3.18.98 \(\int \frac {(96 x^2-3 x^3+e^x (-32 x^3+x^4)) \log (e^{-x} (-3+e^x x))+(-1728 x+54 x^2+e^x (-576 x+18 x^2)+(54 x-18 e^x x^2) \log (e^{-x} (-3+e^x x))) \log (-\frac {16 \log (e^{-x} (-3+e^x x))}{-32+x})+(-864+27 x+e^x (288 x-9 x^2)) \log (e^{-x} (-3+e^x x)) \log ^2(-\frac {16 \log (e^{-x} (-3+e^x x))}{-32+x})}{(96 x^2-3 x^3+e^x (-32 x^3+x^4)) \log (e^{-x} (-3+e^x x))} \, dx\)

Optimal. Leaf size=30 \[ x+\frac {9 \log ^2\left (\frac {\log \left (-3 e^{-x}+x\right )}{2-\frac {x}{16}}\right )}{x} \]

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Rubi [F]  time = 13.75, 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 (96 x^2-3 x^3+e^x \left (-32 x^3+x^4\right )\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right )+\left (-1728 x+54 x^2+e^x \left (-576 x+18 x^2\right )+\left (54 x-18 e^x x^2\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right )\right ) \log \left (-\frac {16 \log \left (e^{-x} \left (-3+e^x x\right )\right )}{-32+x}\right )+\left (-864+27 x+e^x \left (288 x-9 x^2\right )\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right ) \log ^2\left (-\frac {16 \log \left (e^{-x} \left (-3+e^x x\right )\right )}{-32+x}\right )}{\left (96 x^2-3 x^3+e^x \left (-32 x^3+x^4\right )\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((96*x^2 - 3*x^3 + E^x*(-32*x^3 + x^4))*Log[(-3 + E^x*x)/E^x] + (-1728*x + 54*x^2 + E^x*(-576*x + 18*x^2)
+ (54*x - 18*E^x*x^2)*Log[(-3 + E^x*x)/E^x])*Log[(-16*Log[(-3 + E^x*x)/E^x])/(-32 + x)] + (-864 + 27*x + E^x*(
288*x - 9*x^2))*Log[(-3 + E^x*x)/E^x]*Log[(-16*Log[(-3 + E^x*x)/E^x])/(-32 + x)]^2)/((96*x^2 - 3*x^3 + E^x*(-3
2*x^3 + x^4))*Log[(-3 + E^x*x)/E^x]),x]

[Out]

