Integrand size = 283, antiderivative size = 39 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=x \log \left ((5-x)^2 \left (3+e^{\frac {1}{2} \left (2-\frac {x}{-3+x}\right )}-x-\frac {x}{\log (2)}\right )\right ) \] Output:
ln((exp(1-1/2*x/(-3+x))-x+3-x/ln(2))*(5-x)^2)*x
Time = 0.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.18 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=\frac {1}{2} \left (3+2 x \log \left ((-5+x)^2 \left (e^{\frac {-6+x}{2 (-3+x)}}+\frac {-x (1+\log (2))+\log (8)}{\log (2)}\right )\right )\right ) \] Input:
Integrate[(90*x - 114*x^2 + 46*x^3 - 6*x^4 + E^((-6 + x)/(-6 + 2*x))*(21*x - 21*x^2 + 4*x^3)*Log[2] + (198*x - 186*x^2 + 58*x^3 - 6*x^4)*Log[2] + (9 0*x - 78*x^2 + 22*x^3 - 2*x^4 + E^((-6 + x)/(-6 + 2*x))*(-90 + 78*x - 22*x ^2 + 2*x^3)*Log[2] + (-270 + 324*x - 144*x^2 + 28*x^3 - 2*x^4)*Log[2])*Log [(-25*x + 10*x^2 - x^3 + E^((-6 + x)/(-6 + 2*x))*(25 - 10*x + x^2)*Log[2] + (75 - 55*x + 13*x^2 - x^3)*Log[2])/Log[2]])/(90*x - 78*x^2 + 22*x^3 - 2* x^4 + E^((-6 + x)/(-6 + 2*x))*(-90 + 78*x - 22*x^2 + 2*x^3)*Log[2] + (-270 + 324*x - 144*x^2 + 28*x^3 - 2*x^4)*Log[2]),x]
Output:
(3 + 2*x*Log[(-5 + x)^2*(E^((-6 + x)/(2*(-3 + x))) + (-(x*(1 + Log[2])) + Log[8])/Log[2])])/2
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.
| \(\displaystyle \int \frac {-6 x^4+46 x^3-114 x^2+e^{\frac {x-6}{2 x-6}} \left (4 x^3-21 x^2+21 x\right ) \log (2)+\left (-2 x^4+22 x^3-78 x^2+e^{\frac {x-6}{2 x-6}} \left (2 x^3-22 x^2+78 x-90\right ) \log (2)+\left (-2 x^4+28 x^3-144 x^2+324 x-270\right ) \log (2)+90 x\right ) \log \left (\frac {-x^3+10 x^2+e^{\frac {x-6}{2 x-6}} \left (x^2-10 x+25\right ) \log (2)+\left (-x^3+13 x^2-55 x+75\right ) \log (2)-25 x}{\log (2)}\right )+\left (-6 x^4+58 x^3-186 x^2+198 x\right ) \log (2)+90 x}{-2 x^4+22 x^3-78 x^2+e^{\frac {x-6}{2 x-6}} \left (2 x^3-22 x^2+78 x-90\right ) \log (2)+\left (-2 x^4+28 x^3-144 x^2+324 x-270\right ) \log (2)+90 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {3}{x-3}} \left (6 x^4-46 x^3+114 x^2-e^{\frac {x-6}{2 x-6}} \left (4 x^3-21 x^2+21 x\right ) \log (2)-\left (-2 x^4+22 x^3-78 x^2+e^{\frac {x-6}{2 x-6}} \left (2 x^3-22 x^2+78 x-90\right ) \log (2)+\left (-2 x^4+28 x^3-144 x^2+324 x-270\right ) \log (2)+90 x\right ) \log \left (\frac {-x^3+10 x^2+e^{\frac {x-6}{2 x-6}} \left (x^2-10 x+25\right ) \log (2)+\left (-x^3+13 x^2-55 x+75\right ) \log (2)-25 x}{\log (2)}\right )-\left (-6 x^4+58 x^3-186 x^2+198 x\right ) \log (2)-90 x\right )}{2 (3-x)^2 (5-x) \left (-e^{\frac {3}{x-3}} x (1+\log (2))+e^{\frac {3}{x-3}} \log (8)+e^{\frac {x}{2 x-6}} \log (2)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {e^{-\frac {3}{3-x}} \left (-6 x^4+46 x^3-114 x^2+90 x+2 \left (-x^4+11 x^3-39 x^2+45 x-e^{\frac {6-x}{2 (3-x)}} \left (-x^3+11 x^2-39 x+45\right ) \log (2)-\left (x^4-14 x^3+72 x^2-162 x+135\right ) \log (2)\right ) \log \left (-\frac {x^3-10 x^2+25 x-e^{\frac {6-x}{2 (3-x)}} \left (x^2-10 x+25\right ) \log (2)-\left (-x^3+13 x^2-55 x+75\right ) \log (2)}{\log (2)}\right )+e^{\frac {6-x}{2 (3-x)}} \left (4 x^3-21 x^2+21 x\right ) \log (2)+2 \left (-3 x^4+29 x^3-93 x^2+99 x\right ) \log (2)\right )}{(3-x)^2 (5-x) \left (-e^{-\frac {3}{3-x}} (1+\log (2)) x+e^{-\frac {3}{3-x}} \log (8)+e^{-\frac {x}{2 (3-x)}} \log (2)\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{-\frac {3}{3-x}} \left (-6 x^4+46 x^3-114 x^2+90 x+2 \left (-x^4+11 x^3-39 x^2+45 x-e^{\frac {6-x}{2 (3-x)}} \left (-x^3+11 x^2-39 x+45\right ) \log (2)-\left (x^4-14 x^3+72 x^2-162 x+135\right ) \log (2)\right ) \log \left (-\frac {x^3-10 x^2+25 x-e^{\frac {6-x}{2 (3-x)}} \left (x^2-10 x+25\right ) \log (2)-\left (-x^3+13 x^2-55 x+75\right ) \log (2)}{\log (2)}\right )+e^{\frac {6-x}{2 (3-x)}} \left (4 x^3-21 x^2+21 x\right ) \log (2)+2 \left (-3 x^4+29 x^3-93 x^2+99 x\right ) \log (2)\right )}{(3-x)^2 (5-x) \left (-e^{-\frac {3}{3-x}} (1+\log (2)) x+e^{-\frac {3}{3-x}} \log (8)+e^{-\frac {x}{2 (3-x)}} \log (2)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {e^{\frac {x-6}{2 (x-3)}-\frac {3}{3-x}} x \log (2) \left (-\left ((2+\log (4)) x^3\right )+(25+\log (33554432)) x^2-(93+\log (5070602400912917605986812821504)) x+\log (43556142965880123323311949751266331066368)+90\right )}{(3-x)^2 (5-x) (x (1+\log (2))-\log (8)) \left (-e^{\frac {3}{x-3}} (1+\log (2)) x+e^{\frac {3}{x-3}} \log (8)+e^{\frac {x}{2 (x-3)}} \log (2)\right )}+\frac {2 e^{-\frac {3}{x-3}-\frac {3}{3-x}} \left ((1+\log (2)) \log \left (e^{\frac {x-6}{2 (x-3)}} (x-5)^2-\frac {(x-5)^2 (\log (2) x+x-\log (8))}{\log (2)}\right ) x^2+3 (1+\log (2)) x^2-5 \left (1+\frac {\log (2) \log (256)}{\log (32)}\right ) \log \left (e^{\frac {x-6}{2 (x-3)}} (x-5)^2-\frac {(x-5)^2 (\log (2) x+x-\log (8))}{\log (2)}\right ) x-5 \left (1+\log \left (4 \sqrt [5]{2}\right )\right ) x+5 \log (8) \log \left (e^{\frac {x-6}{2 (x-3)}} (x-5)^2-\frac {(x-5)^2 (\log (2) x+x-\log (8))}{\log (2)}\right )\right )}{(5-x) (x (1+\log (2))-\log (8))}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{2} \int \frac {2 x ((3+\log (8)) x-\log (2048)-5)-\frac {e^{\frac {x}{2 (x-3)}} x \log (2) \left ((2+\log (4)) x^3-(25+\log (33554432)) x^2+(93+\log (5070602400912917605986812821504)) x-\log (43556142965880123323311949751266331066368)-90\right )}{(x-3)^2 \left (e^{\frac {x}{2 (x-3)}} \log (2)-e^{\frac {3}{x-3}} (\log (2) x+x-\log (8))\right )}+\frac {2 \left ((1+\log (2)) \log (32) x^2-5 (\log (32)+\log (2) \log (256)) x+5 \log (8) \log (32)\right ) \log \left (e^{\frac {x-6}{2 (x-3)}} (x-5)^2-\frac {(x-5)^2 (\log (2) x+x-\log (8))}{\log (2)}\right )}{\log (32)}}{(5-x) (x (1+\log (2))-\log (8))}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {2 x ((3+\log (8)) x-\log (2048)-5)}{(5-x) (x (1+\log (2))-\log (8))}+\frac {x \left (-\left ((2+\log (4)) x^3\right )+(25+\log (33554432)) x^2-(93+\log (5070602400912917605986812821504)) x+\log (43556142965880123323311949751266331066368)+90\right )}{(3-x)^2 (5-x) (x (1+\log (2))-\log (8))}+\frac {e^{\frac {3}{x-3}} x \left (-\left ((2+\log (4)) x^3\right )+(25+\log (33554432)) x^2-(93+\log (5070602400912917605986812821504)) x+\log (43556142965880123323311949751266331066368)+90\right )}{(3-x)^2 (5-x) \left (-e^{\frac {3}{x-3}} (1+\log (2)) x+e^{\frac {3}{x-3}} \log (8)+e^{\frac {x}{2 (x-3)}} \log (2)\right )}+\frac {2 \left (\log (2) (5+\log (32)) x^2-5 \log (2) (5+\log (256)) x+5 \log (8) \log (32)\right ) \log \left (e^{\frac {x-6}{2 (x-3)}} (x-5)^2-\frac {(x-5)^2 (\log (2) x+x-\log (8))}{\log (2)}\right )}{(5-x) (x (1+\log (2))-\log (8)) \log (32)}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {2 x ((3+\log (8)) x-\log (2048)-5)}{(5-x) (x (1+\log (2))-\log (8))}+\frac {x \left (-\left ((2+\log (4)) x^3\right )+(25+\log (33554432)) x^2-(93+\log (5070602400912917605986812821504)) x+\log (43556142965880123323311949751266331066368)+90\right )}{(3-x)^2 (5-x) (x (1+\log (2))-\log (8))}+\frac {e^{\frac {3}{x-3}} x \left (-\left ((2+\log (4)) x^3\right )+(25+\log (33554432)) x^2-(93+\log (5070602400912917605986812821504)) x+\log (43556142965880123323311949751266331066368)+90\right )}{(3-x)^2 (5-x) \left (-e^{\frac {3}{x-3}} (1+\log (2)) x+e^{\frac {3}{x-3}} \log (8)+e^{\frac {x}{2 (x-3)}} \log (2)\right )}+\frac {2 \left (\log (2) (5+\log (32)) x^2-5 \log (2) (5+\log (256)) x+5 \log (8) \log (32)\right ) \log \left (e^{\frac {x-6}{2 (x-3)}} (x-5)^2-\frac {(x-5)^2 (\log (2) x+x-\log (8))}{\log (2)}\right )}{(5-x) (x (1+\log (2))-\log (8)) \log (32)}\right )dx\) |
Input:
Int[(90*x - 114*x^2 + 46*x^3 - 6*x^4 + E^((-6 + x)/(-6 + 2*x))*(21*x - 21* x^2 + 4*x^3)*Log[2] + (198*x - 186*x^2 + 58*x^3 - 6*x^4)*Log[2] + (90*x - 78*x^2 + 22*x^3 - 2*x^4 + E^((-6 + x)/(-6 + 2*x))*(-90 + 78*x - 22*x^2 + 2 *x^3)*Log[2] + (-270 + 324*x - 144*x^2 + 28*x^3 - 2*x^4)*Log[2])*Log[(-25* x + 10*x^2 - x^3 + E^((-6 + x)/(-6 + 2*x))*(25 - 10*x + x^2)*Log[2] + (75 - 55*x + 13*x^2 - x^3)*Log[2])/Log[2]])/(90*x - 78*x^2 + 22*x^3 - 2*x^4 + E^((-6 + x)/(-6 + 2*x))*(-90 + 78*x - 22*x^2 + 2*x^3)*Log[2] + (-270 + 324 *x - 144*x^2 + 28*x^3 - 2*x^4)*Log[2]),x]
