Optimal. Leaf size=21 \[ \frac {e^{x (-16+4 \log (x))}}{-1+8 x+x^4} \]
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Rubi [F]
time = 1.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {e^{-16 x+4 x \log (x)} \left (4-96 x-4 x^3-12 x^4+\left (-4+32 x+4 x^4\right ) \log (x)\right )}{1-16 x+64 x^2-2 x^4+16 x^5+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{4 x (-4+\log (x))} \left (4-96 x-4 x^3-12 x^4+\left (-4+32 x+4 x^4\right ) \log (x)\right )}{\left (1-8 x-x^4\right )^2} \, dx\\ &=\int \left (\frac {4 e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2}-\frac {96 e^{4 x (-4+\log (x))} x}{\left (-1+8 x+x^4\right )^2}-\frac {4 e^{4 x (-4+\log (x))} x^3}{\left (-1+8 x+x^4\right )^2}-\frac {12 e^{4 x (-4+\log (x))} x^4}{\left (-1+8 x+x^4\right )^2}+\frac {4 e^{4 x (-4+\log (x))} \log (x)}{-1+8 x+x^4}\right ) \, dx\\ &=4 \int \frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2} \, dx-4 \int \frac {e^{4 x (-4+\log (x))} x^3}{\left (-1+8 x+x^4\right )^2} \, dx+4 \int \frac {e^{4 x (-4+\log (x))} \log (x)}{-1+8 x+x^4} \, dx-12 \int \frac {e^{4 x (-4+\log (x))} x^4}{\left (-1+8 x+x^4\right )^2} \, dx-96 \int \frac {e^{4 x (-4+\log (x))} x}{\left (-1+8 x+x^4\right )^2} \, dx\\ &=4 \int \frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2} \, dx-4 \int \frac {e^{4 x (-4+\log (x))} x^3}{\left (-1+8 x+x^4\right )^2} \, dx+4 \int \frac {e^{4 x (-4+\log (x))} \log (x)}{-1+8 x+x^4} \, dx-12 \int \left (\frac {e^{4 x (-4+\log (x))} (1-8 x)}{\left (-1+8 x+x^4\right )^2}+\frac {e^{4 x (-4+\log (x))}}{-1+8 x+x^4}\right ) \, dx-96 \int \frac {e^{4 x (-4+\log (x))} x}{\left (-1+8 x+x^4\right )^2} \, dx\\ &=4 \int \frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2} \, dx-4 \int \frac {e^{4 x (-4+\log (x))} x^3}{\left (-1+8 x+x^4\right )^2} \, dx+4 \int \frac {e^{4 x (-4+\log (x))} \log (x)}{-1+8 x+x^4} \, dx-12 \int \frac {e^{4 x (-4+\log (x))} (1-8 x)}{\left (-1+8 x+x^4\right )^2} \, dx-12 \int \frac {e^{4 x (-4+\log (x))}}{-1+8 x+x^4} \, dx-96 \int \frac {e^{4 x (-4+\log (x))} x}{\left (-1+8 x+x^4\right )^2} \, dx\\ &=4 \int \frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2} \, dx-4 \int \frac {e^{4 x (-4+\log (x))} x^3}{\left (-1+8 x+x^4\right )^2} \, dx+4 \int \frac {e^{4 x (-4+\log (x))} \log (x)}{-1+8 x+x^4} \, dx-12 \int \frac {e^{4 x (-4+\log (x))}}{-1+8 x+x^4} \, dx-12 \int \left (\frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2}-\frac {8 e^{4 x (-4+\log (x))} x}{\left (-1+8 x+x^4\right )^2}\right ) \, dx-96 \int \frac {e^{4 x (-4+\log (x))} x}{\left (-1+8 x+x^4\right )^2} \, dx\\ &=4 \int \frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2} \, dx-4 \int \frac {e^{4 x (-4+\log (x))} x^3}{\left (-1+8 x+x^4\right )^2} \, dx+4 \int \frac {e^{4 x (-4+\log (x))} \log (x)}{-1+8 x+x^4} \, dx-12 \int \frac {e^{4 x (-4+\log (x))}}{\left (-1+8 x+x^4\right )^2} \, dx-12 \int \frac {e^{4 x (-4+\log (x))}}{-1+8 x+x^4} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.37, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^{-16 x} x^{4 x}}{-1+8 x+x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 21, normalized size = 1.00
method | result | size |
risch | \(\frac {x^{4 x} {\mathrm e}^{-16 x}}{x^{4}+8 x -1}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^{\left (4 \, x \log \left (x\right ) - 16 \, x\right )}}{x^{4} + 8 \, x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 21, normalized size = 1.00 \begin {gather*} \frac {e^{\left (4 \, x \log \left (x\right ) - 16 \, x\right )}}{x^{4} + 8 \, x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 19, normalized size = 0.90 \begin {gather*} \frac {e^{4 x \log {\left (x \right )} - 16 x}}{x^{4} + 8 x - 1} \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 = 0.54, size = 20, normalized size = 0.95 \begin {gather*} \frac {x^{4\,x}\,{\mathrm {e}}^{-16\,x}}{x^4+8\,x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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