\(\int \frac {-720+540 x+3 e^{4+x} x^3+(180 x-6 e^{4+x} x^2) \log (\frac {x}{\log (5)})+3 e^{4+x} x \log ^2(\frac {x}{\log (5)})+(-720 x-180 x^2+3 e^{4+x} x^2+(720+180 x-6 e^{4+x} x) \log (\frac {x}{\log (5)})+3 e^{4+x} \log ^2(\frac {x}{\log (5)})) \log (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log (\frac {x}{\log (5)})}{-x+\log (\frac {x}{\log (5)})})}{-240 x-60 x^2+e^{4+x} x^2+(240+60 x-2 e^{4+x} x) \log (\frac {x}{\log (5)})+e^{4+x} \log ^2(\frac {x}{\log (5)})} \, dx\) [8392]

Optimal result

Integrand size = 209, antiderivative size = 30 \[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=3 x \log \left (-e^{4+x}+\frac {60 (4+x)}{x-\log \left (\frac {x}{\log (5)}\right )}\right ) \]

[Out]

3*ln(15*(4+x)/(1/4*x-1/4*ln(x/ln(5)))-exp(4+x))*x

Rubi [F]

\[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=\int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx \]

[In]

Int[(-720 + 540*x + 3*E^(4 + x)*x^3 + (180*x - 6*E^(4 + x)*x^2)*Log[x/Log[5]] + 3*E^(4 + x)*x*Log[x/Log[5]]^2
+ (-720*x - 180*x^2 + 3*E^(4 + x)*x^2 + (720 + 180*x - 6*E^(4 + x)*x)*Log[x/Log[5]] + 3*E^(4 + x)*Log[x/Log[5]
]^2)*Log[(-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])/(-x + Log[x/Log[5]])])/(-240*x - 60*x^2 + E^(4
+ x)*x^2 + (240 + 60*x - 2*E^(4 + x)*x)*Log[x/Log[5]] + E^(4 + x)*Log[x/Log[5]]^2),x]

[Out]

3*x*Log[(240 + 60*x - E^(4 + x)*x + E^(4 + x)*Log[x/Log[5]])/(x - Log[x] + Log[Log[5]])] + 540*Defer[Int][x/((
-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x - Log[x] + Log[Log[5]])), x] - 540*(1 + Log[Log[5]])*D
efer[Int][x/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x - Log[x] + Log[Log[5]])), x] + 720*Defer
[Int][x^2/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x - Log[x] + Log[Log[5]])), x] - 180*(4 + Lo
g[Log[5]])*Defer[Int][x^2/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x - Log[x] + Log[Log[5]])),
x] + 540*Defer[Int][(x*Log[x])/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x - Log[x] + Log[Log[5]
])), x] + 180*Defer[Int][(x^2*Log[x])/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x - Log[x] + Log
[Log[5]])), x] - 540*Defer[Int][(x*Log[x/Log[5]])/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])*(x -
Log[x] + Log[Log[5]])), x] - 180*Defer[Int][(x^2*Log[x/Log[5]])/((-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/
Log[5]])*(x - Log[x] + Log[Log[5]])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {720-540 x-3 e^{4+x} x^3-\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )-3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )-\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{\left (240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ & = \int \left (\frac {180 \left (-4+3 x+4 x^2+x^3-3 x \log \left (\frac {x}{\log (5)}\right )-x^2 \log \left (\frac {x}{\log (5)}\right )\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+3 \left (x+\log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )\right )\right ) \, dx \\ & = 3 \int \left (x+\log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )\right ) \, dx+180 \int \frac {-4+3 x+4 x^2+x^3-3 x \log \left (\frac {x}{\log (5)}\right )-x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ & = \frac {3 x^2}{2}+3 \int \log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right ) \, dx+180 \int \left (-\frac {4}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+\frac {3 x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+\frac {4 x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+\frac {x^3}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}-\frac {3 x \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}-\frac {x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}\right ) \, dx \\ & = \frac {3 x^2}{2}+3 x \log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )-3 \int \frac {-240+e^{4+x} x^3+e^{4+x} x \log ^2(x)+2 e^{4+x} x^2 \log (\log (5))+x \left (180-60 \log (\log (5))+e^{4+x} \log ^2(\log (5))\right )-2 x \log (x) \left (-30+e^{4+x} (x+\log (\log (5)))\right )}{\left (-240+\left (-60+e^{4+x}\right ) x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+180 \int \frac {x^3}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \frac {x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+540 \int \frac {x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-540 \int \frac {x \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-720 \int \frac {1}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+720 \int \frac {x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ & = \frac {3 x^2}{2}+3 x \log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )-3 \int \left (x+\frac {60 \left (4-x^3+3 x \log (x)+x^2 \log (x)-4 x^2 \left (1+\frac {1}{4} \log (\log (5))\right )-3 x (1+\log (\log (5)))\right )}{\left (240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}\right ) \, dx+180 \int \frac {x^3}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \frac {x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+540 \int \frac {x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-540 \int \frac {x \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-720 \int \frac {1}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+720 \int \frac {x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ & = 3 x \log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )+180 \int \frac {x^3}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \frac {x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \frac {4-x^3+3 x \log (x)+x^2 \log (x)-4 x^2 \left (1+\frac {1}{4} \log (\log (5))\right )-3 x (1+\log (\log (5)))}{\left (240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+540 \int \frac {x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-540 \int \frac {x \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-720 \int \frac {1}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+720 \int \frac {x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ & = 3 x \log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )+180 \int \frac {x^3}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \frac {x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \left (-\frac {4}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+\frac {x^3}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}-\frac {3 x \log (x)}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}-\frac {x^2 \log (x)}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+\frac {3 x (1+\log (\log (5)))}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}+\frac {x^2 (4+\log (\log (5)))}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))}\right ) \, dx+540 \int \frac {x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-540 \int \frac {x \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-720 \int \frac {1}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+720 \int \frac {x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ & = 3 x \log \left (\frac {240+60 x-e^{4+x} x+e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{x-\log (x)+\log (\log (5))}\right )+180 \int \frac {x^2 \log (x)}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-180 \int \frac {x^2 \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+540 \int \frac {x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+540 \int \frac {x \log (x)}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-540 \int \frac {x \log \left (\frac {x}{\log (5)}\right )}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx+720 \int \frac {x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-(540 (1+\log (\log (5)))) \int \frac {x}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx-(180 (4+\log (\log (5)))) \int \frac {x^2}{\left (-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )\right ) (x-\log (x)+\log (\log (5)))} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=\int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx \]

