Integrand size = 281, antiderivative size = 36 \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\frac {\left (2+\frac {4 x}{5 \left (x+(i \pi +x+\log (5-e))^2\right )}\right ) \log \left (\frac {4}{x^2}\right )}{x} \] Output:
(2+4/5*x/((ln(exp(1)-5)+x)^2+x))*ln(4/x^2)/x
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(101\) vs. \(2(36)=72\).
Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.81 \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\frac {2 \left (-5 \pi ^2+5 x^2+5 \log ^2(5-e)+10 i \pi (x+\log (5-e))+x (7+10 \log (5-e))\right ) \log \left (\frac {4}{x^2}\right )}{5 x \left (-\pi ^2+x+x^2+2 x \log (5-e)+\log ^2(5-e)+2 i \pi (x+\log (5-e))\right )} \] Input:
Integrate[(-28*x^2 - 48*x^3 - 20*x^4 + (I*Pi + Log[5 - E])^4*(-20 - 10*Log [4/x^2]) + (-14*x^2 - 28*x^3 - 10*x^4)*Log[4/x^2] + (I*Pi + Log[5 - E])^3* (-80*x - 40*x*Log[4/x^2]) + (I*Pi + Log[5 - E])^2*(-48*x - 120*x^2 + (-20* x - 60*x^2)*Log[4/x^2]) + (I*Pi + Log[5 - E])*(-96*x^2 - 80*x^3 + (-48*x^2 - 40*x^3)*Log[4/x^2]))/(5*x^4 + 10*x^5 + 5*x^6 + (20*x^4 + 20*x^5)*(I*Pi + Log[5 - E]) + (10*x^3 + 30*x^4)*(I*Pi + Log[5 - E])^2 + 20*x^3*(I*Pi + L og[5 - E])^3 + 5*x^2*(I*Pi + Log[5 - E])^4),x]
Output:
(2*(-5*Pi^2 + 5*x^2 + 5*Log[5 - E]^2 + (10*I)*Pi*(x + Log[5 - E]) + x*(7 + 10*Log[5 - E]))*Log[4/x^2])/(5*x*(-Pi^2 + x + x^2 + 2*x*Log[5 - E] + Log[ 5 - E]^2 + (2*I)*Pi*(x + Log[5 - E])))
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2153\) vs. \(2(36)=72\).
Time = 7.92 (sec) , antiderivative size = 2153, normalized size of antiderivative = 59.81, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {2026, 2463, 6, 6, 27, 27, 7293, 2009}
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.
| \(\displaystyle \int \frac {-20 x^4-48 x^3-28 x^2+(\log (5-e)+i \pi )^4 \left (-10 \log \left (\frac {4}{x^2}\right )-20\right )+(\log (5-e)+i \pi )^3 \left (-40 x \log \left (\frac {4}{x^2}\right )-80 x\right )+(\log (5-e)+i \pi )^2 \left (-120 x^2+\left (-60 x^2-20 x\right ) \log \left (\frac {4}{x^2}\right )-48 x\right )+(\log (5-e)+i \pi ) \left (-80 x^3-96 x^2+\left (-40 x^3-48 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+\left (-10 x^4-28 x^3-14 x^2\right ) \log \left (\frac {4}{x^2}\right )}{5 x^6+10 x^5+5 x^4+20 x^3 (\log (5-e)+i \pi )^3+5 x^2 (\log (5-e)+i \pi )^4+\left (20 x^5+20 x^4\right ) (\log (5-e)+i \pi )+\left (30 x^4+10 x^3\right ) (\log (5-e)+i \pi )^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-20 x^4-48 x^3-28 x^2+(\log (5-e)+i \pi )^4 \left (-10 \log \left (\frac {4}{x^2}\right )-20\right )+(\log (5-e)+i \pi )^3 \left (-40 x \log \left (\frac {4}{x^2}\right )-80 x\right )+(\log (5-e)+i \pi )^2 \left (-120 x^2+\left (-60 