Integrand size = 117, antiderivative size = 18 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )^2 \]
Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\left (x+\log \left (4+x+\frac {\log (x)}{x}+\log (4 x)\right )\right )^2 \]
Integrate[(2*x + 2*x^2 + 10*x^3 + 2*x^4 + (-2*x + 2*x^2)*Log[x] + 2*x^3*Lo g[4*x] + (2 + 2*x + 10*x^2 + 2*x^3 + (-2 + 2*x)*Log[x] + 2*x^2*Log[4*x])*L og[(4*x + x^2 + Log[x] + x*Log[4*x])/x])/(4*x^2 + x^3 + x*Log[x] + x^2*Log [4*x]),x]
Time = 0.51 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {7239, 27, 7237}
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.
\(\displaystyle \int \frac {2 x^4+10 x^3+2 x^3 \log (4 x)+2 x^2+\left (2 x^2-2 x\right ) \log (x)+\left (2 x^3+10 x^2+2 x^2 \log (4 x)+2 x+(2 x-2) \log (x)+2\right ) \log \left (\frac {x^2+4 x+x \log (4 x)+\log (x)}{x}\right )+2 x}{x^3+4 x^2+x^2 \log (4 x)+x \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 \left (x^3+5 x^2+x^2 \log (4 x)+x+(x-1) \log (x)+1\right ) \left (x+\log \left (x+\log (4 x)+\frac {\log (x)}{x}+4\right )\right )}{x (\log (x)+x (x+\log (4 x)+4))}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {\left (x^3+\log (4 x) x^2+5 x^2+x-(1-x) \log (x)+1\right ) \left (x+\log \left (x+\log (4 x)+4+\frac {\log (x)}{x}\right )\right )}{x (\log (x)+x (x+\log (4 x)+4))}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \left (x+\log \left (x+\log (4 x)+\frac {\log (x)}{x}+4\right )\right )^2\) |
Int[(2*x + 2*x^2 + 10*x^3 + 2*x^4 + (-2*x + 2*x^2)*Log[x] + 2*x^3*Log[4*x] + (2 + 2*x + 10*x^2 + 2*x^3 + (-2 + 2*x)*Log[x] + 2*x^2*Log[4*x])*Log[(4* x + x^2 + Log[x] + x*Log[4*x])/x])/(4*x^2 + x^3 + x*Log[x] + x^2*Log[4*x]) ,x]
3.20.25.3.1 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_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
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. \(93\) vs. \(2(18)=36\).
Time = 10.86 (sec) , antiderivative size = 94, normalized size of antiderivative = 5.22
method | result | size |
parallelrisch | \(x^{2}+2 \ln \left (\frac {x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x}{x}\right ) x +\ln \left (\frac {x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x}{x}\right )^{2}+8 \ln \left (x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x \right )-8 \ln \left (x \right )-8 \ln \left (\frac {x \ln \left (4 x \right )+\ln \left (x \right )+x^{2}+4 x}{x}\right )\) | \(94\) |
int(((2*x^2*ln(4*x)+(-2+2*x)*ln(x)+2*x^3+10*x^2+2*x+2)*ln((x*ln(4*x)+ln(x) +x^2+4*x)/x)+2*x^3*ln(4*x)+(2*x^2-2*x)*ln(x)+2*x^4+10*x^3+2*x^2+2*x)/(x^2* ln(4*x)+x*ln(x)+x^3+4*x^2),x,method=_RETURNVERBOSE)
x^2+2*ln((x*ln(4*x)+ln(x)+x^2+4*x)/x)*x+ln((x*ln(4*x)+ln(x)+x^2+4*x)/x)^2+ 8*ln(x*ln(4*x)+ln(x)+x^2+4*x)-8*ln(x)-8*ln((x*ln(4*x)+ln(x)+x^2+4*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (18) = 36\).
Time = 0.26 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.06 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=x^{2} + 2 \, x \log \left (\frac {x^{2} + 2 \, x \log \left (2\right ) + {\left (x + 1\right )} \log \left (x\right ) + 4 \, x}{x}\right ) + \log \left (\frac {x^{2} + 2 \, x \log \left (2\right ) + {\left (x + 1\right )} \log \left (x\right ) + 4 \, x}{x}\right )^{2} \]
integrate(((2*x^2*log(4*x)+(-2+2*x)*log(x)+2*x^3+10*x^2+2*x+2)*log((x*log( 4*x)+log(x)+x^2+4*x)/x)+2*x^3*log(4*x)+(2*x^2-2*x)*log(x)+2*x^4+10*x^3+2*x ^2+2*x)/(x^2*log(4*x)+x*log(x)+x^3+4*x^2),x, algorithm=\
x^2 + 2*x*log((x^2 + 2*x*log(2) + (x + 1)*log(x) + 4*x)/x) + log((x^2 + 2* x*log(2) + (x + 1)*log(x) + 4*x)/x)^2
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (17) = 34\).