x - (9*Defer[Int][Log[(-16*Log[-3/E^x + x])/(-32 + x)]/(-32 + x), x])/16 + (9*Defer[Int][Log[(-16*Log[-3/E^x +
x])/(-32 + x)]/x, x])/16 + 18*Defer[Int][Log[(-16*Log[-3/E^x + x])/(-32 + x)]/(x^2*Log[-3/E^x + x]), x] + 54*
Defer[Int][Log[(-16*Log[-3/E^x + x])/(-32 + x)]/(x^2*(-3 + E^x*x)*Log[-3/E^x + x]), x] + 54*Defer[Int][Log[(-1
6*Log[-3/E^x + x])/(-32 + x)]/(x*(-3 + E^x*x)*Log[-3/E^x + x]), x] - 9*Defer[Int][Log[(-16*Log[-3/E^x + x])/(-
32 + x)]^2/x^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1-\frac {18 \left (-\left (\left (3+e^x\right ) (-32+x)\right )+\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}-\frac {9 \log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2}\right ) \, dx\\ &=x-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx-18 \int \frac {\left (-\left (\left (3+e^x\right ) (-32+x)\right )+\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx-18 \int \left (-\frac {3 (1+x) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}+\frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) x^2 \log \left (-3 e^{-x}+x\right )}\right ) \, dx\\ &=x-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx-18 \int \frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {(1+x) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx-18 \int \left (\frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{1024 (-32+x) \log \left (-3 e^{-x}+x\right )}-\frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{32 x^2 \log \left (-3 e^{-x}+x\right )}-\frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{1024 x \log \left (-3 e^{-x}+x\right )}\right ) \, dx+54 \int \left (\frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}+\frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}\right ) \, dx\\ &=x-\frac {9}{512} \int \frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )} \, dx+\frac {9}{512} \int \frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \log \left (-3 e^{-x}+x\right )} \, dx+\frac {9}{16} \int \frac {\left (32-x+x \log \left (-3 e^{-x}+x\right )\right ) \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} \int \left (\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )-\frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{\log \left (-3 e^{-x}+x\right )}+\frac {32 \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \log \left (-3 e^{-x}+x\right )}\right ) \, dx-\frac {9}{512} \int \left (\frac {x \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x}+\frac {32 \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )}-\frac {x \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )}\right ) \, dx+\frac {9}{16} \int \left (\frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x}+\frac {32 \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )}-\frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \log \left (-3 e^{-x}+x\right )}\right ) \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} \int \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right ) \, dx-\frac {9}{512} \int \frac {x \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx-\frac {9}{512} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{\log \left (-3 e^{-x}+x\right )} \, dx+\frac {9}{512} \int \frac {x \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} x \log \left (\frac {16 \log \left (-3 e^{-x}+x\right )}{32-x}\right )-\frac {9}{512} \int \frac {x \left (\frac {3-e^x x}{-32+x}+\frac {3+e^x}{\log \left (-3 e^{-x}+x\right )}\right )}{-3+e^x x} \, dx-\frac {9}{512} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{\log \left (-3 e^{-x}+x\right )} \, dx-\frac {9}{512} \int \left (\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )+\frac {32 \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x}\right ) \, dx+\frac {9}{512} \int \left (\frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{\log \left (-3 e^{-x}+x\right )}+\frac {32 \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )}\right ) \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} x \log \left (\frac {16 \log \left (-3 e^{-x}+x\right )}{32-x}\right )-\frac {9}{512} \int \left (\frac {3 (1+x)}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}+\frac {-32+x-x \log \left (-3 e^{-x}+x\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )}\right ) \, dx-\frac {9}{512} \int \log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right ) \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} \int \frac {x \left (\frac {3-e^x x}{-32+x}+\frac {3+e^x}{\log \left (-3 e^{-x}+x\right )}\right )}{-3+e^x x} \, dx-\frac {9}{512} \int \frac {-32+x-x \log \left (-3 e^{-x}+x\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {27}{512} \int \frac {1+x}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x-\frac {9}{512} \int \left (-\frac {x}{-32+x}+\frac {1}{\log \left (-3 e^{-x}+x\right )}\right ) \, dx+\frac {9}{512} \int \left (\frac {3 (1+x)}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}+\frac {-32+x-x \log \left (-3 e^{-x}+x\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )}\right ) \, dx-\frac {27}{512} \int \left (\frac {1}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}+\frac {x}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}\right ) \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} \int \frac {x}{-32+x} \, dx-\frac {9}{512} \int \frac {1}{\log \left (-3 e^{-x}+x\right )} \, dx+\frac {9}{512} \int \frac {-32+x-x \log \left (-3 e^{-x}+x\right )}{(-32+x) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {27}{512} \int \frac {1}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {27}{512} \int \frac {x}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+\frac {27}{512} \int \frac {1+x}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x+\frac {9}{512} \int \left (1+\frac {32}{-32+x}\right ) \, dx+\frac {9}{512} \int \left (-\frac {x}{-32+x}+\frac {1}{\log \left (-3 e^{-x}+x\right )}\right ) \, dx-\frac {9}{512} \int \frac {1}{\log \left (-3 e^{-x}+x\right )} \, dx+\frac {27}{512} \int \left (\frac {1}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}+\frac {x}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )}\right ) \, dx-\frac {27}{512} \int \frac {1}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {27}{512} \int \frac {x}{\left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=\frac {521 x}{512}+\frac {9}{16} \log (32-x)-\frac {9}{512} \int \frac {x}{-32+x} \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=\frac {521 x}{512}+\frac {9}{16} \log (32-x)-\frac {9}{512} \int \left (1+\frac {32}{-32+x}\right ) \, dx-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ &=x-\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{-32+x} \, dx+\frac {9}{16} \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x} \, dx-9 \int \frac {\log ^2\left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2} \, dx+18 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x^2 \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx+54 \int \frac {\log \left (-\frac {16 \log \left (-3 e^{-x}+x\right )}{-32+x}\right )}{x \left (-3+e^x x\right ) \log \left (-3 e^{-x}+x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (96 x^2-3 x^3+e^x \left (-32 x^3+x^4\right )\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right )+\left (-1728 x+54 x^2+e^x \left (-576 x+18 x^2\right )+\left (54 x-18 e^x x^2\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right )\right ) \log \left (-\frac {16 \log \left (e^{-x} \left (-3+e^x x\right )\right )}{-32+x}\right )+\left (-864+27 x+e^x \left (288 x-9 x^2\right )\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right ) \log ^2\left (-\frac {16 \log \left (e^{-x} \left (-3+e^x x\right )\right )}{-32+x}\right )}{\left (96 x^2-3 x^3+e^x \left (-32 x^3+x^4\right )\right ) \log \left (e^{-x} \left (-3+e^x x\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((96*x^2 - 3*x^3 + E^x*(-32*x^3 + x^4))*Log[(-3 + E^x*x)/E^x] + (-1728*x + 54*x^2 + E^x*(-576*x + 18
*x^2) + (54*x - 18*E^x*x^2)*Log[(-3 + E^x*x)/E^x])*Log[(-16*Log[(-3 + E^x*x)/E^x])/(-32 + x)] + (-864 + 27*x +
E^x*(288*x - 9*x^2))*Log[(-3 + E^x*x)/E^x]*Log[(-16*Log[(-3 + E^x*x)/E^x])/(-32 + x)]^2)/((96*x^2 - 3*x^3 + E
^x*(-32*x^3 + x^4))*Log[(-3 + E^x*x)/E^x]),x]

