Integrand size = 111, antiderivative size = 27 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx=\left (x+\log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )^2 \]
Leaf count is larger than twice the leaf count of optimal. \(85\) vs. \(2(27)=54\).
Time = 0.35 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.15 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx=-\frac {\left (x \left (1+x+\log ^2(5)\right )+\log ^2(5) \log \left (\log \left (\frac {x^2}{4}\right )\right )\right ) \left (x \left (1+x-\log ^2(5)\right )+\log ^2(5) \log \left (\log \left (\frac {x^2}{4}\right )\right )-2 \log ^2(5) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{\log ^4(5)} \]
Integrate[(4*x*Log[5]^2 + (2*x^2 + 4*x^3 + 2*x^2*Log[5]^2)*Log[x^2/4] + (4 *Log[5]^2 + (2*x + 4*x^2 + 2*x*Log[5]^2)*Log[x^2/4])*Log[E^((-1 + x + x^2) /Log[5]^2)*Log[x^2/4]])/(x*Log[5]^2*Log[x^2/4]),x]
-(((x*(1 + x + Log[5]^2) + Log[5]^2*Log[Log[x^2/4]])*(x*(1 + x - Log[5]^2) + Log[5]^2*Log[Log[x^2/4]] - 2*Log[5]^2*Log[E^((-1 + x + x^2)/Log[5]^2)*L og[x^2/4]]))/Log[5]^4)
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.
\(\displaystyle \int \frac {\left (\left (4 x^2+2 x+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+4 \log ^2(5)\right ) \log \left (e^{\frac {x^2+x-1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\left (4 x^3+2 x^2+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+4 x \log ^2(5)}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {2 \left (2 \log ^2(5) x+\left (2 x^3+\log ^2(5) x^2+x^2\right ) \log \left (\frac {x^2}{4}\right )+\left (\left (2 x^2+\log ^2(5) x+x\right ) \log \left (\frac {x^2}{4}\right )+2 \log ^2(5)\right ) \log \left (e^{-\frac {-x^2-x+1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{x \log \left (\frac {x^2}{4}\right )}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {2 \int \frac {2 \log ^2(5) x+\left (2 x^3+\log ^2(5) x^2+x^2\right ) \log \left (\frac {x^2}{4}\right )+\left (\left (2 x^2+\log ^2(5) x+x\right ) \log \left (\frac {x^2}{4}\right )+2 \log ^2(5)\right ) \log \left (e^{-\frac {-x^2-x+1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \frac {2 \int \frac {\left (x \left (2 x+\log ^2(5)+1\right ) \log \left (\frac {x^2}{4}\right )+2 \log ^2(5)\right ) \left (x+\log \left (e^{\frac {x^2+x-1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{x \log \left (\frac {x^2}{4}\right )}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {2 \int \left (\frac {\log \left (e^{\frac {x^2+x-1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right ) \left (2 \log \left (\frac {x^2}{4}\right ) x^2+\left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right ) x+2 \log ^2(5)\right )}{x \log \left (\frac {x^2}{4}\right )}+\frac {2 \log \left (\frac {x^2}{4}\right ) x^2+\left (1+\log ^2(5)\right ) \log \left (\frac {x^2}{4}\right ) x+2 \log ^2(5)}{\log \left (\frac {x^2}{4}\right )}\right )dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \left (2 \log ^2(5) \int \frac {\log \left (e^{\frac {x^2+x-1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log \left (\frac {x^2}{4}\right )}dx-\frac {2 x \left (1+\log ^2(5)\right ) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}+\frac {2 x \log ^2(5) \operatorname {ExpIntegralEi}\left (\frac {1}{2} \log \left (\frac {x^2}{4}\right )\right )}{\sqrt {x^2}}-4 \operatorname {LogIntegral}\left (\frac {x^2}{4}\right )-\frac {x^4}{2 \log ^2(5)}+\frac {2 x^3}{3}-\frac {x^3}{3 \log ^2(5)}-\frac {2}{3} x^3 \left (1+\frac {1}{\log ^2(5)}\right )+x^2 \log \left (e^{-\frac {-x^2-x+1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )+\frac {1}{2} x^2 \left (1+\log ^2(5)\right )-\frac {1}{2} x^2 \left (1+\frac {1}{\log ^2(5)}\right )+x \left (1+\log ^2(5)\right ) \log \left (e^{-\frac {-x^2-x+1}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )\right )}{\log ^2(5)}\) |
Int[(4*x*Log[5]^2 + (2*x^2 + 4*x^3 + 2*x^2*Log[5]^2)*Log[x^2/4] + (4*Log[5 ]^2 + (2*x + 4*x^2 + 2*x*Log[5]^2)*Log[x^2/4])*Log[E^((-1 + x + x^2)/Log[5 ]^2)*Log[x^2/4]])/(x*Log[5]^2*Log[x^2/4]),x]
3.26.18.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(105\) vs. \(2(24)=48\).
