Optimal. Leaf size=31 \[ e^{\frac {4 e^4}{1+x^2}}+\log \left (2 \log \left (3-\frac {x^2}{(3+x)^2}\right )\right ) \]
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Rubi [F] time = 1.33, 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 {-6 x-12 x^3-6 x^5+e^{4+\frac {4 e^4}{1+x^2}} \left (-648 x-648 x^2-192 x^3-16 x^4\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+x^2}\right )}{\left (81+81 x+186 x^2+164 x^3+129 x^4+85 x^5+24 x^6+2 x^7\right ) \log \left (\frac {27+18 x+2 x^2}{9+6 x+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 \left (-\frac {8 e^{4+\frac {4 e^4}{1+x^2}} x}{\left (1+x^2\right )^2}-\frac {6 x}{\left (81+81 x+24 x^2+2 x^3\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx\\ &=-\left (6 \int \frac {x}{\left (81+81 x+24 x^2+2 x^3\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx\right )-8 \int \frac {e^{4+\frac {4 e^4}{1+x^2}} x}{\left (1+x^2\right )^2} \, dx\\ &=e^{\frac {4 e^4}{1+x^2}}-6 \int \left (\frac {1}{3 (3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}+\frac {-9-2 x}{3 \left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx\\ &=e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx-2 \int \frac {-9-2 x}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx\\ &=e^{\frac {4 e^4}{1+x^2}}-2 \int \left (-\frac {9}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}-\frac {2 x}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx\\ &=e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+4 \int \frac {x}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+18 \int \frac {1}{\left (27+18 x+2 x^2\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx\\ &=e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+4 \int \left (\frac {1-\sqrt {3}}{\left (18-6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}+\frac {1+\sqrt {3}}{\left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx+18 \int \left (-\frac {2}{3 \sqrt {3} \left (-18+6 \sqrt {3}-4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}-\frac {2}{3 \sqrt {3} \left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )}\right ) \, dx\\ &=e^{\frac {4 e^4}{1+x^2}}-2 \int \frac {1}{(3+x) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx-\left (4 \sqrt {3}\right ) \int \frac {1}{\left (-18+6 \sqrt {3}-4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx-\left (4 \sqrt {3}\right ) \int \frac {1}{\left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+\left (4 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\left (18-6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx+\left (4 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\left (18+6 \sqrt {3}+4 x\right ) \log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.32, size = 33, normalized size = 1.06 \begin {gather*} e^{\frac {4 e^4}{1+x^2}}+\log \left (\log \left (\frac {27+18 x+2 x^2}{(3+x)^2}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 47, normalized size = 1.52 \begin {gather*} {\left (e^{4} \log \left (\log \left (\frac {2 \, x^{2} + 18 \, x + 27}{x^{2} + 6 \, x + 9}\right )\right ) + e^{\left (\frac {4 \, {\left (x^{2} + e^{4} + 1\right )}}{x^{2} + 1}\right )}\right )} e^{\left (-4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.42, size = 59, normalized size = 1.90 \begin {gather*} e^{\left (\frac {4 \, x^{2}}{x^{2} + 1} + \frac {4 \, e^{4}}{x^{2} + 1} + \frac {4}{x^{2} + 1} - 4\right )} + \log \left (\log \left (\frac {2 \, x^{2} + 18 \, x + 27}{x^{2} + 6 \, x + 9}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 37, normalized size = 1.19
method | result | size |
default | \(\ln \left (\ln \left (\frac {2 x^{2}+18 x +27}{x^{2}+6 x +9}\right )\right )+{\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}\) | \(37\) |
risch | \({\mathrm e}^{\frac {4 \,{\mathrm e}^{4}}{x^{2}+1}}+\ln \left (\ln \left (x^{2}+9 x +\frac {27}{2}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+9 x +\frac {27}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (3+x \right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (i \left (3+x \right )\right )^{2} \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (3+x \right )\right ) \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (3+x \right )^{2}\right )^{3}-\pi \,\mathrm {csgn}\left (i \left (x^{2}+9 x +\frac {27}{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{2}+9 x +\frac {27}{2}\right )}{\left (3+x \right )^{2}}\right )^{3}+2 i \ln \relax (2)-4 i \ln \left (3+x \right )\right )}{2}\right )\) | \(223\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 32, normalized size = 1.03 \begin {gather*} e^{\left (\frac {4 \, e^{4}}{x^{2} + 1}\right )} + \log \left (\log \left (2 \, x^{2} + 18 \, x + 27\right ) - 2 \, \log \left (x + 3\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.73, size = 36, normalized size = 1.16 \begin {gather*} \ln \left (\ln \left (\frac {2\,x^2+18\,x+27}{x^2+6\,x+9}\right )\right )+{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^4}{x^2+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.71, size = 32, normalized size = 1.03 \begin {gather*} e^{\frac {4 e^{4}}{x^{2} + 1}} + \log {\left (\log {\left (\frac {2 x^{2} + 18 x + 27}{x^{2} + 6 x + 9} \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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