Integrand size = 88, antiderivative size = 28 \[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=x-4 e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \log \left (x-\frac {\log ^2(2)}{4}\right ) \]
Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=x-4 e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \log \left (x-\frac {\log ^2(2)}{4}\right ) \]
Integrate[((-4*x^2 + x*Log[2]^2)*Log[x] + E^(5 + Log[3/Log[x]]^2)*(16*x*Lo g[x] + (-32*x + 8*Log[2]^2)*Log[(4*x - Log[2]^2)/4]*Log[3/Log[x]]))/((-4*x ^2 + x*Log[2]^2)*Log[x]),x]
Leaf count is larger than twice the leaf count of optimal. \(84\) vs. \(2(28)=56\).
Time = 1.90 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2026, 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.
\(\displaystyle \int \frac {\left (x \log ^2(2)-4 x^2\right ) \log (x)+e^{\log ^2\left (\frac {3}{\log (x)}\right )+5} \left (\left (8 \log ^2(2)-32 x\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )+16 x \log (x)\right )}{\left (x \log ^2(2)-4 x^2\right ) \log (x)} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (x \log ^2(2)-4 x^2\right ) \log (x)+e^{\log ^2\left (\frac {3}{\log (x)}\right )+5} \left (\left (8 \log ^2(2)-32 x\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )+16 x \log (x)\right )}{x \left (\log ^2(2)-4 x\right ) \log (x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (1-\frac {8 e^{\log ^2\left (\frac {3}{\log (x)}\right )+5} \left (-4 x \log \left (x-\frac {\log ^2(2)}{4}\right ) \log \left (\frac {3}{\log (x)}\right )+\log ^2(2) \log \left (x-\frac {\log ^2(2)}{4}\right ) \log \left (\frac {3}{\log (x)}\right )+2 x \log (x)\right )}{x \left (4 x-\log ^2(2)\right ) \log (x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-\frac {4 e^{\log ^2\left (\frac {3}{\log (x)}\right )+5} \left (4 x \log \left (x-\frac {\log ^2(2)}{4}\right ) \log \left (\frac {3}{\log (x)}\right )-\log ^2(2) \log \left (x-\frac {\log ^2(2)}{4}\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (4 x-\log ^2(2)\right ) \log \left (\frac {3}{\log (x)}\right )}\) |
Int[((-4*x^2 + x*Log[2]^2)*Log[x] + E^(5 + Log[3/Log[x]]^2)*(16*x*Log[x] + (-32*x + 8*Log[2]^2)*Log[(4*x - Log[2]^2)/4]*Log[3/Log[x]]))/((-4*x^2 + x *Log[2]^2)*Log[x]),x]
x - (4*E^(5 + Log[3/Log[x]]^2)*(4*x*Log[x - Log[2]^2/4]*Log[3/Log[x]] - Lo g[2]^2*Log[x - Log[2]^2/4]*Log[3/Log[x]]))/((4*x - Log[2]^2)*Log[3/Log[x]] )
3.5.28.3.1 Defintions of rubi rules used
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 34.67 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {\ln \left (2\right )^{2}}{2}-4 \ln \left (-\frac {\ln \left (2\right )^{2}}{4}+x \right ) {\mathrm e}^{\ln \left (\frac {3}{\ln \left (x \right )}\right )^{2}+5}+x\) | \(32\) |
risch | \(-4 \ln \left (-\frac {\ln \left (2\right )^{2}}{4}+x \right ) \ln \left (x \right )^{-2 \ln \left (3\right )} {\mathrm e}^{\ln \left (3\right )^{2}+5+\ln \left (\ln \left (x \right )\right )^{2}}+x\) | \(33\) |