Time = 4.93 (sec) , antiderivative size = 106, normalized size of antiderivative = 3.93
method | result | size |
parallelrisch | \(\frac {\ln \left (5\right )^{4} \ln \left ({\mathrm e}^{\frac {x^{2}+x -1}{\ln \left (5\right )^{2}}} \ln \left (\frac {x^{2}}{4}\right )\right )+2 \ln \left (5\right )^{2} \ln \left ({\mathrm e}^{\frac {x^{2}+x -1}{\ln \left (5\right )^{2}}} \ln \left (\frac {x^{2}}{4}\right )\right ) x +\ln \left (5\right )^{2} \ln \left ({\mathrm e}^{\frac {x^{2}+x -1}{\ln \left (5\right )^{2}}} \ln \left (\frac {x^{2}}{4}\right )\right )^{2}-x \ln \left (5\right )^{2}-\ln \left (5\right )^{4} \ln \left (\ln \left (\frac {x^{2}}{4}\right )\right )}{\ln \left (5\right )^{2}}\) | \(106\) |
risch | \(\text {Expression too large to display}\) | \(4661\) |
int((((2*x*ln(5)^2+4*x^2+2*x)*ln(1/4*x^2)+4*ln(5)^2)*ln(exp((x^2+x-1)/ln(5 )^2)*ln(1/4*x^2))+(2*x^2*ln(5)^2+4*x^3+2*x^2)*ln(1/4*x^2)+4*x*ln(5)^2)/x/l n(5)^2/ln(1/4*x^2),x,method=_RETURNVERBOSE)
1/ln(5)^2*(ln(5)^4*ln(exp((x^2+x-1)/ln(5)^2)*ln(1/4*x^2))+2*ln(5)^2*ln(exp ((x^2+x-1)/ln(5)^2)*ln(1/4*x^2))*x+ln(5)^2*ln(exp((x^2+x-1)/ln(5)^2)*ln(1/ 4*x^2))^2-x*ln(5)^2-ln(5)^4*ln(ln(1/4*x^2)))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx=x^{2} + 2 \, x \log \left (e^{\left (\frac {x^{2} + x - 1}{\log \left (5\right )^{2}}\right )} \log \left (\frac {1}{4} \, x^{2}\right )\right ) + \log \left (e^{\left (\frac {x^{2} + x - 1}{\log \left (5\right )^{2}}\right )} \log \left (\frac {1}{4} \, x^{2}\right )\right )^{2} \]
integrate((((2*x*log(5)^2+4*x^2+2*x)*log(1/4*x^2)+4*log(5)^2)*log(exp((x^2 +x-1)/log(5)^2)*log(1/4*x^2))+(2*x^2*log(5)^2+4*x^3+2*x^2)*log(1/4*x^2)+4* x*log(5)^2)/x/log(5)^2/log(1/4*x^2),x, algorithm=\
x^2 + 2*x*log(e^((x^2 + x - 1)/log(5)^2)*log(1/4*x^2)) + log(e^((x^2 + x - 1)/log(5)^2)*log(1/4*x^2))^2
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.89 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx=x^{2} + 2 x \log {\left (e^{\frac {x^{2} + x - 1}{\log {\left (5 \right )}^{2}}} \log {\left (\frac {x^{2}}{4} \right )} \right )} + \log {\left (e^{\frac {x^{2} + x - 1}{\log {\left (5 \right )}^{2}}} \log {\left (\frac {x^{2}}{4} \right )} \right )}^{2} \]
integrate((((2*x*ln(5)**2+4*x**2+2*x)*ln(1/4*x**2)+4*ln(5)**2)*ln(exp((x** 2+x-1)/ln(5)**2)*ln(1/4*x**2))+(2*x**2*ln(5)**2+4*x**3+2*x**2)*ln(1/4*x**2 )+4*x*ln(5)**2)/x/ln(5)**2/ln(1/4*x**2),x)
x**2 + 2*x*log(exp((x**2 + x - 1)/log(5)**2)*log(x**2/4)) + log(exp((x**2 + x - 1)/log(5)**2)*log(x**2/4))**2
Leaf count of result is larger than twice the leaf count of optimal. 148 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.48 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx=\frac {3 \, x^{2} \log \left (5\right )^{2} + 4 \, x^{3} + 3 \, x^{2} + \frac {3 \, \log \left (5\right )^{4} \log \left (-\log \left (2\right ) + \log \left (x\right )\right )^{2} + 2 \, {\left (\log \left (5\right )^{2} + 3\right )} x^{3} + 3 \, x^{4} + 3 \, {\left (2 \, \log \left (5\right )^{2} \log \left (2\right ) + \log \left (5\right )^{2} - 1\right )} x^{2} - 6 \, {\left (\log \left (5\right )^{2} - {\left (\log \left (5\right )^{4} + \log \left (5\right )^{2}\right )} \log \left (2\right ) + 1\right )} x + 6 \, {\left (\log \left (5\right )^{4} \log \left (2\right ) + x^{2} \log \left (5\right )^{2} + {\left (\log \left (5\right )^{4} + \log \left (5\right )^{2}\right )} x - \log \left (5\right )^{2}\right )} \log \left (-\log \left (2\right ) + \log \left (x\right )\right )}{\log \left (5\right )^{2}}}{3 \, \log \left (5\right )^{2}} \]
