Integrand size = 104, antiderivative size = 26 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=\frac {4-e^{(24-x+\log (x))^2}}{\log \left (5+2 x^2\right )} \]
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=\frac {4-e^{(-24+x)^2+\log ^2(x)} x^{48-2 x}}{\log \left (5+2 x^2\right )} \]
Integrate[(-16*x^2 + E^(576 - 48*x + x^2 + (48 - 2*x)*Log[x] + Log[x]^2)*( 4*x^2 + (-240 + 250*x - 106*x^2 + 100*x^3 - 4*x^4 + (-10 + 10*x - 4*x^2 + 4*x^3)*Log[x])*Log[5 + 2*x^2]))/((5*x + 2*x^3)*Log[5 + 2*x^2]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{x^2-48 x+\log ^2(x)+(48-2 x) \log (x)+576} \left (4 x^2+\left (-4 x^4+100 x^3-106 x^2+\left (4 x^3-4 x^2+10 x-10\right ) \log (x)+250 x-240\right ) \log \left (2 x^2+5\right )\right )-16 x^2}{\left (2 x^3+5 x\right ) \log ^2\left (2 x^2+5\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {e^{x^2-48 x+\log ^2(x)+(48-2 x) \log (x)+576} \left (4 x^2+\left (-4 x^4+100 x^3-106 x^2+\left (4 x^3-4 x^2+10 x-10\right ) \log (x)+250 x-240\right ) \log \left (2 x^2+5\right )\right )-16 x^2}{x \left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (-\frac {16 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {2 x^{47-2 x} e^{(x-24)^2+\log ^2(x)} \left (-2 x^2+2 x^2 \log (x) \log \left (2 x^2+5\right )+53 x^2 \log \left (2 x^2+5\right )-5 x \log (x) \log \left (2 x^2+5\right )-125 x \log \left (2 x^2+5\right )+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )+2 x^4 \log \left (2 x^2+5\right )-2 x^3 \log (x) \log \left (2 x^2+5\right )-50 x^3 \log \left (2 x^2+5\right )\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x^{1-2 x} \left (-8 x^{2 x}+2 x^{48} e^{(x-24)^2+\log ^2(x)}-\left (2 x^3-2 x^2+5 x-5\right ) x^{46} e^{(x-24)^2+\log ^2(x)} (x-\log (x)-24) \log \left (2 x^2+5\right )\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {x^{1-2 x} \left (-8 x^{2 x}+2 e^{(x-24)^2+\log ^2(x)} x^{48}-e^{(x-24)^2+\log ^2(x)} \left (-2 x^3+2 x^2-5 x+5\right ) (-x+\log (x)+24) \log \left (2 x^2+5\right ) x^{46}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle 2 \int \left (-\frac {e^{(x-24)^2+\log ^2(x)} \left (2 \log \left (2 x^2+5\right ) x^4-2 \log (x) \log \left (2 x^2+5\right ) x^3-50 \log \left (2 x^2+5\right ) x^3+2 \log (x) \log \left (2 x^2+5\right ) x^2+53 \log \left (2 x^2+5\right ) x^2-2 x^2-5 \log (x) \log \left (2 x^2+5\right ) x-125 \log \left (2 x^2+5\right ) x+5 \log (x) \log \left (2 x^2+5\right )+120 \log \left (2 x^2+5\right )\right ) x^{47-2 x}}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}-\frac {8 x}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {x^{-2 x} \left (-8 x^{2 x+1}+2 e^{(x-24)^2+\log ^2(x)} x^{49}-e^{(x-24)^2+\log ^2(x)} \left (2 x^3-2 x^2+5 x-5\right ) (x-\log (x)-24) \log \left (2 x^2+5\right ) x^{47}\right )}{\left (2 x^2+5\right ) \log ^2\left (2 x^2+5\right )}dx\) |
Int[(-16*x^2 + E^(576 - 48*x + x^2 + (48 - 2*x)*Log[x] + Log[x]^2)*(4*x^2 + (-240 + 250*x - 106*x^2 + 100*x^3 - 4*x^4 + (-10 + 10*x - 4*x^2 + 4*x^3) *Log[x])*Log[5 + 2*x^2]))/((5*x + 2*x^3)*Log[5 + 2*x^2]^2),x]
3.7.44.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 1.69 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46