[In]

Integrate[(-720 + 540*x + 3*E^(4 + x)*x^3 + (180*x - 6*E^(4 + x)*x^2)*Log[x/Log[5]] + 3*E^(4 + x)*x*Log[x/Log[
5]]^2 + (-720*x - 180*x^2 + 3*E^(4 + x)*x^2 + (720 + 180*x - 6*E^(4 + x)*x)*Log[x/Log[5]] + 3*E^(4 + x)*Log[x/
Log[5]]^2)*Log[(-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])/(-x + Log[x/Log[5]])])/(-240*x - 60*x^2 +
E^(4 + x)*x^2 + (240 + 60*x - 2*E^(4 + x)*x)*Log[x/Log[5]] + E^(4 + x)*Log[x/Log[5]]^2),x]

[Out]

Integrate[(-720 + 540*x + 3*E^(4 + x)*x^3 + (180*x - 6*E^(4 + x)*x^2)*Log[x/Log[5]] + 3*E^(4 + x)*x*Log[x/Log[
5]]^2 + (-720*x - 180*x^2 + 3*E^(4 + x)*x^2 + (720 + 180*x - 6*E^(4 + x)*x)*Log[x/Log[5]] + 3*E^(4 + x)*Log[x/
Log[5]]^2)*Log[(-240 - 60*x + E^(4 + x)*x - E^(4 + x)*Log[x/Log[5]])/(-x + Log[x/Log[5]])])/(-240*x - 60*x^2 +
E^(4 + x)*x^2 + (240 + 60*x - 2*E^(4 + x)*x)*Log[x/Log[5]] + E^(4 + x)*Log[x/Log[5]]^2), x]

Maple [A] (verified)

Time = 11.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47

method	result	size
parallelrisch	\(3 \ln \left (-\frac {{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-x \,{\mathrm e}^{4+x}+60 x +240}{\ln \left (\frac {x}{\ln \left (5\right )}\right )-x}\right ) x\)	\(44\)
risch	\(3 x \ln \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )-3 x \ln \left (-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x \right )-3 i \pi x {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right )}^{2}-\frac {3 i \pi x \,\operatorname {csgn}\left (\frac {i}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right ) \operatorname {csgn}\left (i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )\right ) \operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right )}{2}+\frac {3 i \pi x \,\operatorname {csgn}\left (\frac {i}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right ) {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right )}^{2}}{2}+\frac {3 i \pi x \,\operatorname {csgn}\left (i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )\right ) {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right )}^{2}}{2}+\frac {3 i \pi x {\operatorname {csgn}\left (\frac {i \left (\left ({\mathrm e}^{4+x}-60\right ) x -{\mathrm e}^{4+x} \ln \left (\frac {x}{\ln \left (5\right )}\right )-240\right )}{-\ln \left (\frac {x}{\ln \left (5\right )}\right )+x}\right )}^{3}}{2}+3 i \pi x\)	\(370\)

