Optimal. Leaf size=24 \[ \frac {e^3 (5+x)}{4+x^2-\frac {x}{\log (x \log (x))}} \]
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Rubi [F] time = 13.40, 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^3 (-5-x)+e^3 (-5-x) \log (x)+5 e^3 \log (x) \log (x \log (x))+e^3 \left (4-10 x-x^2\right ) \log (x) \log ^2(x \log (x))}{x^2 \log (x)+\left (-8 x-2 x^3\right ) \log (x) \log (x \log (x))+\left (16+8 x^2+x^4\right ) \log (x) \log ^2(x \log (x))} \, 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^3 \left (-5-x-\log (x) \left (5+x-5 \log (x \log (x))+\left (-4+10 x+x^2\right ) \log ^2(x \log (x))\right )\right )}{\log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx\\ &=e^3 \int \frac {-5-x-\log (x) \left (5+x-5 \log (x \log (x))+\left (-4+10 x+x^2\right ) \log ^2(x \log (x))\right )}{\log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx\\ &=e^3 \int \left (\frac {4-10 x-x^2}{\left (4+x^2\right )^2}-\frac {(5+x) \left (16+8 x^2+x^4+16 \log (x)-4 x \log (x)+8 x^2 \log (x)+x^3 \log (x)+x^4 \log (x)\right )}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {20+8 x-15 x^2-2 x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx\\ &=e^3 \int \frac {4-10 x-x^2}{\left (4+x^2\right )^2} \, dx-e^3 \int \frac {(5+x) \left (16+8 x^2+x^4+16 \log (x)-4 x \log (x)+8 x^2 \log (x)+x^3 \log (x)+x^4 \log (x)\right )}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx+e^3 \int \frac {20+8 x-15 x^2-2 x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}-\frac {1}{8} e^3 \int 0 \, dx-e^3 \int \frac {(5+x) \left (\left (4+x^2\right )^2+\left (16-4 x+8 x^2+x^3+x^4\right ) \log (x)\right )}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx+e^3 \int \left (\frac {16 (5+x)}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}+\frac {-15-2 x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}+e^3 \int \frac {-15-2 x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx-e^3 \int \left (\frac {80}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}-\frac {4 x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {36 x^2}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {13 x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {6 x^4}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {x^5}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {80}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {16 x}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {40 x^2}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {8 x^3}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {5 x^4}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}+\frac {x^5}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2}\right ) \, dx+\left (16 e^3\right ) \int \frac {5+x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx+e^3 \int \left (-\frac {15}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}-\frac {2 x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx+\left (4 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (5 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (6 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (8 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (13 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (16 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx+\left (16 e^3\right ) \int \left (\frac {5}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}+\frac {x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )}\right ) \, dx-\left (36 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (40 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \log (x) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )^2} \, dx\\ &=\frac {e^3 (5+x)}{4+x^2}-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-e^3 \int \frac {x^5}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (2 e^3\right ) \int \frac {x}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx+\left (4 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (5 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (6 e^3\right ) \int \frac {x^4}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (8 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (13 e^3\right ) \int \frac {x^3}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (15 e^3\right ) \int \frac {1}{\left (4+x^2\right ) \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx+\left (16 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx-\left (16 e^3\right ) \int \frac {x}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (36 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (40 e^3\right ) \int \frac {x^2}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx+\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \left (-x+4 \log (x \log (x))+x^2 \log (x \log (x))\right )} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx-\left (80 e^3\right ) \int \frac {1}{\left (4+x^2\right )^2 \log (x) \left (x-\left (4+x^2\right ) \log (x \log (x))\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [A] time = 0.33, size = 29, normalized size = 1.21 \begin {gather*} \frac {e^3 (5+x) \log (x \log (x))}{-x+\left (4+x^2\right ) \log (x \log (x))} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 28, normalized size = 1.17 \begin {gather*} \frac {{\left (x + 5\right )} e^{3} \log \left (x \log \relax (x)\right )}{{\left (x^{2} + 4\right )} \log \left (x \log \relax (x)\right ) - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 56, normalized size = 2.33 \begin {gather*} \frac {x e^{3} \log \relax (x) + x e^{3} \log \left (\log \relax (x)\right ) + 5 \, e^{3} \log \relax (x) + 5 \, e^{3} \log \left (\log \relax (x)\right )}{x^{2} \log \relax (x) + x^{2} \log \left (\log \relax (x)\right ) - x + 4 \, \log \relax (x) + 4 \, \log \left (\log \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.24, size = 212, normalized size = 8.83
method | result | size |
risch | \(\frac {{\mathrm e}^{3} \left (5+x \right )}{x^{2}+4}+\frac {2 i {\mathrm e}^{3} \left (5+x \right ) x}{\left (x^{2}+4\right ) \left (\pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )-\pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-\pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+\pi \,x^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+4 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )-4 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}-4 \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2}+4 \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+2 i x^{2} \ln \relax (x )+2 i x^{2} \ln \left (\ln \relax (x )\right )-2 i x +8 i \ln \relax (x )+8 i \ln \left (\ln \relax (x )\right )\right )}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 50, normalized size = 2.08 \begin {gather*} \frac {{\left (x e^{3} + 5 \, e^{3}\right )} \log \relax (x) + {\left (x e^{3} + 5 \, e^{3}\right )} \log \left (\log \relax (x)\right )}{{\left (x^{2} + 4\right )} \log \relax (x) + {\left (x^{2} + 4\right )} \log \left (\log \relax (x)\right ) - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.59, size = 33, normalized size = 1.38 \begin {gather*} \frac {\ln \left (x\,\ln \relax (x)\right )\,{\mathrm {e}}^3\,\left (x+5\right )}{4\,\ln \left (x\,\ln \relax (x)\right )-x+x^2\,\ln \left (x\,\ln \relax (x)\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.54, size = 53, normalized size = 2.21 \begin {gather*} \frac {x^{2} e^{3} + 5 x e^{3}}{- x^{3} - 4 x + \left (x^{4} + 8 x^{2} + 16\right ) \log {\left (x \log {\relax (x )} \right )}} - \frac {- x e^{3} - 5 e^{3}}{x^{2} + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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