Optimal. Leaf size=29 \[ \frac {4}{2-x+x \left (4+\log \left (e^{2 e^{4 x}-2 x} x\right )\right )} \]
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Rubi [A] time = 0.20, antiderivative size = 27, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 92, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6688, 12, 6686} \begin {gather*} \frac {4}{3 x+x \log \left (e^{2 e^{4 x}-2 x} x\right )+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (-4-\left (-2+8 e^{4 x}\right ) x-\log \left (e^{2 e^{4 x}-2 x} x\right )\right )}{\left (2+3 x+x \log \left (e^{2 e^{4 x}-2 x} x\right )\right )^2} \, dx\\ &=4 \int \frac {-4-\left (-2+8 e^{4 x}\right ) x-\log \left (e^{2 e^{4 x}-2 x} x\right )}{\left (2+3 x+x \log \left (e^{2 e^{4 x}-2 x} x\right )\right )^2} \, dx\\ &=\frac {4}{2+3 x+x \log \left (e^{2 e^{4 x}-2 x} x\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 27, normalized size = 0.93 \begin {gather*} \frac {4}{2+3 x+x \log \left (e^{2 e^{4 x}-2 x} x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 25, normalized size = 0.86 \begin {gather*} \frac {4}{x \log \left (x e^{\left (-2 \, x + 2 \, e^{\left (4 \, x\right )}\right )}\right ) + 3 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 26, normalized size = 0.90 \begin {gather*} -\frac {4}{2 \, x^{2} - 2 \, x e^{\left (4 \, x\right )} - x \log \relax (x) - 3 \, x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.35, size = 399, normalized size = 13.76
method | result | size |
risch | \(\frac {8 i}{\pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )-\pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )^{2}+\pi x \mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{4 x}}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}}\right )-2 \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{{\mathrm e}^{4 x}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}}\right )^{2}+\pi x \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}}\right )^{3}-\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )^{2}+\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right )+\pi x \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )^{3}-\pi x \mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-2 x}\right )-\pi x \,\mathrm {csgn}\left (i {\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right ) \mathrm {csgn}\left (i x \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )^{2}+\pi x \mathrm {csgn}\left (i x \,{\mathrm e}^{2 \,{\mathrm e}^{4 x}-2 x}\right )^{3}-\pi x \mathrm {csgn}\left (i {\mathrm e}^{x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )+2 \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}-\pi x \mathrm {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}+4 i x \ln \left ({\mathrm e}^{{\mathrm e}^{4 x}}\right )+6 i x -4 i x \ln \left ({\mathrm e}^{x}\right )+2 i x \ln \relax (x )+4 i}\) | \(399\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 26, normalized size = 0.90 \begin {gather*} -\frac {4}{2 \, x^{2} - 2 \, x e^{\left (4 \, x\right )} - x \log \relax (x) - 3 \, x - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.38, size = 23, normalized size = 0.79 \begin {gather*} \frac {4}{x\,\left (2\,{\mathrm {e}}^{4\,x}+\ln \relax (x)+3\right )-2\,x^2+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 24, normalized size = 0.83 \begin {gather*} \frac {4}{x \log {\left (x e^{- 2 x} e^{2 e^{4 x}} \right )} + 3 x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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