[In]

int(((3*exp(4+x)*ln(x/ln(5))^2+(-6*x*exp(4+x)+180*x+720)*ln(x/ln(5))+3*x^2*exp(4+x)-180*x^2-720*x)*ln((-exp(4+
x)*ln(x/ln(5))+x*exp(4+x)-60*x-240)/(ln(x/ln(5))-x))+3*x*exp(4+x)*ln(x/ln(5))^2+(-6*x^2*exp(4+x)+180*x)*ln(x/l
n(5))+3*x^3*exp(4+x)+540*x-720)/(exp(4+x)*ln(x/ln(5))^2+(-2*x*exp(4+x)+60*x+240)*ln(x/ln(5))+x^2*exp(4+x)-60*x
^2-240*x),x,method=_RETURNVERBOSE)

[Out]

3*ln(-(exp(4+x)*ln(x/ln(5))-x*exp(4+x)+60*x+240)/(ln(x/ln(5))-x))*x

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=3 \, x \log \left (-\frac {x e^{\left (x + 4\right )} - e^{\left (x + 4\right )} \log \left (\frac {x}{\log \left (5\right )}\right ) - 60 \, x - 240}{x - \log \left (\frac {x}{\log \left (5\right )}\right )}\right ) \]

[In]

integrate(((3*exp(4+x)*log(x/log(5))^2+(-6*x*exp(4+x)+180*x+720)*log(x/log(5))+3*x^2*exp(4+x)-180*x^2-720*x)*l
og((-exp(4+x)*log(x/log(5))+x*exp(4+x)-60*x-240)/(log(x/log(5))-x))+3*x*exp(4+x)*log(x/log(5))^2+(-6*x^2*exp(4
+x)+180*x)*log(x/log(5))+3*x^3*exp(4+x)+540*x-720)/(exp(4+x)*log(x/log(5))^2+(-2*x*exp(4+x)+60*x+240)*log(x/lo
g(5))+x^2*exp(4+x)-60*x^2-240*x),x, algorithm="fricas")

[Out]

3*x*log(-(x*e^(x + 4) - e^(x + 4)*log(x/log(5)) - 60*x - 240)/(x - log(x/log(5))))

Sympy [F(-2)]

Exception generated. \[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(((3*exp(4+x)*ln(x/ln(5))**2+(-6*x*exp(4+x)+180*x+720)*ln(x/ln(5))+3*x**2*exp(4+x)-180*x**2-720*x)*ln
((-exp(4+x)*ln(x/ln(5))+x*exp(4+x)-60*x-240)/(ln(x/ln(5))-x))+3*x*exp(4+x)*ln(x/ln(5))**2+(-6*x**2*exp(4+x)+18
0*x)*ln(x/ln(5))+3*x**3*exp(4+x)+540*x-720)/(exp(4+x)*ln(x/ln(5))**2+(-2*x*exp(4+x)+60*x+240)*ln(x/ln(5))+x**2
*exp(4+x)-60*x**2-240*x),x)

[Out]

Exception raised: TypeError >> '>' not supported between instances of 'Poly' and 'int'

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=3 \, x \log \left (-{\left (x e^{4} + e^{4} \log \left (\log \left (5\right )\right )\right )} e^{x} + e^{\left (x + 4\right )} \log \left (x\right ) + 60 \, x + 240\right ) - 3 \, x \log \left (x - \log \left (x\right ) + \log \left (\log \left (5\right )\right )\right ) \]

[In]

integrate(((3*exp(4+x)*log(x/log(5))^2+(-6*x*exp(4+x)+180*x+720)*log(x/log(5))+3*x^2*exp(4+x)-180*x^2-720*x)*l
og((-exp(4+x)*log(x/log(5))+x*exp(4+x)-60*x-240)/(log(x/log(5))-x))+3*x*exp(4+x)*log(x/log(5))^2+(-6*x^2*exp(4
+x)+180*x)*log(x/log(5))+3*x^3*exp(4+x)+540*x-720)/(exp(4+x)*log(x/log(5))^2+(-2*x*exp(4+x)+60*x+240)*log(x/lo
g(5))+x^2*exp(4+x)-60*x^2-240*x),x, algorithm="maxima")

[Out]

3*x*log(-(x*e^4 + e^4*log(log(5)))*e^x + e^(x + 4)*log(x) + 60*x + 240) - 3*x*log(x - log(x) + log(log(5)))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (29) = 58\).

