Optimal. Leaf size=29 \[ e^{e^{16 x^2}}+x+(5+x) \left (x+\frac {2}{\log (x)}\right )-\log \left (x^2\right ) \]
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Rubi [A] time = 0.68, antiderivative size = 33, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 11, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.193, Rules used = {6742, 6715, 2282, 2194, 6688, 14, 2353, 2297, 2298, 2302, 30} \begin {gather*} x^2+e^{e^{16 x^2}}+6 x+\frac {2 x}{\log (x)}-2 \log (x)+\frac {10}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2194
Rule 2282
Rule 2297
Rule 2298
Rule 2302
Rule 2353
Rule 6688
Rule 6715
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (32 e^{e^{16 x^2}+16 x^2} x+\frac {2 \left (-5-x+x \log (x)-\log ^2(x)+3 x \log ^2(x)+x^2 \log ^2(x)\right )}{x \log ^2(x)}\right ) \, dx\\ &=2 \int \frac {-5-x+x \log (x)-\log ^2(x)+3 x \log ^2(x)+x^2 \log ^2(x)}{x \log ^2(x)} \, dx+32 \int e^{e^{16 x^2}+16 x^2} x \, dx\\ &=2 \int \frac {-5-x+x \log (x)+\left (-1+3 x+x^2\right ) \log ^2(x)}{x \log ^2(x)} \, dx+16 \operatorname {Subst}\left (\int e^{e^{16 x}+16 x} \, dx,x,x^2\right )\\ &=2 \int \left (\frac {-1+3 x+x^2}{x}+\frac {-5-x}{x \log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx+\operatorname {Subst}\left (\int e^x \, dx,x,e^{16 x^2}\right )\\ &=e^{e^{16 x^2}}+2 \int \frac {-1+3 x+x^2}{x} \, dx+2 \int \frac {-5-x}{x \log ^2(x)} \, dx+2 \int \frac {1}{\log (x)} \, dx\\ &=e^{e^{16 x^2}}+2 \text {li}(x)+2 \int \left (3-\frac {1}{x}+x\right ) \, dx+2 \int \left (-\frac {1}{\log ^2(x)}-\frac {5}{x \log ^2(x)}\right ) \, dx\\ &=e^{e^{16 x^2}}+6 x+x^2-2 \log (x)+2 \text {li}(x)-2 \int \frac {1}{\log ^2(x)} \, dx-10 \int \frac {1}{x \log ^2(x)} \, dx\\ &=e^{e^{16 x^2}}+6 x+x^2+\frac {2 x}{\log (x)}-2 \log (x)+2 \text {li}(x)-2 \int \frac {1}{\log (x)} \, dx-10 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=e^{e^{16 x^2}}+6 x+x^2+\frac {10}{\log (x)}+\frac {2 x}{\log (x)}-2 \log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.18, size = 28, normalized size = 0.97 \begin {gather*} e^{e^{16 x^2}}+x (6+x)+\frac {2 (5+x)}{\log (x)}-2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 67, normalized size = 2.31 \begin {gather*} \frac {{\left ({\left (x^{2} + 6 \, x\right )} e^{\left (16 \, x^{2}\right )} \log \relax (x) - 2 \, e^{\left (16 \, x^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (x + 5\right )} e^{\left (16 \, x^{2}\right )} + e^{\left (16 \, x^{2} + e^{\left (16 \, x^{2}\right )}\right )} \log \relax (x)\right )} e^{\left (-16 \, x^{2}\right )}}{\log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 80, normalized size = 2.76 \begin {gather*} \frac {{\left (x^{2} e^{\left (16 \, x^{2}\right )} \log \relax (x) + 6 \, x e^{\left (16 \, x^{2}\right )} \log \relax (x) - 2 \, e^{\left (16 \, x^{2}\right )} \log \relax (x)^{2} + 2 \, x e^{\left (16 \, x^{2}\right )} + e^{\left (16 \, x^{2} + e^{\left (16 \, x^{2}\right )}\right )} \log \relax (x) + 10 \, e^{\left (16 \, x^{2}\right )}\right )} e^{\left (-16 \, x^{2}\right )}}{\log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 28, normalized size = 0.97
method | result | size |
risch | \(6 x +x^{2}-2 \ln \relax (x )+\frac {2 x +10}{\ln \relax (x )}+{\mathrm e}^{{\mathrm e}^{16 x^{2}}}\) | \(28\) |
default | \(6 x +x^{2}-2 \ln \relax (x )+\frac {2 x}{\ln \relax (x )}+\frac {10}{\ln \relax (x )}+{\mathrm e}^{{\mathrm e}^{16 x^{2}}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.40, size = 37, normalized size = 1.28 \begin {gather*} x^{2} + 6 \, x + \frac {10}{\log \relax (x)} + 2 \, {\rm Ei}\left (\log \relax (x)\right ) + e^{\left (e^{\left (16 \, x^{2}\right )}\right )} - 2 \, \Gamma \left (-1, -\log \relax (x)\right ) - 2 \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.30, size = 33, normalized size = 1.14 \begin {gather*} 8\,x+{\mathrm {e}}^{{\mathrm {e}}^{16\,x^2}}-2\,\ln \relax (x)+\frac {2\,x-2\,x\,\ln \relax (x)+10}{\ln \relax (x)}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 27, normalized size = 0.93 \begin {gather*} x^{2} + 6 x + \frac {2 x + 10}{\log {\relax (x )}} + e^{e^{16 x^{2}}} - 2 \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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