integrate((((2*x*log(5)^2+4*x^2+2*x)*log(1/4*x^2)+4*log(5)^2)*log(exp((x^2 +x-1)/log(5)^2)*log(1/4*x^2))+(2*x^2*log(5)^2+4*x^3+2*x^2)*log(1/4*x^2)+4* x*log(5)^2)/x/log(5)^2/log(1/4*x^2),x, algorithm=\
1/3*(3*x^2*log(5)^2 + 4*x^3 + 3*x^2 + (3*log(5)^4*log(-log(2) + log(x))^2 + 2*(log(5)^2 + 3)*x^3 + 3*x^4 + 3*(2*log(5)^2*log(2) + log(5)^2 - 1)*x^2 - 6*(log(5)^2 - (log(5)^4 + log(5)^2)*log(2) + 1)*x + 6*(log(5)^4*log(2) + x^2*log(5)^2 + (log(5)^4 + log(5)^2)*x - log(5)^2)*log(-log(2) + log(x))) /log(5)^2)/log(5)^2
Leaf count of result is larger than twice the leaf count of optimal. 132 vs. \(2 (24) = 48\).
Time = 0.56 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.89 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx=-\frac {\log \left (5\right )^{2} \log \left (-2 \, \log \left (2\right ) + \log \left (x^{2}\right )\right )^{2} - \frac {2 \, {\left (\log \left (5\right )^{2} + 1\right )} x^{3}}{\log \left (5\right )^{2}} - \frac {x^{4}}{\log \left (5\right )^{2}} - 2 \, {\left (\log \left (5\right )^{2} \log \left (-2 \, \log \left (2\right ) + \log \left (x^{2}\right )\right ) + {\left (\log \left (5\right )^{2} + 1\right )} x + x^{2}\right )} \log \left (\log \left (\frac {1}{4} \, x^{2}\right )\right ) - \frac {{\left (\log \left (5\right )^{4} + 2 \, \log \left (5\right )^{2} - 1\right )} x^{2}}{\log \left (5\right )^{2}} + \frac {2 \, {\left (\log \left (5\right )^{2} + 1\right )} x}{\log \left (5\right )^{2}} + 2 \, \log \left (2 \, \log \left (2\right ) - \log \left (x^{2}\right )\right )}{\log \left (5\right )^{2}} \]
integrate((((2*x*log(5)^2+4*x^2+2*x)*log(1/4*x^2)+4*log(5)^2)*log(exp((x^2 +x-1)/log(5)^2)*log(1/4*x^2))+(2*x^2*log(5)^2+4*x^3+2*x^2)*log(1/4*x^2)+4* x*log(5)^2)/x/log(5)^2/log(1/4*x^2),x, algorithm=\
-(log(5)^2*log(-2*log(2) + log(x^2))^2 - 2*(log(5)^2 + 1)*x^3/log(5)^2 - x ^4/log(5)^2 - 2*(log(5)^2*log(-2*log(2) + log(x^2)) + (log(5)^2 + 1)*x + x ^2)*log(log(1/4*x^2)) - (log(5)^4 + 2*log(5)^2 - 1)*x^2/log(5)^2 + 2*(log( 5)^2 + 1)*x/log(5)^2 + 2*log(2*log(2) - log(x^2)))/log(5)^2
Time = 12.80 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {4 x \log ^2(5)+\left (2 x^2+4 x^3+2 x^2 \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )+\left (4 \log ^2(5)+\left (2 x+4 x^2+2 x \log ^2(5)\right ) \log \left (\frac {x^2}{4}\right )\right ) \log \left (e^{\frac {-1+x+x^2}{\log ^2(5)}} \log \left (\frac {x^2}{4}\right )\right )}{x \log ^2(5) \log \left (\frac {x^2}{4}\right )} \, dx={\left (x+\ln \left ({\mathrm {e}}^{\frac {x}{{\ln \left (5\right )}^2}}\,{\mathrm {e}}^{\frac {x^2}{{\ln \left (5\right )}^2}}\,{\mathrm {e}}^{-\frac {1}{{\ln \left (5\right )}^2}}\,\ln \left (\frac {x^2}{4}\right )\right )\right )}^2 \]