x^2-20 x\right ) \log \left (\frac {4}{x^2}\right )-48 x\right )+(\log (5-e)+i \pi ) \left (-80 x^3-96 x^2+\left (-40 x^3-48 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+\left (-10 x^4-28 x^3-14 x^2\right ) \log \left (\frac {4}{x^2}\right )}{x^2 \left (5 x^4+10 x^3 (1+2 i \pi +2 \log (5-e))+5 x^2 \left (1-6 \pi ^2+6 \log ^2(5-e)+4 \log (5-e)+4 i \pi (1+3 \log (5-e))\right )-10 x (1+i (2 \pi -2 i \log (5-e))) (\pi -i \log (5-e))^2+5 (\pi -i \log (5-e))^4\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {-20 x^4-48 x^3-28 x^2+(\log (5-e)+i \pi )^4 \left (-10 \log \left (\frac {4}{x^2}\right )-20\right )+(\log (5-e)+i \pi )^3 \left (-40 x \log \left (\frac {4}{x^2}\right )-80 x\right )+(\log (5-e)+i \pi )^2 \left (-120 x^2+\left (-60 x^2-20 x\right ) \log \left (\frac {4}{x^2}\right )-48 x\right )+(\log (5-e)+i \pi ) \left (-80 x^3-96 x^2+\left (-40 x^3-48 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+\left (-10 x^4-28 x^3-14 x^2\right ) \log \left (\frac {4}{x^2}\right )}{5 x^2 \left (x^2+2 i \pi x+x+2 x \log (5-e)-\pi ^2+\log ^2(5-e)+2 i \pi \log (5-e)\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-20 x^4-48 x^3-28 x^2+(\log (5-e)+i \pi )^4 \left (-10 \log \left (\frac {4}{x^2}\right )-20\right )+(\log (5-e)+i \pi )^3 \left (-40 x \log \left (\frac {4}{x^2}\right )-80 x\right )+(\log (5-e)+i \pi )^2 \left (-120 x^2+\left (-60 x^2-20 x\right ) \log \left (\frac {4}{x^2}\right )-48 x\right )+(\log (5-e)+i \pi ) \left (-80 x^3-96 x^2+\left (-40 x^3-48 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+\left (-10 x^4-28 x^3-14 x^2\right ) \log \left (\frac {4}{x^2}\right )}{5 x^2 \left (x^2+(1+2 i \pi ) x+2 x \log (5-e)-\pi ^2+\log ^2(5-e)+2 i \pi \log (5-e)\right )^2}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-20 x^4-48 x^3-28 x^2+(\log (5-e)+i \pi )^4 \left (-10 \log \left (\frac {4}{x^2}\right )-20\right )+(\log (5-e)+i \pi )^3 \left (-40 x \log \left (\frac {4}{x^2}\right )-80 x\right )+(\log (5-e)+i \pi )^2 \left (-120 x^2+\left (-60 x^2-20 x\right ) \log \left (\frac {4}{x^2}\right )-48 x\right )+(\log (5-e)+i \pi ) \left (-80 x^3-96 x^2+\left (-40 x^3-48 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+\left (-10 x^4-28 x^3-14 x^2\right ) \log \left (\frac {4}{x^2}\right )}{5 x^2 \left (x^2+x (1+2 i \pi +2 \log (5-e))-\pi ^2+\log ^2(5-e)+2 i \pi \log (5-e)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\frac {2 \left (10 x^4+24 x^3+14 x^2+\left (5 x^4+14 x^3+7 x^2\right ) \log \left (\frac {4}{x^2}\right )+5 (i \pi +\log (5-e))^4 \left (\log \left (\frac {4}{x^2}\right )+2\right )+20 (i \pi +\log (5-e))^3 \left (\log \left (\frac {4}{x^2}\right ) x+2 x\right )+2 (i \pi +\log (5-e))^2 \left (30 x^2+12 x+5 \left (3 x^2+x\right ) \log \left (\frac {4}{x^2}\right )\right )+4 (i \pi +\log (5-e)) \left (10 x^3+12 x^2+\left (5 x^3+6 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )\right )}{x^2 \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \int \frac {10 x^4+24 x^3+14 x^2+\left (5 x^4+14 x^3+7 x^2\right ) \log \left (\frac {4}{x^2}\right )+5 (i \pi +\log (5-e))^4 \left (\log \left (\frac {4}{x^2}\right )+2\right )+20 (i \pi +\log (5-e))^3 \left (\log \left (\frac {4}{x^2}\right ) x+2 x\right )+2 (i \pi +\log (5-e))^2 \left (30 x^2+12 x+5 \left (3 x^2+x\right ) \log \left (\frac {4}{x^2}\right )\right )+4 (i \pi +\log (5-e)) \left (10 x^3+12 x^2+\left (5 x^3+6 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )}{x^2 \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {2}{5} \int \left (\frac {10 x^2}{\left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2}+\frac {24 \left (1+\frac {5}{3} (i \pi +\log (5-e))\right ) x}{\left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2}+\frac {14 \left (1-\frac {6}{7} (\pi -i \log (5-e)) (5 \pi -i (4+5 \log (5-e)))\right )}{\left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2}-\frac {24 (\pi -i \log (5-e))^2 \left (1+\frac {5}{3} (i \pi +\log (5-e))\right )}{\left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2 x}+\frac {\left (5 x^4+2 (7+10 i \pi +10 \log (5-e)) x^3+\left (7-30 \pi ^2+24 \log (5-e)+30 \log ^2(5-e)+12 i \pi (2+5 \log (5-e))\right ) x^2-10 (1+i (2 \pi -2 i \log (5-e))) (\pi -i \log (5-e))^2 x+5 (\pi -i \log (5-e))^4\right ) \log \left (\frac {4}{x^2}\right )}{\left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2 x^2}+\frac {10 (\pi -i \log (5-e))^4}{\left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )^2 x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2}{5} \left (\frac {8 (7-6 (\pi -i \log (5-e)) (5 \pi -i (4+5 \log (5-e)))) \text {arctanh}\left (\frac {2 x+2 \log (5-e)+2 i \pi +1}{\sqrt {1+4 i \pi +4 \log (5-e)}}\right )}{(1+4 i \pi +4 \log (5-e))^{3/2}}+\frac {20 \left (1-2 \pi ^4-8 i \pi ^3 (1-\log (5-e))+8 \log (5-e)+18 \log ^2(5-e)+8 \log ^3(5-e)-2 \log ^4(5-e)-6 \pi ^2 \left (3+4 \log (5-e)-2 \log ^2(5-e)\right )+4 i \pi \left (2+9 \log (5-e)+6 \log ^2(5-e)-2 \log ^3(5-e)\right )\right ) \text {arctanh}\left (\frac {2 x+2 \log (5-e)+2 i \pi +1}{\sqrt {1+4 i \pi +4 \log (5-e)}}\right )}{(\pi -i \log (5-e))^2 (1+4 i \pi +4 \log (5-e))^{3/2}}+\frac {8 (3+5 i \pi +5 \log (5-e)) (2 \pi -i (1+2 \log (5-e))) \left (4 \pi (1-\log (5-e))-i \left (1+2 \pi ^2+4 \log (5-e)-2 \log ^2(5-e)\right )\right ) \text {arctanh}\left (\frac {2 x+2 \log (5-e)+2 i \pi +1}{\sqrt {1+4 i \pi +4 \log (5-e)}}\right )}{(\pi -i \log (5-e))^2 (1+4 i \pi +4 \log (5-e))^{3/2}}-\frac {16 (1+2 i \pi +2 \log (5-e)) (3+5 i \pi +5 \log (5-e)) \text {arctanh}\left (\frac {2 x+2 \log (5-e)+2 i \pi +1}{\sqrt {1+4 i \pi +4 \log (5-e)}}\right )}{(1+4 i \pi +4 \log (5-e))^{3/2}}-\frac {40 (\pi -i \log (5-e))^2 \text {arctanh}\left (\frac {2 x+2 \log (5-e)+2 i \pi +1}{\sqrt {1+4 i \pi +4 \log (5-e)}}\right )}{(1+4 i \pi +4 \log (5-e))^{3/2}}+\frac {8 (3+5 i \pi +5 \log (5-e)) \left ((1+2 i \pi +2 \log (5-e)) x+(1+2 i \pi +2 \log (5-e))^2+2 (\pi -i \log (5-e))^2\right )}{(1+4 i \pi +4 \log (5-e)) \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}-\frac {10 (\pi -i \log (5-e))^2 \left ((1+2 i \pi +2 \log (5-e)) x+(1+2 i \pi +2 \log (5-e))^2+2 (\pi -i \log (5-e))^2\right )}{x (1+4 i \pi +4 \log (5-e)) \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}-\frac {5 \log \left (\frac {4}{x^2}\right )}{x}+\frac {8 x (1+2 i \pi +2 \log (5-e)) \log \left (\frac {4}{x^2}\right )}{(1+4 i \pi +4 \log (5-e)) \left (1+2 i \pi +2 \log (5-e)-\sqrt {1+4 i \pi +4 \log (5-e)}\right ) \left (2 x-\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}-\frac {8 x \log \left (\frac {4}{x^2}\right )}{(1+4 i \pi +4 \log (5-e)) \left (2 x-\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}+\frac {8 x (1+2 i \pi +2 \log (5-e)) \log \left (\frac {4}{x^2}\right )}{(1+4 i \pi +4 \log (5-e)) \left (1+2 i \pi +2 \log (5-e)+\sqrt {1+4 i \pi +4 \log (5-e)}\right ) \left (2 x+\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}-\frac {8 x \log \left (\frac {4}{x^2}\right )}{(1+4 i \pi +4 \log (5-e)) \left (2 x+\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}-\frac {8 (3+5 i \pi +5 \log (5-e)) \log (x)}{(\pi -i \log (5-e))^2}+\frac {20 (1+2 i \pi +2 \log (5-e)) \log (x)}{(\pi -i \log (5-e))^2}+\frac {4 (3+5 i \pi +5 \log (5-e)) \log \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}{(\pi -i \log (5-e))^2}-\frac {10 (1+2 i \pi +2 \log (5-e)) \log \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}{(\pi -i \log (5-e))^2}+\frac {8 (1+2 i \pi +2 \log (5-e)) \log \left (2 x-\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}{(1+4 i \pi +4 \log (5-e)) \left (1+2 i \pi +2 \log (5-e)-\sqrt {1+4 i \pi +4 \log (5-e)}\right )}-\frac {8 \log \left (2 x-\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}{1+4 i \pi +4 \log (5-e)}+\frac {8 (1+2 i \pi +2 \log (5-e)) \log \left (2 x+\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}{(1+4 i \pi +4 \log (5-e)) \left (1+2 i \pi +2 \log (5-e)+\sqrt {1+4 i \pi +4 \log (5-e)}\right )}-\frac {8 \log \left (2 x+\sqrt {1+4 i \pi +4 \log (5-e)}+2 \log (5-e)+2 i \pi +1\right )}{1+4 i \pi +4 \log (5-e)}+\frac {10}{x}-\frac {8 (3+5 i \pi +5 \log (5-e)) \left (2 (\pi -i \log (5-e))^2-x (1+2 i \pi +2 \log (5-e))\right )}{(1+4 i \pi +4 \log (5-e)) \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}-\frac {10 x \left (2 (\pi -i \log (5-e))^2-x (1+2 i \pi +2 \log (5-e))\right )}{(1+4 i \pi +4 \log (5-e)) \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}-\frac {2 (2 x+2 \log (5-e)+2 i \pi +1) (7-6 (\pi -i \log (5-e)) (5 \pi -i (4+5 \log (5-e))))}{(1+4 i \pi +4 \log (5-e)) \left (x^2+(1+2 i \pi +2 \log (5-e)) x-(\pi -i \log (5-e))^2\right )}-\frac {20 \left (1-\pi ^2+4 \log (5-e)+\log ^2(5-e)+2 i \pi (2+\log (5-e))\right )}{x (1+4 i \pi +4 \log (5-e))}\right )\) |
Input:
Int[(-28*x^2 - 48*x^3 - 20*x^4 + (I*Pi + Log[5 - E])^4*(-20 - 10*Log[4/x^2 ]) + (-14*x^2 - 28*x^3 - 10*x^4)*Log[4/x^2] + (I*Pi + Log[5 - E])^3*(-80*x - 40*x*Log[4/x^2]) + (I*Pi + Log[5 - E])^2*(-48*x - 120*x^2 + (-20*x - 60 *x^2)*Log[4/x^2]) + (I*Pi + Log[5 - E])*(-96*x^2 - 80*x^3 + (-48*x^2 - 40* x^3)*Log[4/x^2]))/(5*x^4 + 10*x^5 + 5*x^6 + (20*x^4 + 20*x^5)*(I*Pi + Log[ 5 - E]) + (10*x^3 + 30*x^4)*(I*Pi + Log[5 - E])^2 + 20*x^3*(I*Pi + Log[5 - E])^3 + 5*x^2*(I*Pi + Log[5 - E])^4),x]
Output:
(-2*(10/x - (40*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I) *Pi + 4*Log[5 - E]]]*(Pi - I*Log[5 - E])^2)/(1 + (4*I)*Pi + 4*Log[5 - E])^ (3/2) - (16*ArcTanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]]]*(1 + (2*I)*Pi + 2*Log[5 - E])*(3 + (5*I)*Pi + 5*Log[5 - E ]))/(1 + (4*I)*Pi + 4*Log[5 - E])^(3/2) - (20*(1 - Pi^2 + 4*Log[5 - E] + L og[5 - E]^2 + (2*I)*Pi*(2 + Log[5 - E])))/(x*(1 + (4*I)*Pi + 4*Log[5 - E]) ) - (10*x*(2*(Pi - I*Log[5 - E])^2 - x*(1 + (2*I)*Pi + 2*Log[5 - E])))/((1 + (4*I)*Pi + 4*Log[5 - E])*(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) - (8*(3 + (5*I)*Pi + 5*Log[5 - E])*(2*(Pi - I*Log[5 - E ])^2 - x*(1 + (2*I)*Pi + 2*Log[5 - E])))/((1 + (4*I)*Pi + 4*Log[5 - E])*(x ^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) - (10*(Pi - I*Log[5 - E])^2*(2*(Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E] ) + (1 + (2*I)*Pi + 2*Log[5 - E])^2))/(x*(1 + (4*I)*Pi + 4*Log[5 - E])*(x^ 2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) + (8*(3 + (5 *I)*Pi + 5*Log[5 - E])*(2*(Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[ 5 - E]) + (1 + (2*I)*Pi + 2*Log[5 - E])^2))/((1 + (4*I)*Pi + 4*Log[5 - E]) *(x^2 - (Pi - I*Log[5 - E])^2 + x*(1 + (2*I)*Pi + 2*Log[5 - E]))) + (8*Arc Tanh[(1 + (2*I)*Pi + 2*x + 2*Log[5 - E])/Sqrt[1 + (4*I)*Pi + 4*Log[5 - E]] ]*(3 + (5*I)*Pi + 5*Log[5 - E])*(2*Pi - I*(1 + 2*Log[5 - E]))*(4*Pi*(1 - L og[5 - E]) - I*(1 + 2*Pi^2 + 4*Log[5 - E] - 2*Log[5 - E]^2)))/((Pi - I*...
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(59\) vs. \(2(28)=56\).
