50\).
time = 1.72, antiderivative size = 56, normalized size of antiderivative = 2.24, number of steps
used = 18, number of rules used = 10, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6874, 2225,
2326, 6820, 2343, 2346, 2209, 2367, 2334, 2335} \begin {gather*} \frac {x^4}{\log ^2(x)}+\frac {16}{x^2}-\frac {2 e^{x-4} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+e^{2 x-8}+\frac {8 x}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2225
Rule 2326
Rule 2334
Rule 2335
Rule 2343
Rule 2346
Rule 2367
Rule 6820
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 e^{-8+2 x}-\frac {2 e^{-4+x} \left (-x^3+2 x^3 \log (x)+x^4 \log (x)-4 \log ^2(x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+\frac {2 \left (-x^6-4 x^3 \log (x)+2 x^6 \log (x)+4 x^3 \log ^2(x)-16 \log ^3(x)\right )}{x^3 \log ^3(x)}\right ) \, dx\\ &=2 \int e^{-8+2 x} \, dx-2 \int \frac {e^{-4+x} \left (-x^3+2 x^3 \log (x)+x^4 \log (x)-4 \log ^2(x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)} \, dx+2 \int \frac {-x^6-4 x^3 \log (x)+2 x^6 \log (x)+4 x^3 \log ^2(x)-16 \log ^3(x)}{x^3 \log ^3(x)} \, dx\\ &=e^{-8+2 x}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+2 \int \left (-\frac {16}{x^3}-\frac {x^3}{\log ^3(x)}+\frac {2 \left (-2+x^3\right )}{\log ^2(x)}+\frac {4}{\log (x)}\right ) \, dx\\ &=e^{-8+2 x}+\frac {16}{x^2}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}-2 \int \frac {x^3}{\log ^3(x)} \, dx+4 \int \frac {-2+x^3}{\log ^2(x)} \, dx+8 \int \frac {1}{\log (x)} \, dx\\ &=e^{-8+2 x}+\frac {16}{x^2}+\frac {x^4}{\log ^2(x)}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+8 \text {li}(x)+4 \int \left (-\frac {2}{\log ^2(x)}+\frac {x^3}{\log ^2(x)}\right ) \, dx-4 \int \frac {x^3}{\log ^2(x)} \, dx\\ &=e^{-8+2 x}+\frac {16}{x^2}+\frac {x^4}{\log ^2(x)}+\frac {4 x^4}{\log (x)}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+8 \text {li}(x)+4 \int \frac {x^3}{\log ^2(x)} \, dx-8 \int \frac {1}{\log ^2(x)} \, dx-16 \int \frac {x^3}{\log (x)} \, dx\\ &=e^{-8+2 x}+\frac {16}{x^2}+\frac {x^4}{\log ^2(x)}+\frac {8 x}{\log (x)}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+8 \text {li}(x)-8 \int \frac {1}{\log (x)} \, dx+16 \int \frac {x^3}{\log (x)} \, dx-16 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=e^{-8+2 x}+\frac {16}{x^2}-16 \text {Ei}(4 \log (x))+\frac {x^4}{\log ^2(x)}+\frac {8 x}{\log (x)}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}+16 \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )\\ &=e^{-8+2 x}+\frac {16}{x^2}+\frac {x^4}{\log ^2(x)}+\frac {8 x}{\log (x)}-\frac {2 e^{-4+x} \left (x^4 \log (x)+4 x \log ^2(x)\right )}{x^2 \log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.30, size = 36, normalized size = 1.44 \begin {gather*} \frac {\left (e^4 x^3+\left (4 e^4-e^x x\right ) \log (x)\right )^2}{e^8 x^2 \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 48, normalized size = 1.92
method | result | size |
risch | \(\frac {x^{2} {\mathrm e}^{2 x -8}-8 x \,{\mathrm e}^{x -4}+16}{x^{2}}+\frac {x \left (x^{3}-2 x \,{\mathrm e}^{x -4} \ln \left (x \right )+8 \ln \left (x \right )\right )}{\ln \left (x \right )^{2}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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. 54 vs.
\(2 (25) = 50\).
time = 0.37, size = 54, normalized size = 2.16 \begin {gather*} \frac {x^{6} + {\left (x^{2} e^{\left (2 \, x - 8\right )} - 8 \, x e^{\left (x - 4\right )} + 16\right )} \log \left (x\right )^{2} - 2 \, {\left (x^{4} e^{\left (x - 4\right )} - 4 \, x^{3}\right )} \log \left (x\right )}{x^{2} \log \left (x\right )^{2}} \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. 53 vs.
\(2 (17) = 34\).
time = 0.12, size = 53, normalized size = 2.12 \begin {gather*} \frac {x^{4} + 8 x \log {\left (x \right )}}{\log {\left (x \right )}^{2}} + \frac {x e^{2 x - 8} \log {\left (x \right )} + \left (- 2 x^{3} - 8 \log {\left (x \right )}\right ) e^{x - 4}}{x \log {\left (x \right )}} + \frac {16}{x^{2}} \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. 70 vs.
\(2 (25) = 50\).
time = 0.46, size = 70, normalized size = 2.80 \begin {gather*} \frac {{\left (x^{6} e^{12} - 2 \, x^{4} e^{\left (x + 8\right )} \log \left (x\right ) + 8 \, x^{3} e^{12} \log \left (x\right ) + x^{2} e^{\left (2 \, x + 4\right )} \log \left (x\right )^{2} - 8 \, x e^{\left (x + 8\right )} \log \left (x\right )^{2} + 16 \, e^{12} \log \left (x\right )^{2}\right )} e^{\left (-12\right )}}{x^{2} \log \left (x\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.88, size = 49, normalized size = 1.96 \begin {gather*} {\mathrm {e}}^{2\,x-8}+\frac {8\,x}{\ln \left (x\right )}-\frac {8\,{\mathrm {e}}^{x-4}}{x}+\frac {x^4}{{\ln \left (x\right )}^2}+\frac {16}{x^2}-\frac {2\,x^2\,{\mathrm {e}}^{x-4}}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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