Output:
$Aborted
Time = 12.92 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.62
| method | result | size |
| parallelrisch | \(x \ln \left (\frac {\left (x^{2}-10 x +25\right ) \ln \left (2\right ) {\mathrm e}^{\frac {-6+x}{2 x -6}}+\left (-x^{3}+13 x^{2}-55 x +75\right ) \ln \left (2\right )-x^{3}+10 x^{2}-25 x}{\ln \left (2\right )}\right )\) | \(63\) |
| norman | \(\frac {x^{2} \ln \left (\frac {\left (x^{2}-10 x +25\right ) \ln \left (2\right ) {\mathrm e}^{\frac {-6+x}{2 x -6}}+\left (-x^{3}+13 x^{2}-55 x +75\right ) \ln \left (2\right )-x^{3}+10 x^{2}-25 x}{\ln \left (2\right )}\right )-3 x \ln \left (\frac {\left (x^{2}-10 x +25\right ) \ln \left (2\right ) {\mathrm e}^{\frac {-6+x}{2 x -6}}+\left (-x^{3}+13 x^{2}-55 x +75\right ) \ln \left (2\right )-x^{3}+10 x^{2}-25 x}{\ln \left (2\right )}\right )}{-3+x}\) | \(137\) |
| risch | \(x \ln \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )+2 \ln \left (-5+x \right ) x -i \pi x \operatorname {csgn}\left (i \left (-5+x \right )^{2} \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right )^{2}-\frac {i \pi x \operatorname {csgn}\left (i \left (-5+x \right )\right )^{2} \operatorname {csgn}\left (i \left (-5+x \right )^{2}\right )}{2}+i \pi x \,\operatorname {csgn}\left (i \left (-5+x \right )\right ) \operatorname {csgn}\left (i \left (-5+x \right )^{2}\right )^{2}-\frac {i \pi x \operatorname {csgn}\left (i \left (-5+x \right )^{2}\right )^{3}}{2}-\frac {i \pi x \,\operatorname {csgn}\left (i \left (-5+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right ) \operatorname {csgn}\left (i \left (-5+x \right )^{2} \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right )}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (-5+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (-5+x \right )^{2} \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right )^{2}}{2}+\frac {i \pi x \,\operatorname {csgn}\left (i \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right ) \operatorname {csgn}\left (i \left (-5+x \right )^{2} \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right )^{2}}{2}+\frac {i \pi x \operatorname {csgn}\left (i \left (-5+x \right )^{2} \left (\left (x -{\mathrm e}^{\frac {-6+x}{2 x -6}}-3\right ) \ln \left (2\right )+x \right )\right )^{3}}{2}+i \pi x -x \ln \left (\ln \left (2\right )\right )\) | \(357\) |
Input:
