Integrand size = 87, antiderivative size = 29 \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=-x+e^{\frac {5 \left (3+\frac {1}{10 x}\right )}{x}} \left (x+\frac {x}{\log (x)}\right ) \] Output:
exp(5*(1/10/x+3)/x)*(x+x/ln(x))-x
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\left (-1+e^{\frac {1+30 x}{2 x^2}}\right ) x+\frac {e^{\frac {1+30 x}{2 x^2}} x}{\log (x)} \] Input:
Integrate[(-(E^((1 + 30*x)/(2*x^2))*x^2) + E^((1 + 30*x)/(2*x^2))*(-1 - 15 *x + x^2)*Log[x] + (-x^2 + E^((1 + 30*x)/(2*x^2))*(-1 - 15*x + x^2))*Log[x ]^2)/(x^2*Log[x]^2),x]
Output:
(-1 + E^((1 + 30*x)/(2*x^2)))*x + (E^((1 + 30*x)/(2*x^2))*x)/Log[x]
Time = 1.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {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.
\(\displaystyle \int \frac {-e^{\frac {30 x+1}{2 x^2}} x^2+\left (e^{\frac {30 x+1}{2 x^2}} \left (x^2-15 x-1\right )-x^2\right ) \log ^2(x)+e^{\frac {30 x+1}{2 x^2}} \left (x^2-15 x-1\right ) \log (x)}{x^2 \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\frac {1}{2 x^2}+\frac {15}{x}} \left (-x^2+x^2 \log ^2(x)+x^2 \log (x)-15 x \log ^2(x)-\log ^2(x)-15 x \log (x)-\log (x)\right )}{x^2 \log ^2(x)}-1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{\frac {1}{2 x^2}+\frac {15}{x}} \left (15 x \log ^2(x)+\log ^2(x)+15 x \log (x)+\log (x)\right )}{\left (\frac {1}{x^3}+\frac {15}{x^2}\right ) x^2 \log ^2(x)}-x\) |
Input:
Int[(-(E^((1 + 30*x)/(2*x^2))*x^2) + E^((1 + 30*x)/(2*x^2))*(-1 - 15*x + x ^2)*Log[x] + (-x^2 + E^((1 + 30*x)/(2*x^2))*(-1 - 15*x + x^2))*Log[x]^2)/( x^2*Log[x]^2),x]
Output:
-x + (E^(1/(2*x^2) + 15/x)*(Log[x] + 15*x*Log[x] + Log[x]^2 + 15*x*Log[x]^ 2))/((x^(-3) + 15/x^2)*x^2*Log[x]^2)
Time = 1.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21
method | result | size |
risch | \(x \,{\mathrm e}^{\frac {30 x +1}{2 x^{2}}}-x +\frac {x \,{\mathrm e}^{\frac {30 x +1}{2 x^{2}}}}{\ln \left (x \right )}\) | \(35\) |
parallelrisch | \(-\frac {-{\mathrm e}^{\frac {30 x +1}{2 x^{2}}} \ln \left (x \right ) x +x \ln \left (x \right )-x \,{\mathrm e}^{\frac {30 x +1}{2 x^{2}}}}{\ln \left (x \right )}\) | \(42\) |
default | \(-x +\frac {x^{2} {\mathrm e}^{\frac {30 x +1}{2 x^{2}}}+{\mathrm e}^{\frac {30 x +1}{2 x^{2}}} \ln \left (x \right ) x^{2}}{\ln \left (x \right ) x}\) | \(46\) |
parts | \(-x +\frac {x^{2} {\mathrm e}^{\frac {30 x +1}{2 x^{2}}}+{\mathrm e}^{\frac {30 x +1}{2 x^{2}}} \ln \left (x \right ) x^{2}}{\ln \left (x \right ) x}\) | \(46\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {30 x +1}{2 x^{2}}}+{\mathrm e}^{\frac {30 x +1}{2 x^{2}}} \ln \left (x \right ) x^{2}-x^{2} \ln \left (x \right )}{x \ln \left (x \right )}\) | \(49\) |
Input:
int((((x^2-15*x-1)*exp(1/2*(30*x+1)/x^2)-x^2)*ln(x)^2+(x^2-15*x-1)*exp(1/2 *(30*x+1)/x^2)*ln(x)-x^2*exp(1/2*(30*x+1)/x^2))/x^2/ln(x)^2,x,method=_RETU RNVERBOSE)
Output:
x*exp(1/2*(30*x+1)/x^2)-x+x*exp(1/2*(30*x+1)/x^2)/ln(x)
Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\frac {x e^{\left (\frac {30 \, x + 1}{2 \, x^{2}}\right )} + {\left (x e^{\left (\frac {30 \, x + 1}{2 \, x^{2}}\right )} - x\right )} \log \left (x\right )}{\log \left (x\right )} \] Input:
