Optimal. Leaf size=18 \[ \frac {\left (8+2^{-x^2}\right )^2}{9+\log (x)} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(18)=36\).
time = 1.40, 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 = {6820, 6874,
2339, 30, 2326} \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 2326
Rule 2339
Rule 6820
Rule 6874
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 \text {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 [A]
time = 0.43, size = 25, normalized size = 1.39 \begin {gather*} \frac {2^{-2 x^2} \left (1+2^{3+x^2}\right )^2}{9+\log (x)} \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. \(40\) vs.
\(2(19)=38\).
time = 0.76, size = 41, normalized size = 2.28
method | result | size |
risch | \(\frac {2^{-2 x^{2}}+16 \left (\frac {1}{2}\right )^{x^{2}}+64}{9+\ln \left (x \right )}\) | \(24\) |
default | \(\frac {16 \,{\mathrm e}^{-x^{2} \ln \left (2\right )}}{9+\ln \left (x \right )}+\frac {{\mathrm e}^{-2 x^{2} \ln \left (2\right )}}{9+\ln \left (x \right )}+\frac {64}{9+\ln \left (x \right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 35, normalized size = 1.94 \begin {gather*} \frac {\frac {1}{2^{2 \, x^{2}}} + \frac {16}{2^{\left (x^{2}\right )}}}{\log \left (x\right ) + 9} + \frac {64}{\log \left (x\right ) + 9} \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. 41 vs.
\(2 (18) = 36\).
time = 0.35, 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 \left (x\right ) + 9 \cdot 2^{2 \, x^{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. 48 vs.
\(2 (17) = 34\).
time = 0.14, size = 48, normalized size = 2.67 \begin {gather*} \frac {\left (\log {\left (x \right )} + 9\right ) e^{- 2 x^{2} \log {\left (2 \right )}} + \left (16 \log {\left (x \right )} + 144\right ) e^{- x^{2} \log {\left (2 \right )}}}{\log {\left (x \right )}^{2} + 18 \log {\left (x \right )} + 81} + \frac {64}{\log {\left (x \right )} + 9} \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*} \text {could not integrate} \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 \left (x\right )+9\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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