Integrand size = 66, antiderivative size = 33 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=e^{e^4}-x^2+\left (1+x^2\right )^2 \log ^2\left (\frac {e^{-x^2}}{x}\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=\log ^2\left (\frac {1}{x}\right )+x^2 \left (2+x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )+2 \log \left (\frac {1}{x}\right ) \log (x)-2 \log \left (\frac {e^{-x^2}}{x}\right ) \left (x^2+\log (x)\right )-x^2 \left (1+x^2+2 \log (x)\right ) \]
Integrate[(-2*x^2 + (-2 - 8*x^2 - 10*x^4 - 4*x^6)*Log[1/(E^x^2*x)] + (4*x^ 2 + 4*x^4)*Log[1/(E^x^2*x)]^2)/x,x]
Log[x^(-1)]^2 + x^2*(2 + x^2)*Log[1/(E^x^2*x)]^2 + 2*Log[x^(-1)]*Log[x] - 2*Log[1/(E^x^2*x)]*(x^2 + Log[x]) - x^2*(1 + x^2 + 2*Log[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.
\(\displaystyle \int \frac {-2 x^2+\left (4 x^4+4 x^2\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )+\left (-4 x^6-10 x^4-8 x^2-2\right ) \log \left (\frac {e^{-x^2}}{x}\right )}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (4 x \left (x^2+1\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )-\frac {2 \left (2 x^2+1\right ) \left (x^2+1\right )^2 \log \left (\frac {e^{-x^2}}{x}\right )}{x}-2 x\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int x \log ^2\left (\frac {e^{-x^2}}{x}\right )dx-2 \int \frac {\log \left (\frac {e^{-x^2}}{x}\right )}{x}dx+4 \int x^3 \log ^2\left (\frac {e^{-x^2}}{x}\right )dx-\frac {x^8}{6}-\frac {17 x^6}{18}-\frac {21 x^4}{8}-3 x^2-4 x^2 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {2}{3} x^6 \log \left (\frac {e^{-x^2}}{x}\right )-\frac {5}{2} x^4 \log \left (\frac {e^{-x^2}}{x}\right )\) |
Int[(-2*x^2 + (-2 - 8*x^2 - 10*x^4 - 4*x^6)*Log[1/(E^x^2*x)] + (4*x^2 + 4* x^4)*Log[1/(E^x^2*x)]^2)/x,x]
3.8.65.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 3.65 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67
method | result | size |
parallelrisch | \(\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )^{2} x^{4}+2 x^{2} \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )^{2}-x^{2}+\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )^{2}\) | \(55\) |
default | \(\frac {4 x^{6} \ln \left (x \right )}{3}+x^{4} \ln \left (x \right )^{2}+\frac {3 x^{4} \ln \left (x \right )}{2}+2 x^{2} \ln \left (x \right )^{2}-\ln \left (x \right )^{2}-3 x^{4}-x^{2}-\frac {x^{6}}{2}-4 x^{2} \ln \left (x \right )-\frac {65}{144}+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4} \ln \left (x \right )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{2} \ln \left (x \right )+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )-\frac {5 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{4}}{2}-4 x^{2} \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )+\frac {\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4}}{2}-8 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \left (\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{2}}{2}+\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}}{4}-\frac {x^{6}}{12}-\frac {x^{4}}{4}\right )+{\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )}^{2} \left (x^{2}+1\right )^{2}-2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) \ln \left (x \right )-\frac {2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{6}}{3}+2 \ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{2}+\ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{4}-\frac {2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{6}}{3}-2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}\) | \(401\) |