Time = 0.84 (sec) , antiderivative size = 109, normalized size of antiderivative = 3.63 \[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=3 \, {\left (x + 4\right )} \log \left (-{\left (x + 4\right )} e^{\left (x + 4\right )} + e^{\left (x + 4\right )} \log \left (x\right ) - e^{\left (x + 4\right )} \log \left (\log \left (5\right )\right ) + 60 \, x + 4 \, e^{\left (x + 4\right )} + 240\right ) - 3 \, {\left (x + 4\right )} \log \left (x - \log \left (x\right ) + \log \left (\log \left (5\right )\right )\right ) - 12 \, \log \left (-{\left (x + 4\right )} e^{\left (x + 4\right )} + e^{\left (x + 4\right )} \log \left (x\right ) - e^{\left (x + 4\right )} \log \left (\log \left (5\right )\right ) + 60 \, x + 4 \, e^{\left (x + 4\right )} + 240\right ) + 12 \, \log \left (x - \log \left (x\right ) + \log \left (\log \left (5\right )\right )\right ) \]

[In]

integrate(((3*exp(4+x)*log(x/log(5))^2+(-6*x*exp(4+x)+180*x+720)*log(x/log(5))+3*x^2*exp(4+x)-180*x^2-720*x)*l
og((-exp(4+x)*log(x/log(5))+x*exp(4+x)-60*x-240)/(log(x/log(5))-x))+3*x*exp(4+x)*log(x/log(5))^2+(-6*x^2*exp(4
+x)+180*x)*log(x/log(5))+3*x^3*exp(4+x)+540*x-720)/(exp(4+x)*log(x/log(5))^2+(-2*x*exp(4+x)+60*x+240)*log(x/lo
g(5))+x^2*exp(4+x)-60*x^2-240*x),x, algorithm="giac")

[Out]

3*(x + 4)*log(-(x + 4)*e^(x + 4) + e^(x + 4)*log(x) - e^(x + 4)*log(log(5)) + 60*x + 4*e^(x + 4) + 240) - 3*(x
+ 4)*log(x - log(x) + log(log(5))) - 12*log(-(x + 4)*e^(x + 4) + e^(x + 4)*log(x) - e^(x + 4)*log(log(5)) + 6
0*x + 4*e^(x + 4) + 240) + 12*log(x - log(x) + log(log(5)))

Mupad [B] (verification not implemented)

Time = 14.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-720+540 x+3 e^{4+x} x^3+\left (180 x-6 e^{4+x} x^2\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} x \log ^2\left (\frac {x}{\log (5)}\right )+\left (-720 x-180 x^2+3 e^{4+x} x^2+\left (720+180 x-6 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+3 e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )\right ) \log \left (\frac {-240-60 x+e^{4+x} x-e^{4+x} \log \left (\frac {x}{\log (5)}\right )}{-x+\log \left (\frac {x}{\log (5)}\right )}\right )}{-240 x-60 x^2+e^{4+x} x^2+\left (240+60 x-2 e^{4+x} x\right ) \log \left (\frac {x}{\log (5)}\right )+e^{4+x} \log ^2\left (\frac {x}{\log (5)}\right )} \, dx=3\,x\,\ln \left (\frac {60\,x-x\,{\mathrm {e}}^4\,{\mathrm {e}}^x+\ln \left (\frac {x}{\ln \left (5\right )}\right )\,{\mathrm {e}}^4\,{\mathrm {e}}^x+240}{x-\ln \left (\frac {x}{\ln \left (5\right )}\right )}\right ) \]

[In]

int((540*x + log(x/log(5))*(180*x - 6*x^2*exp(x + 4)) + log((60*x - x*exp(x + 4) + log(x/log(5))*exp(x + 4) +
240)/(x - log(x/log(5))))*(3*log(x/log(5))^2*exp(x + 4) - 720*x + 3*x^2*exp(x + 4) + log(x/log(5))*(180*x - 6*
x*exp(x + 4) + 720) - 180*x^2) + 3*x^3*exp(x + 4) + 3*x*log(x/log(5))^2*exp(x + 4) - 720)/(log(x/log(5))^2*exp
(x + 4) - 240*x + x^2*exp(x + 4) + log(x/log(5))*(60*x - 2*x*exp(x + 4) + 240) - 60*x^2),x)

[Out]

3*x*log((60*x - x*exp(4)*exp(x) + log(x/log(5))*exp(4)*exp(x) + 240)/(x - log(x/log(5))))