int((((2*x^3-22*x^2+78*x-90)*ln(2)*exp((-6+x)/(2*x-6))+(-2*x^4+28*x^3-144* x^2+324*x-270)*ln(2)-2*x^4+22*x^3-78*x^2+90*x)*ln(((x^2-10*x+25)*ln(2)*exp ((-6+x)/(2*x-6))+(-x^3+13*x^2-55*x+75)*ln(2)-x^3+10*x^2-25*x)/ln(2))+(4*x^ 3-21*x^2+21*x)*ln(2)*exp((-6+x)/(2*x-6))+(-6*x^4+58*x^3-186*x^2+198*x)*ln( 2)-6*x^4+46*x^3-114*x^2+90*x)/((2*x^3-22*x^2+78*x-90)*ln(2)*exp((-6+x)/(2* x-6))+(-2*x^4+28*x^3-144*x^2+324*x-270)*ln(2)-2*x^4+22*x^3-78*x^2+90*x),x, method=_RETURNVERBOSE)
Output:
ln(((x^2-10*x+25)*ln(2)*exp(1/2*(-6+x)/(-3+x))+(-x^3+13*x^2-55*x+75)*ln(2) -x^3+10*x^2-25*x)/ln(2))*x
Time = 0.10 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.54 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=x \log \left (-\frac {x^{3} - {\left (x^{2} - 10 \, x + 25\right )} e^{\left (\frac {x - 6}{2 \, {\left (x - 3\right )}}\right )} \log \left (2\right ) - 10 \, x^{2} + {\left (x^{3} - 13 \, x^{2} + 55 \, x - 75\right )} \log \left (2\right ) + 25 \, x}{\log \left (2\right )}\right ) \] Input:
integrate((((2*x^3-22*x^2+78*x-90)*log(2)*exp((-6+x)/(2*x-6))+(-2*x^4+28*x ^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3-78*x^2+90*x)*log(((x^2-10*x+25)* log(2)*exp((-6+x)/(2*x-6))+(-x^3+13*x^2-55*x+75)*log(2)-x^3+10*x^2-25*x)/l og(2))+(4*x^3-21*x^2+21*x)*log(2)*exp((-6+x)/(2*x-6))+(-6*x^4+58*x^3-186*x ^2+198*x)*log(2)-6*x^4+46*x^3-114*x^2+90*x)/((2*x^3-22*x^2+78*x-90)*log(2) *exp((-6+x)/(2*x-6))+(-2*x^4+28*x^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3 -78*x^2+90*x),x, algorithm="fricas")
Output:
x*log(-(x^3 - (x^2 - 10*x + 25)*e^(1/2*(x - 6)/(x - 3))*log(2) - 10*x^2 + (x^3 - 13*x^2 + 55*x - 75)*log(2) + 25*x)/log(2))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
Time = 0.74 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.44 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=x \log {\left (\frac {- x^{3} + 10 x^{2} - 25 x + \left (x^{2} - 10 x + 25\right ) e^{\frac {x - 6}{2 x - 6}} \log {\left (2 \right )} + \left (- x^{3} + 13 x^{2} - 55 x + 75\right ) \log {\left (2 \right )}}{\log {\left (2 \right )}} \right )} \] Input:
integrate((((2*x**3-22*x**2+78*x-90)*ln(2)*exp((-6+x)/(2*x-6))+(-2*x**4+28 *x**3-144*x**2+324*x-270)*ln(2)-2*x**4+22*x**3-78*x**2+90*x)*ln(((x**2-10* x+25)*ln(2)*exp((-6+x)/(2*x-6))+(-x**3+13*x**2-55*x+75)*ln(2)-x**3+10*x**2 -25*x)/ln(2))+(4*x**3-21*x**2+21*x)*ln(2)*exp((-6+x)/(2*x-6))+(-6*x**4+58* x**3-186*x**2+198*x)*ln(2)-6*x**4+46*x**3-114*x**2+90*x)/((2*x**3-22*x**2+ 78*x-90)*ln(2)*exp((-6+x)/(2*x-6))+(-2*x**4+28*x**3-144*x**2+324*x-270)*ln (2)-2*x**4+22*x**3-78*x**2+90*x),x)
Output:
x*log((-x**3 + 10*x**2 - 25*x + (x**2 - 10*x + 25)*exp((x - 6)/(2*x - 6))* log(2) + (-x**3 + 13*x**2 - 55*x + 75)*log(2))/log(2))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (32) = 64\).
