Optimal. Leaf size=18 \[ \frac {\left (8+2^{-x^2}\right )^2}{9+\log (x)} \]
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Rubi [B] time = 2.18, antiderivative size = 87, normalized size of antiderivative = 4.83, number of steps used = 7, number of rules used = 5, integrand size = 90, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6688, 6742, 2302, 30, 2288} \begin {gather*} \frac {2^{-2 x^2-1} \left (x^2 \log (16) \log (x)+36 x^2 \log (2)\right )}{x^2 \log (4) (\log (x)+9)^2}+\frac {2^{2-x^2} \left (x^2 \log (16) \log (x)+36 x^2 \log (2)\right )}{x^2 \log (2) (\log (x)+9)^2}+\frac {64}{\log (x)+9} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2288
Rule 2302
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4^{-x^2} \left (1+2^{3+x^2}\right ) \left (-1-2^{3+x^2}-36 x^2 \log (2)-x^2 \log (16) \log (x)\right )}{x (9+\log (x))^2} \, dx\\ &=\int \left (-\frac {64}{x (9+\log (x))^2}+\frac {4^{-x^2} \left (-1-36 x^2 \log (2)-x^2 \log (16) \log (x)\right )}{x (9+\log (x))^2}-\frac {2^{3-x^2} \left (2+36 x^2 \log (2)+x^2 \log (16) \log (x)\right )}{x (9+\log (x))^2}\right ) \, dx\\ &=-\left (64 \int \frac {1}{x (9+\log (x))^2} \, dx\right )+\int \frac {4^{-x^2} \left (-1-36 x^2 \log (2)-x^2 \log (16) \log (x)\right )}{x (9+\log (x))^2} \, dx-\int \frac {2^{3-x^2} \left (2+36 x^2 \log (2)+x^2 \log (16) \log (x)\right )}{x (9+\log (x))^2} \, dx\\ &=\frac {2^{2-x^2} \left (36 x^2 \log (2)+x^2 \log (16) \log (x)\right )}{x^2 \log (2) (9+\log (x))^2}+\frac {2^{-1-2 x^2} \left (36 x^2 \log (2)+x^2 \log (16) \log (x)\right )}{x^2 \log (4) (9+\log (x))^2}-64 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,9+\log (x)\right )\\ &=\frac {64}{9+\log (x)}+\frac {2^{2-x^2} \left (36 x^2 \log (2)+x^2 \log (16) \log (x)\right )}{x^2 \log (2) (9+\log (x))^2}+\frac {2^{-1-2 x^2} \left (36 x^2 \log (2)+x^2 \log (16) \log (x)\right )}{x^2 \log (4) (9+\log (x))^2}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.68, size = 87, normalized size = 4.83 \begin {gather*} -\frac {2^{-1-2 x^2} \left (-36 \log (2) \left (\log (2)+2^{3+x^2} \left (1+2^{2+x^2}\right ) \log (4)\right )-\left (2^{3+x^2} \log (4) \log (16)+\log (2) \left (2^{7+2 x^2} \log (4)+\log (16)\right )\right ) \log (x)\right )}{\log (2) \log (4) (9+\log (x))^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 41, normalized size = 2.28 \begin {gather*} \frac {64 \cdot 2^{2 \, x^{2}} + 16 \cdot 2^{\left (x^{2}\right )} + 1}{2^{2 \, x^{2}} \log \relax (x) + 9 \cdot 2^{2 \, x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (\frac {8 \, x^{2} \log \relax (2)}{2^{\left (x^{2}\right )}} + \frac {x^{2} \log \relax (2)}{2^{2 \, x^{2}}}\right )} \log \relax (x) + \frac {16 \, {\left (18 \, x^{2} \log \relax (2) + 1\right )}}{2^{\left (x^{2}\right )}} + \frac {36 \, x^{2} \log \relax (2) + 1}{2^{2 \, x^{2}}} + 64}{x \log \relax (x)^{2} + 18 \, x \log \relax (x) + 81 \, x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 24, normalized size = 1.33
method | result | size |
risch | \(\frac {2^{-2 x^{2}}+16 \left (\frac {1}{2}\right )^{x^{2}}+64}{9+\ln \relax (x )}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 35, normalized size = 1.94 \begin {gather*} \frac {\frac {1}{2^{2 \, x^{2}}} + \frac {16}{2^{\left (x^{2}\right )}}}{\log \relax (x) + 9} + \frac {64}{\log \relax (x) + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.67, size = 27, normalized size = 1.50 \begin {gather*} \frac {{\left (8\,2^{x^2}+1\right )}^2}{2^{2\,x^2}\,\left (\ln \relax (x)+9\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 48, normalized size = 2.67 \begin {gather*} \frac {\left (\log {\relax (x )} + 9\right ) e^{- 2 x^{2} \log {\relax (2 )}} + \left (16 \log {\relax (x )} + 144\right ) e^{- x^{2} \log {\relax (2 )}}}{\log {\relax (x )}^{2} + 18 \log {\relax (x )} + 81} + \frac {64}{\log {\relax (x )} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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