Time = 2.92 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(\frac {2 \left (5 \ln \left ({\mathrm e}-5\right )^{2}+10 x \ln \left ({\mathrm e}-5\right )+5 x^{2}+7 x \right ) \ln \left (\frac {4}{x^{2}}\right )}{5 \left (\ln \left ({\mathrm e}-5\right )^{2}+2 x \ln \left ({\mathrm e}-5\right )+x^{2}+x \right ) x}\) | \(60\) |
| norman | \(\frac {2 x^{2} \ln \left (\frac {4}{x^{2}}\right )+\left (\frac {14}{5}+4 \ln \left ({\mathrm e}-5\right )\right ) x \ln \left (\frac {4}{x^{2}}\right )+2 \ln \left ({\mathrm e}-5\right )^{2} \ln \left (\frac {4}{x^{2}}\right )}{x \left (\ln \left ({\mathrm e}-5\right )^{2}+2 x \ln \left ({\mathrm e}-5\right )+x^{2}+x \right )}\) | \(71\) |
| parallelrisch | \(\frac {20 \ln \left ({\mathrm e}-5\right )^{2} \ln \left (\frac {4}{x^{2}}\right )+40 \ln \left (\frac {4}{x^{2}}\right ) x \ln \left ({\mathrm e}-5\right )+20 x^{2} \ln \left (\frac {4}{x^{2}}\right )+28 x \ln \left (\frac {4}{x^{2}}\right )}{10 x \left (\ln \left ({\mathrm e}-5\right )^{2}+2 x \ln \left ({\mathrm e}-5\right )+x^{2}+x \right )}\) | \(78\) |
Input:
int(((-10*ln(4/x^2)-20)*ln(exp(1)-5)^4+(-40*x*ln(4/x^2)-80*x)*ln(exp(1)-5) ^3+((-60*x^2-20*x)*ln(4/x^2)-120*x^2-48*x)*ln(exp(1)-5)^2+((-40*x^3-48*x^2 )*ln(4/x^2)-80*x^3-96*x^2)*ln(exp(1)-5)+(-10*x^4-28*x^3-14*x^2)*ln(4/x^2)- 20*x^4-48*x^3-28*x^2)/(5*x^2*ln(exp(1)-5)^4+20*x^3*ln(exp(1)-5)^3+(30*x^4+ 10*x^3)*ln(exp(1)-5)^2+(20*x^5+20*x^4)*ln(exp(1)-5)+5*x^6+10*x^5+5*x^4),x, method=_RETURNVERBOSE)
Output:
2/5*(5*ln(exp(1)-5)^2+10*x*ln(exp(1)-5)+5*x^2+7*x)/(ln(exp(1)-5)^2+2*x*ln( exp(1)-5)+x^2+x)/x*ln(4/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.11 \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\frac {2 \, {\left (10 \, x \log \left (\frac {4}{x^{2}}\right ) \log \left (e - 5\right ) + 5 \, \log \left (\frac {4}{x^{2}}\right ) \log \left (e - 5\right )^{2} + {\left (5 \, x^{2} + 7 \, x\right )} \log \left (\frac {4}{x^{2}}\right )\right )}}{5 \, {\left (x^{3} + 2 \, x^{2} \log \left (e - 5\right ) + x \log \left (e - 5\right )^{2} + x^{2}\right )}} \] Input:
integrate(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log (exp(1)-5)^3+((-60*x^2-20*x)*log(4/x^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-4 0*x^3-48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14*x ^2)*log(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp (1)-5)^3+(30*x^4+10*x^3)*log(exp(1)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x ^6+10*x^5+5*x^4),x, algorithm="fricas")
Output:
2/5*(10*x*log(4/x^2)*log(e - 5) + 5*log(4/x^2)*log(e - 5)^2 + (5*x^2 + 7*x )*log(4/x^2))/(x^3 + 2*x^2*log(e - 5) + x*log(e - 5)^2 + x^2)
Timed out. \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\text {Timed out} \] Input:
integrate(((-10*ln(4/x**2)-20)*ln(exp(1)-5)**4+(-40*x*ln(4/x**2)-80*x)*ln( exp(1)-5)**3+((-60*x**2-20*x)*ln(4/x**2)-120*x**2-48*x)*ln(exp(1)-5)**2+(( -40*x**3-48*x**2)*ln(4/x**2)-80*x**3-96*x**2)*ln(exp(1)-5)+(-10*x**4-28*x* *3-14*x**2)*ln(4/x**2)-20*x**4-48*x**3-28*x**2)/(5*x**2*ln(exp(1)-5)**4+20 *x**3*ln(exp(1)-5)**3+(30*x**4+10*x**3)*ln(exp(1)-5)**2+(20*x**5+20*x**4)* ln(exp(1)-5)+5*x**6+10*x**5+5*x**4),x)
Output:
Timed out
Exception generated. \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log (exp(1)-5)^3+((-60*x^2-20*x)*log(4/x^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-4 0*x^3-48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14*x ^2)*log(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp (1)-5)^3+(30*x^4+10*x^3)*log(exp(1)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x ^6+10*x^5+5*x^4),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*log(%e-5)+1>0)', see `assume?` for more
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (29) = 58\).
