Integrand size = 128, antiderivative size = 30 \[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\sqrt {\left (-1-e^x-x-x^2-\frac {x^2}{\log (x)}\right )^2} \]
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(30)=60\).
Time = 0.70 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\frac {\left (x^2+\left (e^x+x+x^2\right ) \log (x)\right ) \sqrt {\frac {\left (x^2+\left (1+e^x+x+x^2\right ) \log (x)\right )^2}{\log ^2(x)}}}{x^2+\left (1+e^x+x+x^2\right ) \log (x)} \]
Integrate[((-x + 2*x*Log[x] + (1 + E^x + 2*x)*Log[x]^2)*Sqrt[(x^4 + (2*x^2 + 2*E^x*x^2 + 2*x^3 + 2*x^4)*Log[x] + (1 + E^(2*x) + 2*x + 3*x^2 + 2*x^3 + x^4 + E^x*(2 + 2*x + 2*x^2))*Log[x]^2)/Log[x]^2])/(x^2*Log[x] + (1 + E^x + x + x^2)*Log[x]^2),x]
((x^2 + (E^x + x + x^2)*Log[x])*Sqrt[(x^2 + (1 + E^x + x + x^2)*Log[x])^2/ Log[x]^2])/(x^2 + (1 + E^x + x + x^2)*Log[x])
Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(30)=60\).
Time = 2.42 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.40, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {7239, 7270, 25, 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.
| \(\displaystyle \int \frac {\left (-x+\left (2 x+e^x+1\right ) \log ^2(x)+2 x \log (x)\right ) \sqrt {\frac {x^4+\left (x^4+2 x^3+3 x^2+e^x \left (2 x^2+2 x+2\right )+2 x+e^{2 x}+1\right ) \log ^2(x)+\left (2 x^4+2 x^3+2 e^x x^2+2 x^2\right ) \log (x)}{\log ^2(x)}}}{\left (x^2+x+e^x+1\right ) \log ^2(x)+x^2 \log (x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right ) \left (-x+\left (2 x+e^x+1\right ) \log ^2(x)+2 x \log (x)\right )}{\log ^3(x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}dx\) |
| \(\Big \downarrow \) 7270 |
| \(\displaystyle \frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right ) \int -\frac {-\left (\left (2 x+e^x+1\right ) \log ^2(x)\right )-2 x \log (x)+x}{\log ^2(x)}dx}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right ) \int \frac {-\left (\left (2 x+e^x+1\right ) \log ^2(x)\right )-2 x \log (x)+x}{\log ^2(x)}dx}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right ) \int \left (\frac {-2 x \log ^2(x)-\log ^2(x)-2 x \log (x)+x}{\log ^2(x)}-e^x\right )dx}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (-x^2-\frac {x^2}{\log (x)}-x-e^x\right ) \left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )}{\log (x) \sqrt {\frac {\left (x^2+\left (x^2+x+e^x+1\right ) \log (x)\right )^2}{\log ^2(x)}}}\) |
Int[((-x + 2*x*Log[x] + (1 + E^x + 2*x)*Log[x]^2)*Sqrt[(x^4 + (2*x^2 + 2*E ^x*x^2 + 2*x^3 + 2*x^4)*Log[x] + (1 + E^(2*x) + 2*x + 3*x^2 + 2*x^3 + x^4 + E^x*(2 + 2*x + 2*x^2))*Log[x]^2)/Log[x]^2])/(x^2*Log[x] + (1 + E^x + x + x^2)*Log[x]^2),x]
-(((-E^x - x - x^2 - x^2/Log[x])*(x^2 + (1 + E^x + x + x^2)*Log[x]))/(Log[ x]*Sqrt[(x^2 + (1 + E^x + x + x^2)*Log[x])^2/Log[x]^2]))
3.12.47.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Leaf count of result is larger than twice the leaf count of optimal. \(232\) vs. \(2(27)=54\).
