Integrand size = 123, antiderivative size = 29 \[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=\log \left (\left (\frac {13}{4}+e^{2 x+\left (-7+\frac {5}{x}-\log (2)\right )^2}-x\right )^2\right ) \] Output:
ln((exp(2*x+(5/x-7-ln(2))^2)-x+13/4)^2)
\[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=\int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx \] Input:
Integrate[(-8*x^3 + E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[ 2] + x^2*Log[2]^2)/x^2)*(-400 + 560*x + 16*x^3 + 80*x*Log[2]))/(13*x^3 + 4 *E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/ x^2)*x^3 - 4*x^4),x]
Output:
Integrate[(-8*x^3 + E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[ 2] + x^2*Log[2]^2)/x^2)*(-400 + 560*x + 16*x^3 + 80*x*Log[2]))/(13*x^3 + 4 *E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/ x^2)*x^3 - 4*x^4), 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.
| \(\displaystyle \int \frac {\left (16 x^3+560 x+80 x \log (2)-400\right ) \exp \left (\frac {2 x^3+49 x^2+x^2 \log ^2(2)+\left (14 x^2-10 x\right ) \log (2)-70 x+25}{x^2}\right )-8 x^3}{4 x^3 \exp \left (\frac {2 x^3+49 x^2+x^2 \log ^2(2)+\left (14 x^2-10 x\right ) \log (2)-70 x+25}{x^2}\right )-4 x^4+13 x^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 \left (x^3+5 x (7+\log (2))-25\right )}{x^3}+\frac {2^{\frac {10}{x}+2} e^{70/x} \left (4 x^4-15 x^3+20 x^2 (7+\log (2))-5 x (111+\log (8192))+325\right )}{x^3 \left (65536 e^{\frac {25}{x^2}+2 x+49+\log ^2(2)}-2^{\frac {10}{x}+2} e^{70/x} x+13\ 2^{10/x} e^{70/x}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {80 (7+\log (2)) \text {Subst}\left (\int \frac {\exp \left (\frac {10 \left (4^{1+\frac {5}{x}} e^{70/x}-x\right ) (7+\log (2))}{13\ 4^{5/x} e^{70/x}+65536 e^{2 x+\log ^2(2)+49+\frac {25}{x^2}}}\right )}{x}dx,x,\frac {-65536 e^{\frac {25}{x^2}+2 x+49+\log ^2(2)}+4^{\frac {5}{x}+1} e^{70/x} x-13\ 4^{5/x} e^{70/x}}{x}\right )}{65536 e^{\frac {25}{x^2}+2 x+49+\log ^2(2)}+13\ 4^{5/x} e^{70/x}}+60 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{4^{1+\frac {5}{x}} e^{70/x} x-13\ 4^{5/x} e^{70/x}-65536 e^{2 x+\log ^2(2)+49+\frac {25}{x^2}}}dx+20 (111+\log (8192)) \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^2 \left (4^{1+\frac {5}{x}} e^{70/x} x-13\ 4^{5/x} e^{70/x}-65536 e^{2 x+\log ^2(2)+49+\frac {25}{x^2}}\right )}dx-16 \int \frac {e^{\frac {70+10 \log (2)}{x}} x}{4^{1+\frac {5}{x}} e^{70/x} x-13\ 4^{5/x} e^{70/x}-65536 e^{2 x+\log ^2(2)+49+\frac {25}{x^2}}}dx-1300 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^3 \left (4^{1+\frac {5}{x}} e^{70/x} x-13\ 4^{5/x} e^{70/x}-65536 e^{2 x+\log ^2(2)+49+\frac {25}{x^2}}\right )}dx+\frac {50}{x^2}+4 x-\frac {20 (7+\log (2))}{x}\) |
Input:
Int[(-8*x^3 + E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x ^2*Log[2]^2)/x^2)*(-400 + 560*x + 16*x^3 + 80*x*Log[2]))/(13*x^3 + 4*E^((2 5 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/x^2)*x ^3 - 4*x^4),x]
Output:
$Aborted
Time = 0.40 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69
| method | result | size |
| parallelrisch | \(2 \ln \left (-{\mathrm e}^{\frac {x^{2} \ln \left (2\right )^{2}+\left (14 x^{2}-10 x \right ) \ln \left (2\right )+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}+x -\frac {13}{4}\right )\) | \(49\) |
| norman | \(2 \ln \left (4 x -4 \,{\mathrm e}^{\frac {x^{2} \ln \left (2\right )^{2}+\left (14 x^{2}-10 x \right ) \ln \left (2\right )+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}-13\right )\) | \(51\) |
| risch | \(4 x +\frac {\left (-20 \ln \left (2\right )-140\right ) x +50}{x^{2}}-\frac {2 \left (x^{2} \ln \left (2\right )^{2}+\left (14 x^{2}-10 x \right ) \ln \left (2\right )+2 x^{3}+49 x^{2}-70 x +25\right )}{x^{2}}+2 \ln \left (-x +16384 \left (\frac {1}{1024}\right )^{\frac {1}{x}} {\mathrm e}^{\frac {x^{2} \ln \left (2\right )^{2}+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}+\frac {13}{4}\right )\) | \(102\) |
Input:
int(((80*x*ln(2)+16*x^3+560*x-400)*exp((x^2*ln(2)^2+(14*x^2-10*x)*ln(2)+2* x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x^3*exp((x^2*ln(2)^2+(14*x^2-10*x)*ln(2 )+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x,method=_RETURNVERBOSE)
Output:
2*ln(-exp((x^2*ln(2)^2+(14*x^2-10*x)*ln(2)+2*x^3+49*x^2-70*x+25)/x^2)+x-13 /4)
