Optimal. Leaf size=22 \[ \frac {4-\frac {e^4 \left (\frac {1}{4}+e^4\right )}{\log (x)}}{x} \]
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Rubi [A] time = 0.37, antiderivative size = 27, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 6741, 6742, 2306, 2309, 2178} \begin {gather*} \frac {4}{x}-\frac {e^4 \left (1+4 e^4\right )}{4 x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2306
Rule 2309
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^8+4 e^{12}+\left (e^8+4 e^{12}\right ) \log (x)-16 e^4 \log ^2(x)}{x^2 \log ^2(x)} \, dx}{4 e^4}\\ &=\frac {\int \frac {e^4 \left (e^4 \left (1+4 e^4\right )+e^4 \left (1+4 e^4\right ) \log (x)-16 \log ^2(x)\right )}{x^2 \log ^2(x)} \, dx}{4 e^4}\\ &=\frac {1}{4} \int \frac {e^4 \left (1+4 e^4\right )+e^4 \left (1+4 e^4\right ) \log (x)-16 \log ^2(x)}{x^2 \log ^2(x)} \, dx\\ &=\frac {1}{4} \int \left (-\frac {16}{x^2}+\frac {e^4+4 e^8}{x^2 \log ^2(x)}+\frac {e^4+4 e^8}{x^2 \log (x)}\right ) \, dx\\ &=\frac {4}{x}+\frac {1}{4} \left (e^4 \left (1+4 e^4\right )\right ) \int \frac {1}{x^2 \log ^2(x)} \, dx+\frac {1}{4} \left (e^4 \left (1+4 e^4\right )\right ) \int \frac {1}{x^2 \log (x)} \, dx\\ &=\frac {4}{x}-\frac {e^4 \left (1+4 e^4\right )}{4 x \log (x)}-\frac {1}{4} \left (e^4 \left (1+4 e^4\right )\right ) \int \frac {1}{x^2 \log (x)} \, dx+\frac {1}{4} \left (e^4 \left (1+4 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4}{x}+\frac {1}{4} e^4 \left (1+4 e^4\right ) \text {Ei}(-\log (x))-\frac {e^4 \left (1+4 e^4\right )}{4 x \log (x)}-\frac {1}{4} \left (e^4 \left (1+4 e^4\right )\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\frac {4}{x}-\frac {e^4 \left (1+4 e^4\right )}{4 x \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 1.45 \begin {gather*} \frac {4}{x}-\frac {e^4}{4 x \log (x)}-\frac {e^8}{x \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.25, size = 20, normalized size = 0.91 \begin {gather*} -\frac {4 \, e^{8} + e^{4} - 16 \, \log \relax (x)}{4 \, x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 26, normalized size = 1.18 \begin {gather*} \frac {{\left (16 \, e^{4} \log \relax (x) - 4 \, e^{12} - e^{8}\right )} e^{\left (-4\right )}}{4 \, x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 24, normalized size = 1.09
method | result | size |
risch | \(\frac {4}{x}-\frac {{\mathrm e}^{4} \left (4 \,{\mathrm e}^{4}+1\right )}{4 x \ln \relax (x )}\) | \(24\) |
norman | \(\frac {\left (4 \,{\mathrm e}^{2} \ln \relax (x )-\frac {{\mathrm e}^{8} \left (4 \,{\mathrm e}^{4}+1\right ) {\mathrm e}^{-2}}{4}\right ) {\mathrm e}^{-2}}{x \ln \relax (x )}\) | \(36\) |
default | \(\frac {{\mathrm e}^{-4} \left (\frac {16 \,{\mathrm e}^{4}}{x}-4 \,{\mathrm e}^{12} \expIntegralEi \left (1, \ln \relax (x )\right )-{\mathrm e}^{8} \expIntegralEi \left (1, \ln \relax (x )\right )+4 \,{\mathrm e}^{12} \left (-\frac {1}{x \ln \relax (x )}+\expIntegralEi \left (1, \ln \relax (x )\right )\right )+{\mathrm e}^{8} \left (-\frac {1}{x \ln \relax (x )}+\expIntegralEi \left (1, \ln \relax (x )\right )\right )\right )}{4}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.43, size = 45, normalized size = 2.05 \begin {gather*} \frac {1}{4} \, {\left (4 \, {\rm Ei}\left (-\log \relax (x)\right ) e^{12} + {\rm Ei}\left (-\log \relax (x)\right ) e^{8} - 4 \, e^{12} \Gamma \left (-1, \log \relax (x)\right ) - e^{8} \Gamma \left (-1, \log \relax (x)\right ) + \frac {16 \, e^{4}}{x}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 22, normalized size = 1.00 \begin {gather*} \frac {4}{x}-\frac {{\mathrm {e}}^4+4\,{\mathrm {e}}^8}{4\,x\,\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 19, normalized size = 0.86 \begin {gather*} \frac {4}{x} + \frac {- 4 e^{8} - e^{4}}{4 x \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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