[Out]

Integrate[((96*x^2 - 3*x^3 + E^x*(-32*x^3 + x^4))*Log[(-3 + E^x*x)/E^x] + (-1728*x + 54*x^2 + E^x*(-576*x + 18
*x^2) + (54*x - 18*E^x*x^2)*Log[(-3 + E^x*x)/E^x])*Log[(-16*Log[(-3 + E^x*x)/E^x])/(-32 + x)] + (-864 + 27*x +
E^x*(288*x - 9*x^2))*Log[(-3 + E^x*x)/E^x]*Log[(-16*Log[(-3 + E^x*x)/E^x])/(-32 + x)]^2)/((96*x^2 - 3*x^3 + E
^x*(-32*x^3 + x^4))*Log[(-3 + E^x*x)/E^x]), x]

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fricas [A]  time = 0.68, size = 32, normalized size = 1.07 \begin {gather*} \frac {x^{2} + 9 \, \log \left (-\frac {16 \, \log \left ({\left (x e^{x} - 3\right )} e^{\left (-x\right )}\right )}{x - 32}\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^2+288*x)*exp(x)+27*x-864)*log((exp(x)*x-3)/exp(x))*log(-16*log((exp(x)*x-3)/exp(x))/(x-32))^
2+((-18*exp(x)*x^2+54*x)*log((exp(x)*x-3)/exp(x))+(18*x^2-576*x)*exp(x)+54*x^2-1728*x)*log(-16*log((exp(x)*x-3
)/exp(x))/(x-32))+((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)*log((exp(x)*x-3)/exp(x)))/((x^4-32*x^3)*exp(x)-3*x^3+96*x
^2)/log((exp(x)*x-3)/exp(x)),x, algorithm="fricas")

[Out]

(x^2 + 9*log(-16*log((x*e^x - 3)*e^(-x))/(x - 32))^2)/x

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^2+288*x)*exp(x)+27*x-864)*log((exp(x)*x-3)/exp(x))*log(-16*log((exp(x)*x-3)/exp(x))/(x-32))^
2+((-18*exp(x)*x^2+54*x)*log((exp(x)*x-3)/exp(x))+(18*x^2-576*x)*exp(x)+54*x^2-1728*x)*log(-16*log((exp(x)*x-3
)/exp(x))/(x-32))+((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)*log((exp(x)*x-3)/exp(x)))/((x^4-32*x^3)*exp(x)-3*x^3+96*x
^2)/log((exp(x)*x-3)/exp(x)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (-9 x^{2}+288 x \right ) {\mathrm e}^{x}+27 x -864\right ) \ln \left (\left ({\mathrm e}^{x} x -3\right ) {\mathrm e}^{-x}\right ) \ln \left (-\frac {16 \ln \left (\left ({\mathrm e}^{x} x -3\right ) {\mathrm e}^{-x}\right )}{x -32}\right )^{2}+\left (\left (-18 \,{\mathrm e}^{x} x^{2}+54 x \right ) \ln \left (\left ({\mathrm e}^{x} x -3\right ) {\mathrm e}^{-x}\right )+\left (18 x^{2}-576 x \right ) {\mathrm e}^{x}+54 x^{2}-1728 x \right ) \ln \left (-\frac {16 \ln \left (\left ({\mathrm e}^{x} x -3\right ) {\mathrm e}^{-x}\right )}{x -32}\right )+\left (\left (x^{4}-32 x^{3}\right ) {\mathrm e}^{x}-3 x^{3}+96 x^{2}\right ) \ln \left (\left ({\mathrm e}^{x} x -3\right ) {\mathrm e}^{-x}\right )}{\left (\left (x^{4}-32 x^{3}\right ) {\mathrm e}^{x}-3 x^{3}+96 x^{2}\right ) \ln \left (\left ({\mathrm e}^{x} x -3\right ) {\mathrm e}^{-x}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-9*x^2+288*x)*exp(x)+27*x-864)*ln((exp(x)*x-3)/exp(x))*ln(-16*ln((exp(x)*x-3)/exp(x))/(x-32))^2+((-18*e
xp(x)*x^2+54*x)*ln((exp(x)*x-3)/exp(x))+(18*x^2-576*x)*exp(x)+54*x^2-1728*x)*ln(-16*ln((exp(x)*x-3)/exp(x))/(x
-32))+((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)*ln((exp(x)*x-3)/exp(x)))/((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)/ln((exp(x
)*x-3)/exp(x)),x)