Time = 0.15 (sec) , antiderivative size = 233, normalized size of antiderivative = 7.77
| method | result | size |
| risch | \(\frac {\sqrt {\frac {\left (x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )\right )^{2}}{\ln \left (x \right )^{2}}}\, \ln \left (x \right ) x^{2}}{x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )}+\frac {\sqrt {\frac {\left (x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )\right )^{2}}{\ln \left (x \right )^{2}}}\, \ln \left (x \right ) x}{x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )}+\frac {\sqrt {\frac {\left (x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )\right )^{2}}{\ln \left (x \right )^{2}}}\, \ln \left (x \right ) {\mathrm e}^{x}}{x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )}+\frac {\sqrt {\frac {\left (x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )\right )^{2}}{\ln \left (x \right )^{2}}}\, x^{2}}{x^{2} \ln \left (x \right )+{\mathrm e}^{x} \ln \left (x \right )+x \ln \left (x \right )+x^{2}+\ln \left (x \right )}\) | \(233\) |
int(((exp(x)+2*x+1)*ln(x)^2+2*x*ln(x)-x)*(((exp(x)^2+(2*x^2+2*x+2)*exp(x)+ x^4+2*x^3+3*x^2+2*x+1)*ln(x)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*ln(x)+x^4) /ln(x)^2)^(1/2)/((exp(x)+x^2+x+1)*ln(x)^2+x^2*ln(x)),x,method=_RETURNVERBO SE)
((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+ex p(x)*ln(x)+x*ln(x)+x^2+ln(x))*ln(x)*x^2+((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x ^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))*ln(x )*x+((x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x )+exp(x)*ln(x)+x*ln(x)+x^2+ln(x))*ln(x)*exp(x)+((x^2*ln(x)+exp(x)*ln(x)+x* ln(x)+x^2+ln(x))^2/ln(x)^2)^(1/2)/(x^2*ln(x)+exp(x)*ln(x)+x*ln(x)+x^2+ln(x ))*x^2
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.63 \[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\frac {x^{2} + {\left (x^{2} + x + e^{x}\right )} \log \left (x\right )}{\log \left (x\right )} \]
integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2) *exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*l og(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(x)),x, algor ithm=\
Timed out. \[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\text {Timed out} \]
integrate(((exp(x)+2*x+1)*ln(x)**2+2*x*ln(x)-x)*(((exp(x)**2+(2*x**2+2*x+2 )*exp(x)+x**4+2*x**3+3*x**2+2*x+1)*ln(x)**2+(2*exp(x)*x**2+2*x**4+2*x**3+2 *x**2)*ln(x)+x**4)/ln(x)**2)**(1/2)/((exp(x)+x**2+x+1)*ln(x)**2+x**2*ln(x) ),x)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\frac {x^{2} + {\left (x^{2} + x\right )} \log \left (x\right ) + e^{x} \log \left (x\right )}{\log \left (x\right )} \]
integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2) *exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*l og(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(x)),x, algor ithm=\
\[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\int { \frac {{\left ({\left (2 \, x + e^{x} + 1\right )} \log \left (x\right )^{2} + 2 \, x \log \left (x\right ) - x\right )} \sqrt {\frac {x^{4} + {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, {\left (x^{2} + x + 1\right )} e^{x} + 2 \, x + e^{\left (2 \, x\right )} + 1\right )} \log \left (x\right )^{2} + 2 \, {\left (x^{4} + x^{3} + x^{2} e^{x} + x^{2}\right )} \log \left (x\right )}{\log \left (x\right )^{2}}}}{x^{2} \log \left (x\right ) + {\left (x^{2} + x + e^{x} + 1\right )} \log \left (x\right )^{2}} \,d x } \]
integrate(((exp(x)+2*x+1)*log(x)^2+2*x*log(x)-x)*(((exp(x)^2+(2*x^2+2*x+2) *exp(x)+x^4+2*x^3+3*x^2+2*x+1)*log(x)^2+(2*exp(x)*x^2+2*x^4+2*x^3+2*x^2)*l og(x)+x^4)/log(x)^2)^(1/2)/((exp(x)+x^2+x+1)*log(x)^2+x^2*log(x)),x, algor ithm=\
integrate(((2*x + e^x + 1)*log(x)^2 + 2*x*log(x) - x)*sqrt((x^4 + (x^4 + 2 *x^3 + 3*x^2 + 2*(x^2 + x + 1)*e^x + 2*x + e^(2*x) + 1)*log(x)^2 + 2*(x^4 + x^3 + x^2*e^x + x^2)*log(x))/log(x)^2)/(x^2*log(x) + (x^2 + x + e^x + 1) *log(x)^2), x)
Timed out. \[ \int \frac {\left (-x+2 x \log (x)+\left (1+e^x+2 x\right ) \log ^2(x)\right ) \sqrt {\frac {x^4+\left (2 x^2+2 e^x x^2+2 x^3+2 x^4\right ) \log (x)+\left (1+e^{2 x}+2 x+3 x^2+2 x^3+x^4+e^x \left (2+2 x+2 x^2\right )\right ) \log ^2(x)}{\log ^2(x)}}}{x^2 \log (x)+\left (1+e^x+x+x^2\right ) \log ^2(x)} \, dx=\int \frac {\sqrt {\frac {{\ln \left (x\right )}^2\,\left (2\,x+{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (2\,x^2+2\,x+2\right )+3\,x^2+2\,x^3+x^4+1\right )+\ln \left (x\right )\,\left (2\,x^2\,{\mathrm {e}}^x+2\,x^2+2\,x^3+2\,x^4\right )+x^4}{{\ln \left (x\right )}^2}}\,\left (\left (2\,x+{\mathrm {e}}^x+1\right )\,{\ln \left (x\right )}^2+2\,x\,\ln \left (x\right )-x\right )}{x^2\,\ln \left (x\right )+{\ln \left (x\right )}^2\,\left (x+{\mathrm {e}}^x+x^2+1\right )} \,d x \]
int((((log(x)^2*(2*x + exp(2*x) + exp(x)*(2*x + 2*x^2 + 2) + 3*x^2 + 2*x^3 + x^4 + 1) + log(x)*(2*x^2*exp(x) + 2*x^2 + 2*x^3 + 2*x^4) + x^4)/log(x)^ 2)^(1/2)*(log(x)^2*(2*x + exp(x) + 1) - x + 2*x*log(x)))/(x^2*log(x) + log (x)^2*(x + exp(x) + x^2 + 1)),x)