Time = 0.23 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.83 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=x^{2} + 2 x \log {\left (\frac {x^{2} + x \left (\log {\left (x \right )} + \log {\left (4 \right )}\right ) + 4 x + \log {\left (x \right )}}{x} \right )} + \log {\left (\frac {x^{2} + x \left (\log {\left (x \right )} + \log {\left (4 \right )}\right ) + 4 x + \log {\left (x \right )}}{x} \right )}^{2} \]
integrate(((2*x**2*ln(4*x)+(-2+2*x)*ln(x)+2*x**3+10*x**2+2*x+2)*ln((x*ln(4 *x)+ln(x)+x**2+4*x)/x)+2*x**3*ln(4*x)+(2*x**2-2*x)*ln(x)+2*x**4+10*x**3+2* x**2+2*x)/(x**2*ln(4*x)+x*ln(x)+x**3+4*x**2),x)
x**2 + 2*x*log((x**2 + x*(log(x) + log(4)) + 4*x + log(x))/x) + log((x**2 + x*(log(x) + log(4)) + 4*x + log(x))/x)**2
\[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\int { \frac {2 \, {\left (x^{4} + x^{3} \log \left (4 \, x\right ) + 5 \, x^{3} + x^{2} + {\left (x^{2} - x\right )} \log \left (x\right ) + {\left (x^{3} + x^{2} \log \left (4 \, x\right ) + 5 \, x^{2} + {\left (x - 1\right )} \log \left (x\right ) + x + 1\right )} \log \left (\frac {x^{2} + x \log \left (4 \, x\right ) + 4 \, x + \log \left (x\right )}{x}\right ) + x\right )}}{x^{3} + x^{2} \log \left (4 \, x\right ) + 4 \, x^{2} + x \log \left (x\right )} \,d x } \]
integrate(((2*x^2*log(4*x)+(-2+2*x)*log(x)+2*x^3+10*x^2+2*x+2)*log((x*log( 4*x)+log(x)+x^2+4*x)/x)+2*x^3*log(4*x)+(2*x^2-2*x)*log(x)+2*x^4+10*x^3+2*x ^2+2*x)/(x^2*log(4*x)+x*log(x)+x^3+4*x^2),x, algorithm=\
2*integrate((x^4 + x^3*log(4*x) + 5*x^3 + x^2 + (x^2 - x)*log(x) + (x^3 + x^2*log(4*x) + 5*x^2 + (x - 1)*log(x) + x + 1)*log((x^2 + x*log(4*x) + 4*x + log(x))/x) + x)/(x^3 + x^2*log(4*x) + 4*x^2 + x*log(x)), x)
\[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx=\int { \frac {2 \, {\left (x^{4} + x^{3} \log \left (4 \, x\right ) + 5 \, x^{3} + x^{2} + {\left (x^{2} - x\right )} \log \left (x\right ) + {\left (x^{3} + x^{2} \log \left (4 \, x\right ) + 5 \, x^{2} + {\left (x - 1\right )} \log \left (x\right ) + x + 1\right )} \log \left (\frac {x^{2} + x \log \left (4 \, x\right ) + 4 \, x + \log \left (x\right )}{x}\right ) + x\right )}}{x^{3} + x^{2} \log \left (4 \, x\right ) + 4 \, x^{2} + x \log \left (x\right )} \,d x } \]
integrate(((2*x^2*log(4*x)+(-2+2*x)*log(x)+2*x^3+10*x^2+2*x+2)*log((x*log( 4*x)+log(x)+x^2+4*x)/x)+2*x^3*log(4*x)+(2*x^2-2*x)*log(x)+2*x^4+10*x^3+2*x ^2+2*x)/(x^2*log(4*x)+x*log(x)+x^3+4*x^2),x, algorithm=\
integrate(2*(x^4 + x^3*log(4*x) + 5*x^3 + x^2 + (x^2 - x)*log(x) + (x^3 + x^2*log(4*x) + 5*x^2 + (x - 1)*log(x) + x + 1)*log((x^2 + x*log(4*x) + 4*x + log(x))/x) + x)/(x^3 + x^2*log(4*x) + 4*x^2 + x*log(x)), x)
Time = 11.88 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {2 x+2 x^2+10 x^3+2 x^4+\left (-2 x+2 x^2\right ) \log (x)+2 x^3 \log (4 x)+\left (2+2 x+10 x^2+2 x^3+(-2+2 x) \log (x)+2 x^2 \log (4 x)\right ) \log \left (\frac {4 x+x^2+\log (x)+x \log (4 x)}{x}\right )}{4 x^2+x^3+x \log (x)+x^2 \log (4 x)} \, dx={\left (x+\ln \left (\frac {4\,x+\ln \left (x\right )+x\,\ln \left (4\,x\right )+x^2}{x}\right )\right )}^2 \]