| method | result | size |
| parallelrisch | \(-\frac {-80+20 \,{\mathrm e}^{\ln \left (x \right )^{2}+\left (-2 x +48\right ) \ln \left (x \right )+x^{2}-48 x +576}}{20 \ln \left (2 x^{2}+5\right )}\) | \(38\) |
| risch | \(\frac {4}{\ln \left (2 x^{2}+5\right )}-\frac {x^{-2 x +48} {\mathrm e}^{\ln \left (x \right )^{2}+576+x^{2}-48 x}}{\ln \left (2 x^{2}+5\right )}\) | \(46\) |
int(((((4*x^3-4*x^2+10*x-10)*ln(x)-4*x^4+100*x^3-106*x^2+250*x-240)*ln(2*x ^2+5)+4*x^2)*exp(ln(x)^2+(-2*x+48)*ln(x)+x^2-48*x+576)-16*x^2)/(2*x^3+5*x) /ln(2*x^2+5)^2,x,method=_RETURNVERBOSE)
Time = 0.41 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=-\frac {e^{\left (x^{2} - 2 \, {\left (x - 24\right )} \log \left (x\right ) + \log \left (x\right )^{2} - 48 \, x + 576\right )} - 4}{\log \left (2 \, x^{2} + 5\right )} \]
integrate(((((4*x^3-4*x^2+10*x-10)*log(x)-4*x^4+100*x^3-106*x^2+250*x-240) *log(2*x^2+5)+4*x^2)*exp(log(x)^2+(-2*x+48)*log(x)+x^2-48*x+576)-16*x^2)/( 2*x^3+5*x)/log(2*x^2+5)^2,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=- \frac {e^{x^{2} - 48 x + \left (48 - 2 x\right ) \log {\left (x \right )} + \log {\left (x \right )}^{2} + 576}}{\log {\left (2 x^{2} + 5 \right )}} + \frac {4}{\log {\left (2 x^{2} + 5 \right )}} \]
integrate(((((4*x**3-4*x**2+10*x-10)*ln(x)-4*x**4+100*x**3-106*x**2+250*x- 240)*ln(2*x**2+5)+4*x**2)*exp(ln(x)**2+(-2*x+48)*ln(x)+x**2-48*x+576)-16*x **2)/(2*x**3+5*x)/ln(2*x**2+5)**2,x)
Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=-\frac {x^{48} e^{\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 48 \, x + 576\right )}}{\log \left (2 \, x^{2} + 5\right )} + \frac {4}{\log \left (2 \, x^{2} + 5\right )} \]
integrate(((((4*x^3-4*x^2+10*x-10)*log(x)-4*x^4+100*x^3-106*x^2+250*x-240) *log(2*x^2+5)+4*x^2)*exp(log(x)^2+(-2*x+48)*log(x)+x^2-48*x+576)-16*x^2)/( 2*x^3+5*x)/log(2*x^2+5)^2,x, algorithm=\
Time = 0.45 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=-\frac {e^{\left (x^{2} - 2 \, x \log \left (x\right ) + \log \left (x\right )^{2} - 48 \, x + 48 \, \log \left (x\right ) + 576\right )} - 4}{\log \left (2 \, x^{2} + 5\right )} \]
integrate(((((4*x^3-4*x^2+10*x-10)*log(x)-4*x^4+100*x^3-106*x^2+250*x-240) *log(2*x^2+5)+4*x^2)*exp(log(x)^2+(-2*x+48)*log(x)+x^2-48*x+576)-16*x^2)/( 2*x^3+5*x)/log(2*x^2+5)^2,x, algorithm=\
Time = 11.76 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int \frac {-16 x^2+e^{576-48 x+x^2+(48-2 x) \log (x)+\log ^2(x)} \left (4 x^2+\left (-240+250 x-106 x^2+100 x^3-4 x^4+\left (-10+10 x-4 x^2+4 x^3\right ) \log (x)\right ) \log \left (5+2 x^2\right )\right )}{\left (5 x+2 x^3\right ) \log ^2\left (5+2 x^2\right )} \, dx=\frac {4}{\ln \left (2\,x^2+5\right )}-\frac {x^{48}\,{\mathrm {e}}^{-48\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{576}\,{\mathrm {e}}^{{\ln \left (x\right )}^2}}{x^{2\,x}\,\ln \left (2\,x^2+5\right )} \]
int((exp(log(x)^2 - 48*x - log(x)*(2*x - 48) + x^2 + 576)*(log(2*x^2 + 5)* (250*x - 106*x^2 + 100*x^3 - 4*x^4 + log(x)*(10*x - 4*x^2 + 4*x^3 - 10) - 240) + 4*x^2) - 16*x^2)/(log(2*x^2 + 5)^2*(5*x + 2*x^3)),x)