integrate((((x^2-15*x-1)*exp(1/2*(30*x+1)/x^2)-x^2)*log(x)^2+(x^2-15*x-1)* exp(1/2*(30*x+1)/x^2)*log(x)-x^2*exp(1/2*(30*x+1)/x^2))/x^2/log(x)^2,x, al gorithm="fricas")
Output:
(x*e^(1/2*(30*x + 1)/x^2) + (x*e^(1/2*(30*x + 1)/x^2) - x)*log(x))/log(x)
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=- x + \frac {\left (x \log {\left (x \right )} + x\right ) e^{\frac {15 x + \frac {1}{2}}{x^{2}}}}{\log {\left (x \right )}} \] Input:
integrate((((x**2-15*x-1)*exp(1/2*(30*x+1)/x**2)-x**2)*ln(x)**2+(x**2-15*x -1)*exp(1/2*(30*x+1)/x**2)*ln(x)-x**2*exp(1/2*(30*x+1)/x**2))/x**2/ln(x)** 2,x)
Output:
-x + (x*log(x) + x)*exp((15*x + 1/2)/x**2)/log(x)
Time = 0.10 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=-x + \frac {{\left (x \log \left (x\right ) + x\right )} e^{\left (\frac {15}{x} + \frac {1}{2 \, x^{2}}\right )}}{\log \left (x\right )} \] Input:
integrate((((x^2-15*x-1)*exp(1/2*(30*x+1)/x^2)-x^2)*log(x)^2+(x^2-15*x-1)* exp(1/2*(30*x+1)/x^2)*log(x)-x^2*exp(1/2*(30*x+1)/x^2))/x^2/log(x)^2,x, al gorithm="maxima")
Output:
-x + (x*log(x) + x)*e^(15/x + 1/2/x^2)/log(x)
\[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\int { -\frac {x^{2} e^{\left (\frac {30 \, x + 1}{2 \, x^{2}}\right )} - {\left (x^{2} - 15 \, x - 1\right )} e^{\left (\frac {30 \, x + 1}{2 \, x^{2}}\right )} \log \left (x\right ) + {\left (x^{2} - {\left (x^{2} - 15 \, x - 1\right )} e^{\left (\frac {30 \, x + 1}{2 \, x^{2}}\right )}\right )} \log \left (x\right )^{2}}{x^{2} \log \left (x\right )^{2}} \,d x } \] Input:
integrate((((x^2-15*x-1)*exp(1/2*(30*x+1)/x^2)-x^2)*log(x)^2+(x^2-15*x-1)* exp(1/2*(30*x+1)/x^2)*log(x)-x^2*exp(1/2*(30*x+1)/x^2))/x^2/log(x)^2,x, al gorithm="giac")
Output:
integrate(-(x^2*e^(1/2*(30*x + 1)/x^2) - (x^2 - 15*x - 1)*e^(1/2*(30*x + 1 )/x^2)*log(x) + (x^2 - (x^2 - 15*x - 1)*e^(1/2*(30*x + 1)/x^2))*log(x)^2)/ (x^2*log(x)^2), x)
Timed out. \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\int -\frac {x^2\,{\mathrm {e}}^{\frac {15\,x+\frac {1}{2}}{x^2}}+{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^{\frac {15\,x+\frac {1}{2}}{x^2}}\,\left (-x^2+15\,x+1\right )+x^2\right )+{\mathrm {e}}^{\frac {15\,x+\frac {1}{2}}{x^2}}\,\ln \left (x\right )\,\left (-x^2+15\,x+1\right )}{x^2\,{\ln \left (x\right )}^2} \,d x \] Input:
int(-(x^2*exp((15*x + 1/2)/x^2) + log(x)^2*(exp((15*x + 1/2)/x^2)*(15*x - x^2 + 1) + x^2) + exp((15*x + 1/2)/x^2)*log(x)*(15*x - x^2 + 1))/(x^2*log( x)^2),x)
Output:
int(-(x^2*exp((15*x + 1/2)/x^2) + log(x)^2*(exp((15*x + 1/2)/x^2)*(15*x - x^2 + 1) + x^2) + exp((15*x + 1/2)/x^2)*log(x)*(15*x - x^2 + 1))/(x^2*log( x)^2), x)
Time = 0.18 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-e^{\frac {1+30 x}{2 x^2}} x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right ) \log (x)+\left (-x^2+e^{\frac {1+30 x}{2 x^2}} \left (-1-15 x+x^2\right )\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx=\frac {x \left (e^{\frac {30 x +1}{2 x^{2}}} \mathrm {log}\left (x \right )+e^{\frac {30 x +1}{2 x^{2}}}-\mathrm {log}\left (x \right )\right )}{\mathrm {log}\left (x \right )} \] Input:
int((((x^2-15*x-1)*exp(1/2*(30*x+1)/x^2)-x^2)*log(x)^2+(x^2-15*x-1)*exp(1/ 2*(30*x+1)/x^2)*log(x)-x^2*exp(1/2*(30*x+1)/x^2))/x^2/log(x)^2,x)
Output:
(x*(e**((30*x + 1)/(2*x**2))*log(x) + e**((30*x + 1)/(2*x**2)) - log(x)))/ log(x)