int((((8*ln(2)^2-32*x)*ln(-1/4*ln(2)^2+x)*ln(3/ln(x))+16*x*ln(x))*exp(ln(3 /ln(x))^2+5)+(x*ln(2)^2-4*x^2)*ln(x))/(x*ln(2)^2-4*x^2)/ln(x),x,method=_RE TURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=-4 \, e^{\left (\log \left (\frac {3}{\log \left (x\right )}\right )^{2} + 5\right )} \log \left (-\frac {1}{4} \, \log \left (2\right )^{2} + x\right ) + x \]
integrate((((8*log(2)^2-32*x)*log(-1/4*log(2)^2+x)*log(3/log(x))+16*x*log( x))*exp(log(3/log(x))^2+5)+(x*log(2)^2-4*x^2)*log(x))/(x*log(2)^2-4*x^2)/l og(x),x, algorithm=\
Timed out. \[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=\text {Timed out} \]
integrate((((8*ln(2)**2-32*x)*ln(-1/4*ln(2)**2+x)*ln(3/ln(x))+16*x*ln(x))* exp(ln(3/ln(x))**2+5)+(x*ln(2)**2-4*x**2)*ln(x))/(x*ln(2)**2-4*x**2)/ln(x) ,x)
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
Time = 0.39 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=8 \, e^{\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} + 5\right )} \log \left (2\right ) - 4 \, e^{\left (\log \left (3\right )^{2} - 2 \, \log \left (3\right ) \log \left (\log \left (x\right )\right ) + \log \left (\log \left (x\right )\right )^{2} + 5\right )} \log \left (-\log \left (2\right )^{2} + 4 \, x\right ) + x \]
integrate((((8*log(2)^2-32*x)*log(-1/4*log(2)^2+x)*log(3/log(x))+16*x*log( x))*exp(log(3/log(x))^2+5)+(x*log(2)^2-4*x^2)*log(x))/(x*log(2)^2-4*x^2)/l og(x),x, algorithm=\
8*e^(log(3)^2 - 2*log(3)*log(log(x)) + log(log(x))^2 + 5)*log(2) - 4*e^(lo g(3)^2 - 2*log(3)*log(log(x)) + log(log(x))^2 + 5)*log(-log(2)^2 + 4*x) + x
\[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=\int { \frac {8 \, {\left ({\left (\log \left (2\right )^{2} - 4 \, x\right )} \log \left (-\frac {1}{4} \, \log \left (2\right )^{2} + x\right ) \log \left (\frac {3}{\log \left (x\right )}\right ) + 2 \, x \log \left (x\right )\right )} e^{\left (\log \left (\frac {3}{\log \left (x\right )}\right )^{2} + 5\right )} + {\left (x \log \left (2\right )^{2} - 4 \, x^{2}\right )} \log \left (x\right )}{{\left (x \log \left (2\right )^{2} - 4 \, x^{2}\right )} \log \left (x\right )} \,d x } \]
integrate((((8*log(2)^2-32*x)*log(-1/4*log(2)^2+x)*log(3/log(x))+16*x*log( x))*exp(log(3/log(x))^2+5)+(x*log(2)^2-4*x^2)*log(x))/(x*log(2)^2-4*x^2)/l og(x),x, algorithm=\
Time = 13.65 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-4 x^2+x \log ^2(2)\right ) \log (x)+e^{5+\log ^2\left (\frac {3}{\log (x)}\right )} \left (16 x \log (x)+\left (-32 x+8 \log ^2(2)\right ) \log \left (\frac {1}{4} \left (4 x-\log ^2(2)\right )\right ) \log \left (\frac {3}{\log (x)}\right )\right )}{\left (-4 x^2+x \log ^2(2)\right ) \log (x)} \, dx=x-4\,{\mathrm {e}}^{{\ln \left (3\right )}^2}\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\ln \left (\frac {1}{\ln \left (x\right )}\right )}^2}\,\ln \left (x-\frac {{\ln \left (2\right )}^2}{4}\right )\,{\left (\frac {1}{\ln \left (x\right )}\right )}^{2\,\ln \left (3\right )} \]