Integrand size = 161, antiderivative size = 27 \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=\frac {1+\frac {\log \left (e^5+x\right )}{20 x}}{1-x+\log (4 x)} \]
Time = 0.68 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=\frac {20 x+\log \left (e^5+x\right )}{20 \left (x-x^2+x \log (4 x)\right )} \]
Integrate[(x - 21*x^2 + 20*x^3 + E^5*(-20*x + 20*x^2) + x*Log[4*x] + (-2*x + 2*x^2 + E^5*(-2 + 2*x) + (-E^5 - x)*Log[4*x])*Log[E^5 + x])/(20*x^3 - 4 0*x^4 + 20*x^5 + E^5*(20*x^2 - 40*x^3 + 20*x^4) + (40*x^3 - 40*x^4 + E^5*( 40*x^2 - 40*x^3))*Log[4*x] + (20*E^5*x^2 + 20*x^3)*Log[4*x]^2),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.
\(\displaystyle \int \frac {20 x^3-21 x^2+e^5 \left (20 x^2-20 x\right )+\left (2 x^2-2 x+e^5 (2 x-2)+\left (-x-e^5\right ) \log (4 x)\right ) \log \left (x+e^5\right )+x+x \log (4 x)}{20 x^5-40 x^4+20 x^3+\left (20 x^3+20 e^5 x^2\right ) \log ^2(4 x)+e^5 \left (20 x^4-40 x^3+20 x^2\right )+\left (-40 x^4+40 x^3+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\log (4 x) \left (x-\left (x+e^5\right ) \log \left (x+e^5\right )\right )+(x-1) \left (x \left (20 x+20 e^5-1\right )+2 \left (x+e^5\right ) \log \left (x+e^5\right )\right )}{20 x^2 \left (x+e^5\right ) (-x+\log (4 x)+1)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{20} \int \frac {(1-x) \left (\left (-20 x-20 e^5+1\right ) x-2 \left (x+e^5\right ) \log \left (x+e^5\right )\right )+\log (4 x) \left (x-\left (x+e^5\right ) \log \left (x+e^5\right )\right )}{x^2 \left (x+e^5\right ) (-x+\log (4 x)+1)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{20} \int \left (\frac {20 x^2-21 \left (1-\frac {20 e^5}{21}\right ) x+\log (4 x)-20 e^5+1}{x \left (x+e^5\right ) (-x+\log (4 x)+1)^2}+\frac {(2 x-\log (4 x)-2) \log \left (x+e^5\right )}{x^2 (x-\log (4 x)-1)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{20} \left (-2 \int \frac {\log \left (x+e^5\right )}{x^2 (x-\log (4 x)-1)^2}dx-\int \frac {\log (4 x) \log \left (x+e^5\right )}{x^2 (x-\log (4 x)-1)^2}dx-\frac {\int \frac {1}{x (x-\log (4 x)-1)}dx}{e^5}+\frac {\int \frac {1}{\left (x+e^5\right ) (x-\log (4 x)-1)}dx}{e^5}+2 \int \frac {\log \left (x+e^5\right )}{x (x-\log (4 x)-1)^2}dx+\frac {20}{-x+\log (4 x)+1}\right )\) |
Int[(x - 21*x^2 + 20*x^3 + E^5*(-20*x + 20*x^2) + x*Log[4*x] + (-2*x + 2*x ^2 + E^5*(-2 + 2*x) + (-E^5 - x)*Log[4*x])*Log[E^5 + x])/(20*x^3 - 40*x^4 + 20*x^5 + E^5*(20*x^2 - 40*x^3 + 20*x^4) + (40*x^3 - 40*x^4 + E^5*(40*x^2 - 40*x^3))*Log[4*x] + (20*E^5*x^2 + 20*x^3)*Log[4*x]^2),x]
3.16.87.3.1 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]]
Time = 11.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96
method | result | size |
parallelrisch | \(-\frac {\ln \left ({\mathrm e}^{5}+x \right )+20 x}{20 x \left (x -\ln \left (4 x \right )-1\right )}\) | \(26\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{5}+x \right )}{20 x \left (x -\ln \left (4 x \right )-1\right )}-\frac {1}{x -\ln \left (4 x \right )-1}\) | \(36\) |