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.23 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=-\frac {2 \, x^{2} \log \left (\log \left (2\right )\right ) - 2 \, {\left (x^{2} - 3 \, x\right )} \log \left (-{\left (e^{\left (\frac {3}{2 \, {\left (x - 3\right )}}\right )} \log \left (2\right ) + e^{\left (\frac {3}{2 \, {\left (x - 3\right )}}\right )}\right )} x + e^{\frac {1}{2}} \log \left (2\right ) + 3 \, e^{\left (\frac {3}{2 \, {\left (x - 3\right )}}\right )} \log \left (2\right )\right ) - 4 \, {\left (x^{2} - 3 \, x\right )} \log \left (x - 5\right ) - 6 \, x \log \left (\log \left (2\right )\right ) + 9}{2 \, {\left (x - 3\right )}} \] Input:
integrate((((2*x^3-22*x^2+78*x-90)*log(2)*exp((-6+x)/(2*x-6))+(-2*x^4+28*x ^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3-78*x^2+90*x)*log(((x^2-10*x+25)* log(2)*exp((-6+x)/(2*x-6))+(-x^3+13*x^2-55*x+75)*log(2)-x^3+10*x^2-25*x)/l og(2))+(4*x^3-21*x^2+21*x)*log(2)*exp((-6+x)/(2*x-6))+(-6*x^4+58*x^3-186*x ^2+198*x)*log(2)-6*x^4+46*x^3-114*x^2+90*x)/((2*x^3-22*x^2+78*x-90)*log(2) *exp((-6+x)/(2*x-6))+(-2*x^4+28*x^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3 -78*x^2+90*x),x, algorithm="maxima")
Output:
-1/2*(2*x^2*log(log(2)) - 2*(x^2 - 3*x)*log(-(e^(3/2/(x - 3))*log(2) + e^( 3/2/(x - 3)))*x + e^(1/2)*log(2) + 3*e^(3/2/(x - 3))*log(2)) - 4*(x^2 - 3* x)*log(x - 5) - 6*x*log(log(2)) + 9)/(x - 3)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (32) = 64\).
Time = 3.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.44 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=x \log \left (-x^{3} \log \left (2\right ) + x^{2} e^{\left (-\frac {x}{2 \, {\left (x - 3\right )}} + 1\right )} \log \left (2\right ) - x^{3} + 13 \, x^{2} \log \left (2\right ) - 10 \, x e^{\left (-\frac {x}{2 \, {\left (x - 3\right )}} + 1\right )} \log \left (2\right ) + 10 \, x^{2} - 55 \, x \log \left (2\right ) + 25 \, e^{\left (-\frac {x}{2 \, {\left (x - 3\right )}} + 1\right )} \log \left (2\right ) - 25 \, x + 75 \, \log \left (2\right )\right ) - x \log \left (\log \left (2\right )\right ) \] Input:
integrate((((2*x^3-22*x^2+78*x-90)*log(2)*exp((-6+x)/(2*x-6))+(-2*x^4+28*x ^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3-78*x^2+90*x)*log(((x^2-10*x+25)* log(2)*exp((-6+x)/(2*x-6))+(-x^3+13*x^2-55*x+75)*log(2)-x^3+10*x^2-25*x)/l og(2))+(4*x^3-21*x^2+21*x)*log(2)*exp((-6+x)/(2*x-6))+(-6*x^4+58*x^3-186*x ^2+198*x)*log(2)-6*x^4+46*x^3-114*x^2+90*x)/((2*x^3-22*x^2+78*x-90)*log(2) *exp((-6+x)/(2*x-6))+(-2*x^4+28*x^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3 -78*x^2+90*x),x, algorithm="giac")
Output:
x*log(-x^3*log(2) + x^2*e^(-1/2*x/(x - 3) + 1)*log(2) - x^3 + 13*x^2*log(2 ) - 10*x*e^(-1/2*x/(x - 3) + 1)*log(2) + 10*x^2 - 55*x*log(2) + 25*e^(-1/2 *x/(x - 3) + 1)*log(2) - 25*x + 75*log(2)) - x*log(log(2))