Time = 0.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.92 \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\frac {2 \, {\left (10 \, x^{2} \log \left (2\right ) - 5 \, x^{2} \log \left (x^{2}\right ) + 20 \, x \log \left (2\right ) \log \left (e - 5\right ) - 10 \, x \log \left (x^{2}\right ) \log \left (e - 5\right ) + 10 \, \log \left (2\right ) \log \left (e - 5\right )^{2} - 5 \, \log \left (x^{2}\right ) \log \left (e - 5\right )^{2} + 14 \, x \log \left (2\right ) - 7 \, x \log \left (x^{2}\right )\right )}}{5 \, {\left (x^{3} + 2 \, x^{2} \log \left (e - 5\right ) + x \log \left (e - 5\right )^{2} + x^{2}\right )}} \] Input:
integrate(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log (exp(1)-5)^3+((-60*x^2-20*x)*log(4/x^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-4 0*x^3-48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14*x ^2)*log(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp (1)-5)^3+(30*x^4+10*x^3)*log(exp(1)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x ^6+10*x^5+5*x^4),x, algorithm="giac")
Output:
2/5*(10*x^2*log(2) - 5*x^2*log(x^2) + 20*x*log(2)*log(e - 5) - 10*x*log(x^ 2)*log(e - 5) + 10*log(2)*log(e - 5)^2 - 5*log(x^2)*log(e - 5)^2 + 14*x*lo g(2) - 7*x*log(x^2))/(x^3 + 2*x^2*log(e - 5) + x*log(e - 5)^2 + x^2)
Time = 2.95 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.36 \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\frac {2\,\ln \left (\frac {4}{x^2}\right )}{x}+\frac {4\,\ln \left (\frac {4}{x^2}\right )}{5\,\left (x^2+\left (2\,\ln \left (\mathrm {e}-5\right )+1\right )\,x+{\ln \left (\mathrm {e}-5\right )}^2\right )} \] Input:
int(-(log(4/x^2)*(14*x^2 + 28*x^3 + 10*x^4) + log(exp(1) - 5)*(log(4/x^2)* (48*x^2 + 40*x^3) + 96*x^2 + 80*x^3) + log(exp(1) - 5)^4*(10*log(4/x^2) + 20) + log(exp(1) - 5)^3*(80*x + 40*x*log(4/x^2)) + log(exp(1) - 5)^2*(48*x + log(4/x^2)*(20*x + 60*x^2) + 120*x^2) + 28*x^2 + 48*x^3 + 20*x^4)/(log( exp(1) - 5)*(20*x^4 + 20*x^5) + log(exp(1) - 5)^2*(10*x^3 + 30*x^4) + 5*x^ 2*log(exp(1) - 5)^4 + 20*x^3*log(exp(1) - 5)^3 + 5*x^4 + 10*x^5 + 5*x^6),x )
Output:
(2*log(4/x^2))/x + (4*log(4/x^2))/(5*(log(exp(1) - 5)^2 + x*(2*log(exp(1) - 5) + 1) + x^2))
Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.56 \[ \int \frac {-28 x^2-48 x^3-20 x^4+(i \pi +\log (5-e))^4 \left (-20-10 \log \left (\frac {4}{x^2}\right )\right )+\left (-14 x^2-28 x^3-10 x^4\right ) \log \left (\frac {4}{x^2}\right )+(i \pi +\log (5-e))^3 \left (-80 x-40 x \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e))^2 \left (-48 x-120 x^2+\left (-20 x-60 x^2\right ) \log \left (\frac {4}{x^2}\right )\right )+(i \pi +\log (5-e)) \left (-96 x^2-80 x^3+\left (-48 x^2-40 x^3\right ) \log \left (\frac {4}{x^2}\right )\right )}{5 x^4+10 x^5+5 x^6+\left (20 x^4+20 x^5\right ) (i \pi +\log (5-e))+\left (10 x^3+30 x^4\right ) (i \pi +\log (5-e))^2+20 x^3 (i \pi +\log (5-e))^3+5 x^2 (i \pi +\log (5-e))^4} \, dx=\frac {4 \mathrm {log}\left (e -5\right )^{3} \mathrm {log}\left (\frac {4}{x^{2}}\right )+6 \mathrm {log}\left (e -5\right )^{2} \mathrm {log}\left (\frac {4}{x^{2}}\right ) x +2 \mathrm {log}\left (e -5\right )^{2} \mathrm {log}\left (\frac {4}{x^{2}}\right )-4 \mathrm {log}\left (e -5\right )^{2} \mathrm {log}\left (x \right ) x +\frac {48 \,\mathrm {log}\left (e -5\right ) \mathrm {log}\left (\frac {4}{x^{2}}\right ) x}{5}-8 \,\mathrm {log}\left (e -5\right ) \mathrm {log}\left (x \right ) x^{2}-2 \,\mathrm {log}\left (\frac {4}{x^{2}}\right ) x^{3}+\frac {14 \,\mathrm {log}\left (\frac {4}{x^{2}}\right ) x}{5}-4 \,\mathrm {log}\left (x \right ) x^{3}-4 \,\mathrm {log}\left (x \right ) x^{2}}{x \left (2 \mathrm {log}\left (e -5\right )^{3}+4 \mathrm {log}\left (e -5\right )^{2} x +\mathrm {log}\left (e -5\right )^{2}+2 \,\mathrm {log}\left (e -5\right ) x^{2}+4 \,\mathrm {log}\left (e -5\right ) x +x^{2}+x \right )} \] Input:
int(((-10*log(4/x^2)-20)*log(exp(1)-5)^4+(-40*x*log(4/x^2)-80*x)*log(exp(1 )-5)^3+((-60*x^2-20*x)*log(4/x^2)-120*x^2-48*x)*log(exp(1)-5)^2+((-40*x^3- 48*x^2)*log(4/x^2)-80*x^3-96*x^2)*log(exp(1)-5)+(-10*x^4-28*x^3-14*x^2)*lo g(4/x^2)-20*x^4-48*x^3-28*x^2)/(5*x^2*log(exp(1)-5)^4+20*x^3*log(exp(1)-5) ^3+(30*x^4+10*x^3)*log(exp(1)-5)^2+(20*x^5+20*x^4)*log(exp(1)-5)+5*x^6+10* x^5+5*x^4),x)
Output:
(2*(10*log(e - 5)**3*log(4/x**2) + 15*log(e - 5)**2*log(4/x**2)*x + 5*log( e - 5)**2*log(4/x**2) - 10*log(e - 5)**2*log(x)*x + 24*log(e - 5)*log(4/x* *2)*x - 20*log(e - 5)*log(x)*x**2 - 5*log(4/x**2)*x**3 + 7*log(4/x**2)*x - 10*log(x)*x**3 - 10*log(x)*x**2))/(5*x*(2*log(e - 5)**3 + 4*log(e - 5)**2 *x + log(e - 5)**2 + 2*log(e - 5)*x**2 + 4*log(e - 5)*x + x**2 + x))