Optimal. Leaf size=31 \[ e^{\frac {x}{4 \log (x)}} \left (e^{3+\left (2-\frac {5}{x}+x\right )^4}+x+x^2\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(31)=62\).
time = 3.35, antiderivative size = 146, normalized size of antiderivative = 4.71, number of steps
used = 2, number of rules used = 2, integrand size = 236, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.008, Rules used = {12, 2326}
\begin {gather*} \frac {e^{\frac {x}{4 \log (x)}} \left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )-\left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )+x^7+x^6\right ) \log (x)+x^7+x^6\right )}{x^5 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {e^{\frac {x}{4 \log (x)}} \left (-\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5-x^6-x^7+\left (\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{x^5 \log ^2(x)} \, dx\\ &=\frac {e^{\frac {x}{4 \log (x)}} \left (\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5+x^6+x^7-\left (\exp \left (\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}\right ) x^5+x^6+x^7\right ) \log (x)\right )}{x^5 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.26, size = 62, normalized size = 2.00 \begin {gather*} e^{-71+\frac {x}{4 \log (x)}} \left (e^{\frac {625-1000 x+100 x^2+440 x^3-88 x^5+4 x^6+8 x^7+x^8}{x^4}}+e^{71} x (1+x)\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. \(64\) vs.
\(2(27)=54\).
time = 0.29, size = 65, normalized size = 2.10
method | result | size |
risch | \(\frac {\left (4 x^{2}+4 x +4 \,{\mathrm e}^{\frac {x^{8}+8 x^{7}+4 x^{6}-88 x^{5}-71 x^{4}+440 x^{3}+100 x^{2}-1000 x +625}{x^{4}}}\right ) {\mathrm e}^{\frac {x}{4 \ln \left (x \right )}}}{4}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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. 57 vs.
\(2 (27) = 54\).
time = 0.37, size = 57, normalized size = 1.84 \begin {gather*} {\left (x^{2} + x + e^{\left (\frac {x^{8} + 8 \, x^{7} + 4 \, x^{6} - 88 \, x^{5} - 71 \, x^{4} + 440 \, x^{3} + 100 \, x^{2} - 1000 \, x + 625}{x^{4}}\right )}\right )} e^{\left (\frac {x}{4 \, \log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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. 95 vs.
\(2 (27) = 54\).
time = 0.67, size = 95, normalized size = 3.06 \begin {gather*} x^{2} e^{\left (\frac {x}{4 \, \log \left (x\right )}\right )} + x e^{\left (\frac {x}{4 \, \log \left (x\right )}\right )} + e^{\left (\frac {4 \, x^{8} \log \left (x\right ) + 32 \, x^{7} \log \left (x\right ) + 16 \, x^{6} \log \left (x\right ) - 352 \, x^{5} \log \left (x\right ) + x^{5} - 284 \, x^{4} \log \left (x\right ) + 1760 \, x^{3} \log \left (x\right ) + 400 \, x^{2} \log \left (x\right ) - 4000 \, x \log \left (x\right ) + 2500 \, \log \left (x\right )}{4 \, x^{4} \log \left (x\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.05, size = 78, normalized size = 2.52 \begin {gather*} x\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (x\right )}}+x^2\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (x\right )}}+{\mathrm {e}}^{-88\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-71}\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (x\right )}}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{8\,x^3}\,{\mathrm {e}}^{\frac {100}{x^2}}\,{\mathrm {e}}^{440/x}\,{\mathrm {e}}^{\frac {625}{x^4}}\,{\mathrm {e}}^{-\frac {1000}{x^3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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