int((((-exp(5)-x)*ln(4*x)+(-2+2*x)*exp(5)+2*x^2-2*x)*ln(exp(5)+x)+x*ln(4*x )+(20*x^2-20*x)*exp(5)+20*x^3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*ln(4*x)^2+ ((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*ln(4*x)+(20*x^4-40*x^3+20*x^2)*exp (5)+20*x^5-40*x^4+20*x^3),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=-\frac {20 \, x + \log \left (x + e^{5}\right )}{20 \, {\left (x^{2} - x \log \left (4 \, x\right ) - x\right )}} \]
integrate((((-exp(5)-x)*log(4*x)+(-2+2*x)*exp(5)+2*x^2-2*x)*log(exp(5)+x)+ x*log(4*x)+(20*x^2-20*x)*exp(5)+20*x^3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*l og(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*log(4*x)+(20*x^4-40*x^3+ 20*x^2)*exp(5)+20*x^5-40*x^4+20*x^3),x, algorithm=\
Exception generated. \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=\text {Exception raised: TypeError} \]
integrate((((-exp(5)-x)*ln(4*x)+(-2+2*x)*exp(5)+2*x**2-2*x)*ln(exp(5)+x)+x *ln(4*x)+(20*x**2-20*x)*exp(5)+20*x**3-21*x**2+x)/((20*x**2*exp(5)+20*x**3 )*ln(4*x)**2+((-40*x**3+40*x**2)*exp(5)-40*x**4+40*x**3)*ln(4*x)+(20*x**4- 40*x**3+20*x**2)*exp(5)+20*x**5-40*x**4+20*x**3),x)
Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=-\frac {20 \, x + \log \left (x + e^{5}\right )}{20 \, {\left (x^{2} - x {\left (2 \, \log \left (2\right ) + 1\right )} - x \log \left (x\right )\right )}} \]
integrate((((-exp(5)-x)*log(4*x)+(-2+2*x)*exp(5)+2*x^2-2*x)*log(exp(5)+x)+ x*log(4*x)+(20*x^2-20*x)*exp(5)+20*x^3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*l og(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*log(4*x)+(20*x^4-40*x^3+ 20*x^2)*exp(5)+20*x^5-40*x^4+20*x^3),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
Time = 0.33 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.85 \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=-\frac {20 \, x + \log \left (x + e^{5}\right )}{20 \, {\left ({\left (x + e^{5}\right )}^{2} - 2 \, {\left (x + e^{5}\right )} e^{5} - {\left (x + e^{5}\right )} \log \left (4 \, x\right ) + e^{5} \log \left (4 \, x\right ) - x + e^{10}\right )}} \]
integrate((((-exp(5)-x)*log(4*x)+(-2+2*x)*exp(5)+2*x^2-2*x)*log(exp(5)+x)+ x*log(4*x)+(20*x^2-20*x)*exp(5)+20*x^3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*l og(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*log(4*x)+(20*x^4-40*x^3+ 20*x^2)*exp(5)+20*x^5-40*x^4+20*x^3),x, algorithm=\
-1/20*(20*x + log(x + e^5))/((x + e^5)^2 - 2*(x + e^5)*e^5 - (x + e^5)*log (4*x) + e^5*log(4*x) - x + e^10)
Time = 8.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx=\frac {20\,x+\ln \left (x+{\mathrm {e}}^5\right )}{20\,x\,\left (\ln \left (4\,x\right )-x+1\right )} \]
int((x + x*log(4*x) - exp(5)*(20*x - 20*x^2) - log(x + exp(5))*(2*x - 2*x^ 2 + log(4*x)*(x + exp(5)) - exp(5)*(2*x - 2)) - 21*x^2 + 20*x^3)/(log(4*x) *(exp(5)*(40*x^2 - 40*x^3) + 40*x^3 - 40*x^4) + log(4*x)^2*(20*x^2*exp(5) + 20*x^3) + exp(5)*(20*x^2 - 40*x^3 + 20*x^4) + 20*x^3 - 40*x^4 + 20*x^5), x)