Optimal. Leaf size=29 \[ \frac {2}{-2 x+\frac {4}{\log \left (\frac {x}{\left (16-e^4+\frac {4 x}{3}\right )^2}\right )}} \]
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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 {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{\left (-192+12 e^4\right ) x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx\\ &=\int \frac {96 \left (1-\frac {e^4}{16}\right )-8 x+x \left (48-3 e^4+4 x\right ) \log ^2\left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )}{x \left (48-3 e^4+4 x\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {2 \left (6 \left (16-e^4\right )+\left (56-3 e^4\right ) x-4 x^2\right )}{x^2 \left (48-3 e^4+4 x\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}+\frac {4}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )}\right ) \, dx\\ &=-\frac {1}{x}+2 \int \frac {6 \left (16-e^4\right )+\left (56-3 e^4\right ) x-4 x^2}{x^2 \left (48-3 e^4+4 x\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )} \, dx\\ &=-\frac {1}{x}+2 \int \left (\frac {8}{\left (-48+3 e^4-4 x\right ) \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}+\frac {2}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}+\frac {1}{x \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2}\right ) \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )} \, dx\\ &=-\frac {1}{x}+2 \int \frac {1}{x \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx+4 \int \frac {1}{x^2 \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )} \, dx+16 \int \frac {1}{\left (-48+3 e^4-4 x\right ) \left (-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.07, size = 31, normalized size = 1.07 \begin {gather*} \frac {-1-\frac {2}{-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.04, size = 46, normalized size = 1.59
method | result | size |
risch | \(-\frac {1}{x}-\frac {2}{x \left (\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right ) x -2\right )}\) | \(46\) |
norman | \(-\frac {\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right )}{\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right ) x -2}\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 47, normalized size = 1.62 \begin {gather*} -\frac {2 \, \log \left (3\right ) - 2 \, \log \left (4 \, x - 3 \, e^{4} + 48\right ) + \log \left (x\right )}{2 \, x \log \left (3\right ) - 2 \, x \log \left (4 \, x - 3 \, e^{4} + 48\right ) + x \log \left (x\right ) - 2} \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. 62 vs.
\(2 (25) = 50\).
time = 0.37, size = 62, normalized size = 2.14 \begin {gather*} -\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right ) - 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. 42 vs.
\(2 (20) = 40\).
time = 0.15, size = 42, normalized size = 1.45 \begin {gather*} - \frac {2}{x^{2} \log {\left (\frac {9 x}{16 x^{2} + 384 x + \left (- 24 x - 288\right ) e^{4} + 2304 + 9 e^{8}} \right )} - 2 x} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 66 vs.
\(2 (25) = 50\).
time = 0.88, size = 66, normalized size = 2.28 \begin {gather*} -\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right ) - 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.54, size = 51, normalized size = 1.76 \begin {gather*} -\frac {2}{x\,\left (2\,x\,\ln \left (3\right )-x\,\ln \left (384\,x-288\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8-24\,x\,{\mathrm {e}}^4+16\,x^2+2304\right )+x\,\ln \left (x\right )-2\right )}-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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