[Out]

int((((-9*x^2+288*x)*exp(x)+27*x-864)*ln((exp(x)*x-3)/exp(x))*ln(-16*ln((exp(x)*x-3)/exp(x))/(x-32))^2+((-18*e
xp(x)*x^2+54*x)*ln((exp(x)*x-3)/exp(x))+(18*x^2-576*x)*exp(x)+54*x^2-1728*x)*ln(-16*ln((exp(x)*x-3)/exp(x))/(x
-32))+((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)*ln((exp(x)*x-3)/exp(x)))/((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)/ln((exp(x
)*x-3)/exp(x)),x)

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maxima [B]  time = 1.17, size = 71, normalized size = 2.37 \begin {gather*} \frac {x^{2} + 144 \, \log \relax (2)^{2} + 18 \, {\left (4 \, \log \relax (2) - \log \left (x - 32\right )\right )} \log \left (x - \log \left (x e^{x} - 3\right )\right ) + 9 \, \log \left (x - \log \left (x e^{x} - 3\right )\right )^{2} - 72 \, \log \relax (2) \log \left (x - 32\right ) + 9 \, \log \left (x - 32\right )^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x^2+288*x)*exp(x)+27*x-864)*log((exp(x)*x-3)/exp(x))*log(-16*log((exp(x)*x-3)/exp(x))/(x-32))^
2+((-18*exp(x)*x^2+54*x)*log((exp(x)*x-3)/exp(x))+(18*x^2-576*x)*exp(x)+54*x^2-1728*x)*log(-16*log((exp(x)*x-3
)/exp(x))/(x-32))+((x^4-32*x^3)*exp(x)-3*x^3+96*x^2)*log((exp(x)*x-3)/exp(x)))/((x^4-32*x^3)*exp(x)-3*x^3+96*x
^2)/log((exp(x)*x-3)/exp(x)),x, algorithm="maxima")

[Out]

(x^2 + 144*log(2)^2 + 18*(4*log(2) - log(x - 32))*log(x - log(x*e^x - 3)) + 9*log(x - log(x*e^x - 3))^2 - 72*l
og(2)*log(x - 32) + 9*log(x - 32)^2)/x

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mupad [B]  time = 1.54, size = 26, normalized size = 0.87 \begin {gather*} x+\frac {9\,{\ln \left (-\frac {16\,\ln \left (x-3\,{\mathrm {e}}^{-x}\right )}{x-32}\right )}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(exp(-x)*(x*exp(x) - 3))*(exp(x)*(32*x^3 - x^4) - 96*x^2 + 3*x^3) + log(-(16*log(exp(-x)*(x*exp(x) - 3
)))/(x - 32))*(1728*x + exp(x)*(576*x - 18*x^2) - log(exp(-x)*(x*exp(x) - 3))*(54*x - 18*x^2*exp(x)) - 54*x^2)
- log(exp(-x)*(x*exp(x) - 3))*log(-(16*log(exp(-x)*(x*exp(x) - 3)))/(x - 32))^2*(27*x + exp(x)*(288*x - 9*x^2
) - 864))/(log(exp(-x)*(x*exp(x) - 3))*(exp(x)*(32*x^3 - x^4) - 96*x^2 + 3*x^3)),x)

[Out]

x + (9*log(-(16*log(x - 3*exp(-x)))/(x - 32))^2)/x

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sympy [A]  time = 2.02, size = 26, normalized size = 0.87 \begin {gather*} x + \frac {9 \log {\left (- \frac {16 \log {\left (\left (x e^{x} - 3\right ) e^{- x} \right )}}{x - 32} \right )}^{2}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-9*x**2+288*x)*exp(x)+27*x-864)*ln((exp(x)*x-3)/exp(x))*ln(-16*ln((exp(x)*x-3)/exp(x))/(x-32))**2
+((-18*exp(x)*x**2+54*x)*ln((exp(x)*x-3)/exp(x))+(18*x**2-576*x)*exp(x)+54*x**2-1728*x)*ln(-16*ln((exp(x)*x-3)
/exp(x))/(x-32))+((x**4-32*x**3)*exp(x)-3*x**3+96*x**2)*ln((exp(x)*x-3)/exp(x)))/((x**4-32*x**3)*exp(x)-3*x**3
+96*x**2)/ln((exp(x)*x-3)/exp(x)),x)

[Out]

x + 9*log(-16*log((x*exp(x) - 3)*exp(-x))/(x - 32))**2/x

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