Optimal. Leaf size=18 \[ x \log \left (\frac {3 e^x}{x \left (-4+x^2\right )^2}\right ) \]
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Rubi [A] time = 0.14, antiderivative size = 20, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 9, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6725, 207, 260, 321, 266, 43, 2548, 1810, 206} \begin {gather*} x \log \left (\frac {3 e^x}{x \left (4-x^2\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 206
Rule 207
Rule 260
Rule 266
Rule 321
Rule 1810
Rule 2548
Rule 6725
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4}{-4+x^2}-\frac {4 x}{-4+x^2}-\frac {5 x^2}{-4+x^2}+\frac {x^3}{-4+x^2}+\log \left (\frac {3 e^x}{x \left (-4+x^2\right )^2}\right )\right ) \, dx\\ &=4 \int \frac {1}{-4+x^2} \, dx-4 \int \frac {x}{-4+x^2} \, dx-5 \int \frac {x^2}{-4+x^2} \, dx+\int \frac {x^3}{-4+x^2} \, dx+\int \log \left (\frac {3 e^x}{x \left (-4+x^2\right )^2}\right ) \, dx\\ &=-5 x-2 \tanh ^{-1}\left (\frac {x}{2}\right )+x \log \left (\frac {3 e^x}{x \left (4-x^2\right )^2}\right )-2 \log \left (4-x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{-4+x} \, dx,x,x^2\right )-20 \int \frac {1}{-4+x^2} \, dx-\int \frac {-4+4 x+5 x^2-x^3}{4-x^2} \, dx\\ &=-5 x+8 \tanh ^{-1}\left (\frac {x}{2}\right )+x \log \left (\frac {3 e^x}{x \left (4-x^2\right )^2}\right )-2 \log \left (4-x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \left (1+\frac {4}{-4+x}\right ) \, dx,x,x^2\right )-\int \left (-5+x+\frac {16}{4-x^2}\right ) \, dx\\ &=8 \tanh ^{-1}\left (\frac {x}{2}\right )+x \log \left (\frac {3 e^x}{x \left (4-x^2\right )^2}\right )-16 \int \frac {1}{4-x^2} \, dx\\ &=x \log \left (\frac {3 e^x}{x \left (4-x^2\right )^2}\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.06, size = 18, normalized size = 1.00 \begin {gather*} x \log \left (\frac {3 e^x}{x \left (-4+x^2\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 21, normalized size = 1.17 \begin {gather*} x \log \left (\frac {3 \, e^{x}}{x^{5} - 8 \, x^{3} + 16 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.75, size = 23, normalized size = 1.28 \begin {gather*} x^{2} + x \log \left (\frac {3}{x^{5} - 8 \, x^{3} + 16 \, x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 22, normalized size = 1.22
method | result | size |
default | \(x \ln \left (\frac {3 \,{\mathrm e}^{x}}{x^{5}-8 x^{3}+16 x}\right )\) | \(22\) |
norman | \(x \ln \left (\frac {3 \,{\mathrm e}^{x}}{x^{5}-8 x^{3}+16 x}\right )\) | \(22\) |
risch | \(x \ln \left ({\mathrm e}^{x}\right )-2 x \ln \left (x^{2}-4\right )-x \ln \relax (x )+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (x^{2}-4\right )^{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{\left (x^{2}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{\left (x^{2}-4\right )^{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{\left (x^{2}-4\right )^{2}}\right )^{2}}{2}+\frac {i \pi x \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{\left (x^{2}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (x^{2}-4\right )^{2}}\right )^{2}}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left (x^{2}-4\right )^{2}\right )^{3}}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{\left (x^{2}-4\right )^{2}}\right )^{3}}{2}-i \pi x \,\mathrm {csgn}\left (i \left (x^{2}-4\right )\right ) \mathrm {csgn}\left (i \left (x^{2}-4\right )^{2}\right )^{2}-\frac {i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{\left (x^{2}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (x^{2}-4\right )^{2}}\right )}{2}-\frac {i \pi x \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{x \left (x^{2}-4\right )^{2}}\right )^{3}}{2}-\frac {i \pi x \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (\frac {i}{\left (x^{2}-4\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{x}}{\left (x^{2}-4\right )^{2}}\right )}{2}+\frac {i \pi x \mathrm {csgn}\left (i \left (x^{2}-4\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{2}-4\right )^{2}\right )}{2}+x \ln \relax (3)\) | \(339\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.76, size = 48, normalized size = 2.67 \begin {gather*} x^{2} + x {\left (\log \relax (3) + 5\right )} - 2 \, {\left (x + 2\right )} \log \left (x + 2\right ) - 2 \, {\left (x - 2\right )} \log \left (x - 2\right ) - x \log \relax (x) - 5 \, x + 4 \, \log \left (x + 2\right ) - 4 \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.04, size = 21, normalized size = 1.17 \begin {gather*} x\,\left (x+\ln \left (\frac {1}{x^5-8\,x^3+16\,x}\right )+\ln \relax (3)\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 19, normalized size = 1.06 \begin {gather*} x \log {\left (\frac {3 e^{x}}{x^{5} - 8 x^{3} + 16 x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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