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=2 \, \log \left (-4 \, x + 4 \, e^{\left (\frac {x^{2} \log \left (2\right )^{2} + 2 \, x^{3} + 49 \, x^{2} + 2 \, {\left (7 \, x^{2} - 5 \, x\right )} \log \left (2\right ) - 70 \, x + 25}{x^{2}}\right )} + 13\right ) \] Input:
integrate(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)* log(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2- 10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x, algorithm="fricas ")
Output:
2*log(-4*x + 4*e^((x^2*log(2)^2 + 2*x^3 + 49*x^2 + 2*(7*x^2 - 5*x)*log(2) - 70*x + 25)/x^2) + 13)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=2 \log {\left (- x + e^{\frac {2 x^{3} + x^{2} \log {\left (2 \right )}^{2} + 49 x^{2} - 70 x + \left (14 x^{2} - 10 x\right ) \log {\left (2 \right )} + 25}{x^{2}}} + \frac {13}{4} \right )} \] Input:
integrate(((80*x*ln(2)+16*x**3+560*x-400)*exp((x**2*ln(2)**2+(14*x**2-10*x )*ln(2)+2*x**3+49*x**2-70*x+25)/x**2)-8*x**3)/(4*x**3*exp((x**2*ln(2)**2+( 14*x**2-10*x)*ln(2)+2*x**3+49*x**2-70*x+25)/x**2)-4*x**4+13*x**3),x)
Output:
2*log(-x + exp((2*x**3 + x**2*log(2)**2 + 49*x**2 - 70*x + (14*x**2 - 10*x )*log(2) + 25)/x**2) + 13/4)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (30) = 60\).
Time = 0.19 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=-\frac {20 \, \log \left (2\right )}{x} + 2 \, \log \left (4 \, x - 13\right ) + 2 \, \log \left (\frac {{\left ({\left (4 \, x - 13\right )} e^{\left (\frac {10 \, \log \left (2\right )}{x} + \frac {70}{x}\right )} - 65536 \, e^{\left (\log \left (2\right )^{2} + 2 \, x + \frac {25}{x^{2}} + 49\right )}\right )} e^{\left (-\frac {70}{x}\right )}}{4 \, x - 13}\right ) \] Input:
integrate(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)* log(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2- 10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x, algorithm="maxima ")
Output:
-20*log(2)/x + 2*log(4*x - 13) + 2*log(((4*x - 13)*e^(10*log(2)/x + 70/x) - 65536*e^(log(2)^2 + 2*x + 25/x^2 + 49))*e^(-70/x)/(4*x - 13))
Time = 0.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=2 \, \log \left (4 \, x - 4 \, e^{\left (\frac {x^{2} \log \left (2\right )^{2} + 2 \, x^{3} + 14 \, x^{2} \log \left (2\right ) + 49 \, x^{2} - 10 \, x \log \left (2\right ) - 70 \, x + 25}{x^{2}}\right )} - 13\right ) \] Input:
integrate(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)* log(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2- 10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x, algorithm="giac")
Output:
2*log(4*x - 4*e^((x^2*log(2)^2 + 2*x^3 + 14*x^2*log(2) + 49*x^2 - 10*x*log (2) - 70*x + 25)/x^2) - 13)
Time = 2.46 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=2\,\ln \left (13\,2^{10/x}-4\,2^{10/x}\,x+65536\,{\mathrm {e}}^{{\ln \left (2\right )}^2}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{49}\,{\mathrm {e}}^{\frac {25}{x^2}}\,{\mathrm {e}}^{-\frac {70}{x}}\right )-\frac {20\,\ln \left (2\right )}{x} \] Input:
int((exp((x^2*log(2)^2 - 70*x - log(2)*(10*x - 14*x^2) + 49*x^2 + 2*x^3 + 25)/x^2)*(560*x + 80*x*log(2) + 16*x^3 - 400) - 8*x^3)/(4*x^3*exp((x^2*log (2)^2 - 70*x - log(2)*(10*x - 14*x^2) + 49*x^2 + 2*x^3 + 25)/x^2) + 13*x^3 - 4*x^4),x)
Output:
2*log(13*2^(10/x) - 4*2^(10/x)*x + 65536*exp(log(2)^2)*exp(2*x)*exp(49)*ex p(25/x^2)*exp(-70/x)) - (20*log(2))/x
\[ \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx=\int \frac {\left (80 \,\mathrm {log}\left (2\right ) x +16 x^{3}+560 x -400\right ) {\mathrm e}^{\frac {\mathrm {log}\left (2\right )^{2} x^{2}+\left (14 x^{2}-10 x \right ) \mathrm {log}\left (2\right )+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}-8 x^{3}}{4 x^{3} {\mathrm e}^{\frac {\mathrm {log}\left (2\right )^{2} x^{2}+\left (14 x^{2}-10 x \right ) \mathrm {log}\left (2\right )+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}-4 x^{4}+13 x^{3}}d x \] Input:
int(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2) +2*x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2-10*x)* log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x)
Output:
int(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2) +2*x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2-10*x)* log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x)