parts | \(\frac {4 x^{6} \ln \left (x \right )}{3}+x^{4} \ln \left (x \right )^{2}+\frac {3 x^{4} \ln \left (x \right )}{2}+2 x^{2} \ln \left (x \right )^{2}-\ln \left (x \right )^{2}-3 x^{4}-x^{2}-\frac {x^{6}}{2}-4 x^{2} \ln \left (x \right )-\frac {65}{144}+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )-2 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4} \ln \left (x \right )-4 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{2} \ln \left (x \right )+8 \left (\ln \left ({\mathrm e}^{x^{2}}\right )-x^{2}\right ) \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )-\frac {5 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{4}}{2}-4 x^{2} \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+2 x^{2} \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )+\frac {\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) x^{4}}{2}-8 \left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right ) \left (\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{2}}{2}+\frac {\ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}}{4}-\frac {x^{6}}{12}-\frac {x^{4}}{4}\right )+{\left (\ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right )+\ln \left ({\mathrm e}^{x^{2}}\right )+\ln \left (x \right )\right )}^{2} \left (x^{2}+1\right )^{2}-2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) \ln \left (x \right )-\frac {2 \ln \left (\frac {{\mathrm e}^{-x^{2}}}{x}\right ) x^{6}}{3}+2 \ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{2}+\ln \left ({\mathrm e}^{x^{2}}\right )^{2} x^{4}-\frac {2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{6}}{3}-2 \ln \left ({\mathrm e}^{x^{2}}\right ) x^{4}\) | \(401\) |
risch | \(\text {Expression too large to display}\) | \(7067\) |
int(((4*x^4+4*x^2)*ln(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*ln(1/exp(x^2 )/x)-2*x^2)/x,x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx={\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - x^{2} \]
integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1 /exp(x^2)/x)-2*x^2)/x,x, algorithm=\
Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=- x^{2} + \left (x^{4} + 2 x^{2} + 1\right ) \log {\left (\frac {e^{- x^{2}}}{x} \right )}^{2} \]
integrate(((4*x**4+4*x**2)*ln(1/exp(x**2)/x)**2+(-4*x**6-10*x**4-8*x**2-2) *ln(1/exp(x**2)/x)-2*x**2)/x,x)
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (30) = 60\).
Time = 0.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 5.18 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=-\frac {2}{3} \, x^{6} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - \frac {5}{2} \, x^{4} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{4} + 2 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right )^{2} - 4 \, x^{2} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - x^{2} - {\left (2 \, x^{2} + \log \left (x^{2}\right )\right )} \log \left (x\right ) + \log \left (x\right )^{2} + \frac {1}{6} \, {\left (4 \, x^{6} + 3 \, x^{4}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) + 2 \, {\left (x^{4} + x^{2}\right )} \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) - 2 \, \log \left (x\right ) \log \left (\frac {e^{\left (-x^{2}\right )}}{x}\right ) \]
integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1 /exp(x^2)/x)-2*x^2)/x,x, algorithm=\
-2/3*x^6*log(e^(-x^2)/x) + x^4*log(e^(-x^2)/x)^2 - 5/2*x^4*log(e^(-x^2)/x) - x^4 + 2*x^2*log(e^(-x^2)/x)^2 - 4*x^2*log(e^(-x^2)/x) - x^2 - (2*x^2 + log(x^2))*log(x) + log(x)^2 + 1/6*(4*x^6 + 3*x^4)*log(e^(-x^2)/x) + 2*(x^4 + x^2)*log(e^(-x^2)/x) - 2*log(x)*log(e^(-x^2)/x)
Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx=x^{8} + 2 \, x^{6} + x^{4} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x^{6} + 2 \, x^{4} + x^{2}\right )} \log \left (x\right ) \]
integrate(((4*x^4+4*x^2)*log(1/exp(x^2)/x)^2+(-4*x^6-10*x^4-8*x^2-2)*log(1 /exp(x^2)/x)-2*x^2)/x,x, algorithm=\
Time = 8.85 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.91 \[ \int \frac {-2 x^2+\left (-2-8 x^2-10 x^4-4 x^6\right ) \log \left (\frac {e^{-x^2}}{x}\right )+\left (4 x^2+4 x^4\right ) \log ^2\left (\frac {e^{-x^2}}{x}\right )}{x} \, dx={\ln \left (\frac {{\mathrm {e}}^{-x^2}}{x}\right )}^2\,\left (x^4+2\,x^2+1\right )-x^2 \]