Time = 3.50 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.82 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=x\,\left (\ln \left (10\,x^2-\ln \left (2\right )\,\left (x^3-13\,x^2+55\,x-75\right )-25\,x-x^3+{\mathrm {e}}^{-\frac {6}{2\,x-6}}\,{\mathrm {e}}^{\frac {x}{2\,x-6}}\,\ln \left (2\right )\,\left (x^2-10\,x+25\right )\right )-\ln \left (\ln \left (2\right )\right )\right ) \] Input:
int((90*x + log(-(25*x + log(2)*(55*x - 13*x^2 + x^3 - 75) - 10*x^2 + x^3 - exp((x - 6)/(2*x - 6))*log(2)*(x^2 - 10*x + 25))/log(2))*(90*x - log(2)* (144*x^2 - 324*x - 28*x^3 + 2*x^4 + 270) - 78*x^2 + 22*x^3 - 2*x^4 + exp(( x - 6)/(2*x - 6))*log(2)*(78*x - 22*x^2 + 2*x^3 - 90)) + log(2)*(198*x - 1 86*x^2 + 58*x^3 - 6*x^4) - 114*x^2 + 46*x^3 - 6*x^4 + exp((x - 6)/(2*x - 6 ))*log(2)*(21*x - 21*x^2 + 4*x^3))/(90*x - log(2)*(144*x^2 - 324*x - 28*x^ 3 + 2*x^4 + 270) - 78*x^2 + 22*x^3 - 2*x^4 + exp((x - 6)/(2*x - 6))*log(2) *(78*x - 22*x^2 + 2*x^3 - 90)),x)
Output:
x*(log(10*x^2 - log(2)*(55*x - 13*x^2 + x^3 - 75) - 25*x - x^3 + exp(-6/(2 *x - 6))*exp(x/(2*x - 6))*log(2)*(x^2 - 10*x + 25)) - log(log(2)))
Time = 0.18 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.28 \[ \int \frac {90 x-114 x^2+46 x^3-6 x^4+e^{\frac {-6+x}{-6+2 x}} \left (21 x-21 x^2+4 x^3\right ) \log (2)+\left (198 x-186 x^2+58 x^3-6 x^4\right ) \log (2)+\left (90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)\right ) \log \left (\frac {-25 x+10 x^2-x^3+e^{\frac {-6+x}{-6+2 x}} \left (25-10 x+x^2\right ) \log (2)+\left (75-55 x+13 x^2-x^3\right ) \log (2)}{\log (2)}\right )}{90 x-78 x^2+22 x^3-2 x^4+e^{\frac {-6+x}{-6+2 x}} \left (-90+78 x-22 x^2+2 x^3\right ) \log (2)+\left (-270+324 x-144 x^2+28 x^3-2 x^4\right ) \log (2)} \, dx=\mathrm {log}\left (\frac {e^{\frac {x}{2 x -6}} \mathrm {log}\left (2\right ) x^{2}-10 e^{\frac {x}{2 x -6}} \mathrm {log}\left (2\right ) x +25 e^{\frac {x}{2 x -6}} \mathrm {log}\left (2\right )-e^{\frac {3}{x -3}} \mathrm {log}\left (2\right ) x^{3}+13 e^{\frac {3}{x -3}} \mathrm {log}\left (2\right ) x^{2}-55 e^{\frac {3}{x -3}} \mathrm {log}\left (2\right ) x +75 e^{\frac {3}{x -3}} \mathrm {log}\left (2\right )-e^{\frac {3}{x -3}} x^{3}+10 e^{\frac {3}{x -3}} x^{2}-25 e^{\frac {3}{x -3}} x}{e^{\frac {3}{x -3}} \mathrm {log}\left (2\right )}\right ) x \] Input:
int((((2*x^3-22*x^2+78*x-90)*log(2)*exp((-6+x)/(2*x-6))+(-2*x^4+28*x^3-144 *x^2+324*x-270)*log(2)-2*x^4+22*x^3-78*x^2+90*x)*log(((x^2-10*x+25)*log(2) *exp((-6+x)/(2*x-6))+(-x^3+13*x^2-55*x+75)*log(2)-x^3+10*x^2-25*x)/log(2)) +(4*x^3-21*x^2+21*x)*log(2)*exp((-6+x)/(2*x-6))+(-6*x^4+58*x^3-186*x^2+198 *x)*log(2)-6*x^4+46*x^3-114*x^2+90*x)/((2*x^3-22*x^2+78*x-90)*log(2)*exp(( -6+x)/(2*x-6))+(-2*x^4+28*x^3-144*x^2+324*x-270)*log(2)-2*x^4+22*x^3-78*x^ 2+90*x),x)
Output:
log((e**(x/(2*x - 6))*log(2)*x**2 - 10*e**(x/(2*x - 6))*log(2)*x + 25*e**( x/(2*x - 6))*log(2) - e**(3/(x - 3))*log(2)*x**3 + 13*e**(3/(x - 3))*log(2 )*x**2 - 55*e**(3/(x - 3))*log(2)*x + 75*e**(3/(x - 3))*log(2) - e**(3/(x - 3))*x**3 + 10*e**(3/(x - 3))*x**2 - 25*e**(3/(x - 3))*x)/(e**(3/(x - 3)) *log(2)))*x