Optimal. Leaf size=29 \[ \frac {\left (x-\log (3)+x^2 \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )\right )^2}{x^2} \]
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Rubi [F] time = 1.86, 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 {\left (2 e^2 x+2 x^2\right ) \log (3)+\left (-2 e^2-2 x\right ) \log ^2(3)+\left (-4 e^2 x^3+4 e^2 x^2 \log (3)\right ) \log \left (\frac {12 e^2+12 x}{x}\right )+\left (2 e^2 x^3+2 x^4\right ) \log ^2\left (\frac {12 e^2+12 x}{x}\right )-4 e^2 x^4 \log ^3\left (\frac {12 e^2+12 x}{x}\right )+\left (2 e^2 x^4+2 x^5\right ) \log ^4\left (\frac {12 e^2+12 x}{x}\right )}{e^2 x^3+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (2 e^2 x+2 x^2\right ) \log (3)+\left (-2 e^2-2 x\right ) \log ^2(3)+\left (-4 e^2 x^3+4 e^2 x^2 \log (3)\right ) \log \left (\frac {12 e^2+12 x}{x}\right )+\left (2 e^2 x^3+2 x^4\right ) \log ^2\left (\frac {12 e^2+12 x}{x}\right )-4 e^2 x^4 \log ^3\left (\frac {12 e^2+12 x}{x}\right )+\left (2 e^2 x^4+2 x^5\right ) \log ^4\left (\frac {12 e^2+12 x}{x}\right )}{x^3 \left (e^2+x\right )} \, dx\\ &=\int \frac {2 \left (x-\log (3)+x^2 \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )\right ) \left (\left (e^2+x\right ) \log (3)-2 e^2 x^2 \log \left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \left (e^2+x\right ) \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )\right )}{x^3 \left (e^2+x\right )} \, dx\\ &=2 \int \frac {\left (x-\log (3)+x^2 \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )\right ) \left (\left (e^2+x\right ) \log (3)-2 e^2 x^2 \log \left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \left (e^2+x\right ) \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )\right )}{x^3 \left (e^2+x\right )} \, dx\\ &=2 \int \left (\frac {(x-\log (3)) \log (3)}{x^3}+\frac {2 e^2 (-x+\log (3)) \log \left (12+\frac {12 e^2}{x}\right )}{x \left (e^2+x\right )}+\log ^2\left (12+\frac {12 e^2}{x}\right )+\frac {2 e^2 x \log ^3\left (12+\frac {12 e^2}{x}\right )}{-e^2-x}+x \log ^4\left (12+\frac {12 e^2}{x}\right )\right ) \, dx\\ &=2 \int \log ^2\left (12+\frac {12 e^2}{x}\right ) \, dx+2 \int x \log ^4\left (12+\frac {12 e^2}{x}\right ) \, dx+\left (4 e^2\right ) \int \frac {(-x+\log (3)) \log \left (12+\frac {12 e^2}{x}\right )}{x \left (e^2+x\right )} \, dx+\left (4 e^2\right ) \int \frac {x \log ^3\left (12+\frac {12 e^2}{x}\right )}{-e^2-x} \, dx+(2 \log (3)) \int \frac {x-\log (3)}{x^3} \, dx\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-2 \operatorname {Subst}\left (\int \frac {\log ^4\left (12+12 e^2 x\right )}{x^3} \, dx,x,\frac {1}{x}\right )+\left (4 e^2\right ) \int \frac {\log \left (12+\frac {12 e^2}{x}\right )}{x} \, dx+\left (4 e^2\right ) \int \left (-\log ^3\left (12+\frac {12 e^2}{x}\right )+\frac {e^2 \log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x}\right ) \, dx+\left (4 e^2\right ) \int \frac {(-x+\log (3)) \log \left (\frac {12 e^2+12 x}{x}\right )}{x \left (e^2+x\right )} \, dx\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )-\left (4 e^2\right ) \int \log ^3\left (12+\frac {12 e^2}{x}\right ) \, dx-\left (4 e^2\right ) \int \frac {(-x+\log (3)) \log (x)}{x \left (e^2+x\right )} \, dx+\left (4 e^2\right ) \int \frac {(-x+\log (3)) \log \left (12 e^2+12 x\right )}{x \left (e^2+x\right )} \, dx-\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (12+12 e^2 x\right )}{x} \, dx,x,\frac {1}{x}\right )-\left (48 e^2\right ) \operatorname {Subst}\left (\int \frac {\log ^3\left (12+12 e^2 x\right )}{x^2 \left (12+12 e^2 x\right )} \, dx,x,\frac {1}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx-\left (4 e^2 \left (-\log (x)+\log \left (12 e^2+12 x\right )-\log \left (\frac {12 e^2+12 x}{x}\right )\right )\right ) \int \frac {-x+\log (3)}{x \left (e^2+x\right )} \, dx\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )-4 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x \left (-\frac {1}{e^2}+\frac {x}{12 e^2}\right )^2} \, dx,x,12+\frac {12 e^2}{x}\right )-\left (4 e^2\right ) \int \left (\frac {\left (-e^2-\log (3)\right ) \log (x)}{e^2 \left (e^2+x\right )}+\frac {\log (3) \log (x)}{e^2 x}\right ) \, dx+\left (4 e^2\right ) \int \left (\frac {\left (-e^2-\log (3)\right ) \log \left (12 e^2+12 x\right )}{e^2 \left (e^2+x\right )}+\frac {\log (3) \log \left (12 e^2+12 x\right )}{e^2 x}\right ) \, dx-\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+e^2 x\right )}{x} \, dx,x,\frac {1}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx-\left (12 e^4\right ) \int \frac {\log ^2\left (12+\frac {12 e^2}{x}\right )}{x} \, dx-\left (4 e^2 \left (-\log (x)+\log \left (12 e^2+12 x\right )-\log \left (\frac {12 e^2+12 x}{x}\right )\right )\right ) \int \left (\frac {-e^2-\log (3)}{e^2 \left (e^2+x\right )}+\frac {\log (3)}{e^2 x}\right ) \, dx\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 \log (3) \log (x) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )-4 \left (e^2+\log (3)\right ) \log \left (e^2+x\right ) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )+4 e^2 \text {Li}_2\left (-\frac {e^2}{x}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\log ^3(x)}{\left (-\frac {1}{e^2}+\frac {x}{12 e^2}\right )^2} \, dx,x,12+\frac {12 e^2}{x}\right )+\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x \left (-\frac {1}{e^2}+\frac {x}{12 e^2}\right )} \, dx,x,12+\frac {12 e^2}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx+\left (12 e^4\right ) \operatorname {Subst}\left (\int \frac {\log ^2\left (12+12 e^2 x\right )}{x} \, dx,x,\frac {1}{x}\right )-(4 \log (3)) \int \frac {\log (x)}{x} \, dx+(4 \log (3)) \int \frac {\log \left (12 e^2+12 x\right )}{x} \, dx+\left (4 \left (e^2+\log (3)\right )\right ) \int \frac {\log (x)}{e^2+x} \, dx-\left (4 \left (e^2+\log (3)\right )\right ) \int \frac {\log \left (12 e^2+12 x\right )}{e^2+x} \, dx\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+4 \log (3) (2+\log (12)) \log (x)-2 \log (3) \log ^2(x)+12 e^4 \log \left (-\frac {e^2}{x}\right ) \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 e^2 \left (1+\frac {e^2}{x}\right ) x \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 \log (3) \log (x) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )-4 \left (e^2+\log (3)\right ) \log \left (e^2+x\right ) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )+4 \left (e^2+\log (3)\right ) \log (x) \log \left (1+\frac {x}{e^2}\right )+4 e^2 \text {Li}_2\left (-\frac {e^2}{x}\right )+\frac {1}{3} e^2 \operatorname {Subst}\left (\int \frac {\log ^3(x)}{-\frac {1}{e^2}+\frac {x}{12 e^2}} \, dx,x,12+\frac {12 e^2}{x}\right )-e^2 \operatorname {Subst}\left (\int \frac {\log ^2(x)}{-\frac {1}{e^2}+\frac {x}{12 e^2}} \, dx,x,12+\frac {12 e^2}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx-\left (4 e^4\right ) \operatorname {Subst}\left (\int \frac {\log ^3(x)}{x} \, dx,x,12+\frac {12 e^2}{x}\right )-\left (288 e^6\right ) \operatorname {Subst}\left (\int \frac {\log \left (-e^2 x\right ) \log \left (12+12 e^2 x\right )}{12+12 e^2 x} \, dx,x,\frac {1}{x}\right )+(4 \log (3)) \int \frac {\log \left (1+\frac {x}{e^2}\right )}{x} \, dx-\frac {1}{3} \left (e^2+\log (3)\right ) \operatorname {Subst}\left (\int \frac {12 \log (x)}{x} \, dx,x,12 e^2+12 x\right )-\left (4 \left (e^2+\log (3)\right )\right ) \int \frac {\log \left (1+\frac {x}{e^2}\right )}{x} \, dx\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+4 \log (3) (2+\log (12)) \log (x)-2 \log (3) \log ^2(x)+4 e^2 \left (1+\frac {e^2}{x}\right ) x \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 e^4 \log \left (-\frac {e^2}{x}\right ) \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 \log (3) \log (x) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )-4 \left (e^2+\log (3)\right ) \log \left (e^2+x\right ) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )+4 \left (e^2+\log (3)\right ) \log (x) \log \left (1+\frac {x}{e^2}\right )+4 e^2 \text {Li}_2\left (-\frac {e^2}{x}\right )-4 \log (3) \text {Li}_2\left (-\frac {x}{e^2}\right )+4 \left (e^2+\log (3)\right ) \text {Li}_2\left (-\frac {x}{e^2}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx-\left (4 e^4\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,\log \left (\frac {12 \left (e^2+x\right )}{x}\right )\right )-\left (12 e^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{12}\right ) \log ^2(x)}{x} \, dx,x,12+\frac {12 e^2}{x}\right )+\left (24 e^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{12}\right ) \log (x)}{x} \, dx,x,12+\frac {12 e^2}{x}\right )-\left (24 e^4\right ) \operatorname {Subst}\left (\int \frac {\log (x) \log \left (-e^2 \left (-\frac {1}{e^2}+\frac {x}{12 e^2}\right )\right )}{x} \, dx,x,12+\frac {12 e^2}{x}\right )-\left (4 \left (e^2+\log (3)\right )\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,12 e^2+12 x\right )\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+4 \log (3) (2+\log (12)) \log (x)-2 \log (3) \log ^2(x)-2 \left (e^2+\log (3)\right ) \log ^2\left (12 \left (e^2+x\right )\right )+4 e^2 \left (1+\frac {e^2}{x}\right ) x \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 e^4 \log \left (-\frac {e^2}{x}\right ) \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )-e^4 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 \log (3) \log (x) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )-4 \left (e^2+\log (3)\right ) \log \left (e^2+x\right ) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )+4 \left (e^2+\log (3)\right ) \log (x) \log \left (1+\frac {x}{e^2}\right )+4 e^2 \text {Li}_2\left (-\frac {e^2}{x}\right )-4 \log (3) \text {Li}_2\left (-\frac {x}{e^2}\right )+4 \left (e^2+\log (3)\right ) \text {Li}_2\left (-\frac {x}{e^2}\right )+12 e^4 \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right ) \text {Li}_2\left (\frac {e^2+x}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx-\left (24 e^4\right ) \operatorname {Subst}\left (\int \frac {\log (x) \text {Li}_2\left (\frac {x}{12}\right )}{x} \, dx,x,12+\frac {12 e^2}{x}\right )\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+4 \log (3) (2+\log (12)) \log (x)-2 \log (3) \log ^2(x)-2 \left (e^2+\log (3)\right ) \log ^2\left (12 \left (e^2+x\right )\right )+4 e^2 \left (1+\frac {e^2}{x}\right ) x \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 e^4 \log \left (-\frac {e^2}{x}\right ) \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )-e^4 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 \log (3) \log (x) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )-4 \left (e^2+\log (3)\right ) \log \left (e^2+x\right ) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )+4 \left (e^2+\log (3)\right ) \log (x) \log \left (1+\frac {x}{e^2}\right )+4 e^2 \text {Li}_2\left (-\frac {e^2}{x}\right )-4 \log (3) \text {Li}_2\left (-\frac {x}{e^2}\right )+4 \left (e^2+\log (3)\right ) \text {Li}_2\left (-\frac {x}{e^2}\right )+12 e^4 \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right ) \text {Li}_2\left (\frac {e^2+x}{x}\right )-24 e^4 \log \left (\frac {12 \left (e^2+x\right )}{x}\right ) \text {Li}_3\left (\frac {e^2+x}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx+\left (24 e^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{12}\right )}{x} \, dx,x,12+\frac {12 e^2}{x}\right )\\ &=\frac {(x-\log (3))^2}{x^2}+2 \left (e^2+x\right ) \log ^2\left (12+\frac {12 e^2}{x}\right )-4 e^2 \left (e^2+x\right ) \log ^3\left (12+\frac {12 e^2}{x}\right )+4 e^2 \log (12) \log (x)+4 \log (3) (2+\log (12)) \log (x)-2 \log (3) \log ^2(x)-2 \left (e^2+\log (3)\right ) \log ^2\left (12 \left (e^2+x\right )\right )+4 e^2 \left (1+\frac {e^2}{x}\right ) x \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 e^4 \log \left (-\frac {e^2}{x}\right ) \log ^3\left (\frac {12 \left (e^2+x\right )}{x}\right )-e^4 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+x^2 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )+4 \log (3) \log (x) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )-4 \left (e^2+\log (3)\right ) \log \left (e^2+x\right ) \left (\log (x)+\log \left (\frac {12 \left (e^2+x\right )}{x}\right )-\log \left (12 e^2+12 x\right )\right )+4 \left (e^2+\log (3)\right ) \log (x) \log \left (1+\frac {x}{e^2}\right )+4 e^2 \text {Li}_2\left (-\frac {e^2}{x}\right )-4 \log (3) \text {Li}_2\left (-\frac {x}{e^2}\right )+4 \left (e^2+\log (3)\right ) \text {Li}_2\left (-\frac {x}{e^2}\right )+12 e^4 \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right ) \text {Li}_2\left (\frac {e^2+x}{x}\right )-24 e^4 \log \left (\frac {12 \left (e^2+x\right )}{x}\right ) \text {Li}_3\left (\frac {e^2+x}{x}\right )+24 e^4 \text {Li}_4\left (\frac {e^2+x}{x}\right )+\left (4 e^4\right ) \int \frac {\log ^3\left (12+\frac {12 e^2}{x}\right )}{e^2+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.30, size = 55, normalized size = 1.90 \begin {gather*} \frac {\log ^2(3)-x \log (9)+2 x^2 (x-\log (3)) \log ^2\left (\frac {12 \left (e^2+x\right )}{x}\right )+x^4 \log ^4\left (\frac {12 \left (e^2+x\right )}{x}\right )}{x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 55, normalized size = 1.90 \begin {gather*} \frac {x^{4} \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{4} + 2 \, {\left (x^{3} - x^{2} \log \relax (3)\right )} \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{2} - 2 \, x \log \relax (3) + \log \relax (3)^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 256, normalized size = 8.83 \begin {gather*} \frac {{\left (e^{8} \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{4} - \frac {2 \, {\left (x + e^{2}\right )}^{2} e^{4} \log \relax (3) \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{2}}{x^{2}} + \frac {4 \, {\left (x + e^{2}\right )} e^{4} \log \relax (3) \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{2}}{x} - 2 \, e^{4} \log \relax (3) \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{2} + \frac {2 \, {\left (x + e^{2}\right )} e^{6} \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{2}}{x} - 2 \, e^{6} \log \left (\frac {12 \, {\left (x + e^{2}\right )}}{x}\right )^{2} - \frac {2 \, {\left (x + e^{2}\right )}^{3} e^{2} \log \relax (3)}{x^{3}} + \frac {4 \, {\left (x + e^{2}\right )}^{2} e^{2} \log \relax (3)}{x^{2}} - \frac {2 \, {\left (x + e^{2}\right )} e^{2} \log \relax (3)}{x} + \frac {{\left (x + e^{2}\right )}^{4} \log \relax (3)^{2}}{x^{4}} - \frac {4 \, {\left (x + e^{2}\right )}^{3} \log \relax (3)^{2}}{x^{3}} + \frac {5 \, {\left (x + e^{2}\right )}^{2} \log \relax (3)^{2}}{x^{2}} - \frac {2 \, {\left (x + e^{2}\right )} \log \relax (3)^{2}}{x}\right )} e^{\left (-2\right )}}{\frac {{\left (x + e^{2}\right )}^{2} e^{2}}{x^{2}} - \frac {2 \, {\left (x + e^{2}\right )} e^{2}}{x} + e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.78, size = 0, normalized size = 0.00 \[\int \frac {\left (2 x^{4} {\mathrm e}^{2}+2 x^{5}\right ) \ln \left (\frac {12 \,{\mathrm e}^{2}+12 x}{x}\right )^{4}-4 x^{4} {\mathrm e}^{2} \ln \left (\frac {12 \,{\mathrm e}^{2}+12 x}{x}\right )^{3}+\left (2 x^{3} {\mathrm e}^{2}+2 x^{4}\right ) \ln \left (\frac {12 \,{\mathrm e}^{2}+12 x}{x}\right )^{2}+\left (4 x^{2} {\mathrm e}^{2} \ln \relax (3)-4 x^{3} {\mathrm e}^{2}\right ) \ln \left (\frac {12 \,{\mathrm e}^{2}+12 x}{x}\right )+\left (-2 \,{\mathrm e}^{2}-2 x \right ) \ln \relax (3)^{2}+\left (2 \,{\mathrm e}^{2} x +2 x^{2}\right ) \ln \relax (3)}{x^{3} {\mathrm e}^{2}+x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 528, normalized size = 18.21 \begin {gather*} x^{2} \log \left (x + e^{2}\right )^{4} - 4 \, x^{2} {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \relax (x)^{3} + x^{2} \log \relax (x)^{4} + {\left (2 \, e^{\left (-6\right )} \log \left (x + e^{2}\right ) - 2 \, e^{\left (-6\right )} \log \relax (x) - \frac {{\left (2 \, x - e^{2}\right )} e^{\left (-4\right )}}{x^{2}}\right )} e^{2} \log \relax (3)^{2} + 4 \, {\left (x^{2} {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} - x^{2} \log \relax (x)\right )} \log \left (x + e^{2}\right )^{3} - 4 \, {\left (e^{\left (-2\right )} \log \left (x + e^{2}\right ) - e^{\left (-2\right )} \log \relax (x)\right )} e^{2} \log \relax (3) \log \left (\frac {12 \, e^{2}}{x} + 12\right ) + {\left (\log \relax (3)^{4} + 8 \, \log \relax (3)^{3} \log \relax (2) + 24 \, \log \relax (3)^{2} \log \relax (2)^{2} + 32 \, \log \relax (3) \log \relax (2)^{3} + 16 \, \log \relax (2)^{4}\right )} x^{2} + 2 \, {\left (e^{\left (-4\right )} \log \left (x + e^{2}\right ) - e^{\left (-4\right )} \log \relax (x) - \frac {e^{\left (-2\right )}}{x}\right )} e^{2} \log \relax (3) - 2 \, {\left (e^{\left (-4\right )} \log \left (x + e^{2}\right ) - e^{\left (-4\right )} \log \relax (x) - \frac {e^{\left (-2\right )}}{x}\right )} \log \relax (3)^{2} - 2 \, {\left (6 \, x^{2} {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \relax (x) - 3 \, x^{2} \log \relax (x)^{2} - 3 \, {\left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} x^{2} - x\right )} \log \left (x + e^{2}\right )^{2} + 2 \, {\left (3 \, {\left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} x^{2} + x\right )} \log \relax (x)^{2} + 2 \, {\left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} x - 2 \, {\left (e^{\left (-2\right )} \log \left (x + e^{2}\right ) - e^{\left (-2\right )} \log \relax (x)\right )} \log \relax (3) + 2 \, {\left (\log \left (x + e^{2}\right )^{2} - 2 \, \log \left (x + e^{2}\right ) \log \relax (x) + \log \relax (x)^{2}\right )} \log \relax (3) + 4 \, {\left (3 \, x^{2} {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} \log \relax (x)^{2} - x^{2} \log \relax (x)^{3} + {\left (\log \relax (3)^{3} + 6 \, \log \relax (3)^{2} \log \relax (2) + 12 \, \log \relax (3) \log \relax (2)^{2} + 8 \, \log \relax (2)^{3}\right )} x^{2} + x {\left (\log \relax (3) + 2 \, \log \relax (2)\right )} - {\left (3 \, {\left (\log \relax (3)^{2} + 4 \, \log \relax (3) \log \relax (2) + 4 \, \log \relax (2)^{2}\right )} x^{2} + x\right )} \log \relax (x)\right )} \log \left (x + e^{2}\right ) - 4 \, {\left ({\left (\log \relax (3)^{3} + 6 \, \log \relax (3)^{2} \log \relax (2) + 12 \, \log \relax (3) \log \relax (2)^{2} + 8 \, \log \relax (2)^{3}\right )} x^{2} + x {\left (\log \relax (3) + 2 \, \log \relax (2)\right )}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.73, size = 55, normalized size = 1.90 \begin {gather*} -\frac {\left (\ln \relax (3)-x^2\,{\ln \left (\frac {12\,x+12\,{\mathrm {e}}^2}{x}\right )}^2\right )\,\left (x^2\,{\ln \left (\frac {12\,x+12\,{\mathrm {e}}^2}{x}\right )}^2+2\,x-\ln \relax (3)\right )}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.31, size = 53, normalized size = 1.83 \begin {gather*} x^{2} \log {\left (\frac {12 x + 12 e^{2}}{x} \right )}^{4} + \left (2 x - 2 \log {\relax (3 )}\right ) \log {\left (\frac {12 x + 12 e^{2}}{x} \right )}^{2} + \frac {- 2 x \log {\relax (3 )} + \log {\relax (3 )}^{2}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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