Integrand size = 87, antiderivative size = 28 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=-2+\log (2)+\frac {-x+\log (x)}{e^5+x}-\log \left ((-3+x) x^2\right ) \] Output:
ln(2)-ln(x^2*(-3+x))-2+(ln(x)-x)/(exp(5)+x)
Time = 0.08 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=\frac {e^5}{e^5+x}-\log (3-x)-2 \log (x)+\frac {\log (x)}{e^5+x} \] Input:
Integrate[(E^10*(6 - 3*x) - 3*x + 7*x^2 - 3*x^3 + E^5*(-3 + 16*x - 7*x^2) + (3*x - x^2)*Log[x])/(-3*x^3 + x^4 + E^10*(-3*x + x^2) + E^5*(-6*x^2 + 2* x^3)),x]
Output:
E^5/(E^5 + x) - Log[3 - x] - 2*Log[x] + Log[x]/(E^5 + x)
Leaf count is larger than twice the leaf count of optimal. \(304\) vs. \(2(28)=56\).
Time = 1.67 (sec) , antiderivative size = 304, normalized size of antiderivative = 10.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2026, 2463, 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.
| \(\displaystyle \int \frac {-3 x^3+7 x^2+e^5 \left (-7 x^2+16 x-3\right )+\left (3 x-x^2\right ) \log (x)-3 x+e^{10} (6-3 x)}{x^4-3 x^3+e^{10} \left (x^2-3 x\right )+e^5 \left (2 x^3-6 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {-3 x^3+7 x^2+e^5 \left (-7 x^2+16 x-3\right )+\left (3 x-x^2\right ) \log (x)-3 x+e^{10} (6-3 x)}{x \left (x^3-\left (3-2 e^5\right ) x^2-e^5 \left (6-e^5\right ) x-3 e^{10}\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {-3 x^3+7 x^2+e^5 \left (-7 x^2+16 x-3\right )+\left (3 x-x^2\right ) \log (x)-3 x+e^{10} (6-3 x)}{\left (3+e^5\right )^2 (x-3) x}-\frac {-3 x^3+7 x^2+e^5 \left (-7 x^2+16 x-3\right )+\left (3 x-x^2\right ) \log (x)-3 x+e^{10} (6-3 x)}{\left (3+e^5\right )^2 x \left (x+e^5\right )}-\frac {-3 x^3+7 x^2+e^5 \left (-7 x^2+16 x-3\right )+\left (3 x-x^2\right ) \log (x)-3 x+e^{10} (6-3 x)}{\left (3+e^5\right ) x \left (x+e^5\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (2+7 e^5\right ) x}{\left (3+e^5\right )^2}+\frac {3 x}{3+e^5}-\frac {\left (7-4 e^5\right ) x}{\left (3+e^5\right )^2}+\frac {3+16 e^5+7 e^{10}}{\left (3+e^5\right ) \left (x+e^5\right )}-\frac {3 e^5 \left (2+e^5\right )}{\left (3+e^5\right ) \left (x+e^5\right )}-\frac {3 e^{10}}{\left (3+e^5\right ) \left (x+e^5\right )}-\frac {7 e^5}{\left (3+e^5\right ) \left (x+e^5\right )}-\frac {3}{\left (3+e^5\right ) \left (x+e^5\right )}-\frac {x \log (x)}{e^5 \left (x+e^5\right )}-\log (3-x)+\frac {3 \log (x)}{e^5 \left (3+e^5\right )}-\frac {6 \log (x)}{3+e^5}+\frac {e^5 \left (1-2 e^5\right ) \log (x)}{\left (3+e^5\right )^2}+\frac {3 \left (1-2 e^5\right ) \log (x)}{\left (3+e^5\right )^2}-\frac {\left (3-7 e^{10}\right ) \log \left (x+e^5\right )}{e^5 \left (3+e^5\right )}-\frac {7 e^5 \log \left (x+e^5\right )}{3+e^5}-\frac {\log \left (x+e^5\right )}{3+e^5}+\frac {\log \left (x+e^5\right )}{e^5}\) |
Input:
Int[(E^10*(6 - 3*x) - 3*x + 7*x^2 - 3*x^3 + E^5*(-3 + 16*x - 7*x^2) + (3*x - x^2)*Log[x])/(-3*x^3 + x^4 + E^10*(-3*x + x^2) + E^5*(-6*x^2 + 2*x^3)), x]
Output:
-(((7 - 4*E^5)*x)/(3 + E^5)^2) + (3*x)/(3 + E^5) - ((2 + 7*E^5)*x)/(3 + E^ 5)^2 - 3/((3 + E^5)*(E^5 + x)) - (7*E^5)/((3 + E^5)*(E^5 + x)) - (3*E^10)/ ((3 + E^5)*(E^5 + x)) - (3*E^5*(2 + E^5))/((3 + E^5)*(E^5 + x)) + (3 + 16* E^5 + 7*E^10)/((3 + E^5)*(E^5 + x)) - Log[3 - x] + (3*(1 - 2*E^5)*Log[x])/ (3 + E^5)^2 + (E^5*(1 - 2*E^5)*Log[x])/(3 + E^5)^2 - (6*Log[x])/(3 + E^5) + (3*Log[x])/(E^5*(3 + E^5)) - (x*Log[x])/(E^5*(E^5 + x)) + Log[E^5 + x]/E ^5 - Log[E^5 + x]/(3 + E^5) - (7*E^5*Log[E^5 + x])/(3 + E^5) - ((3 - 7*E^1 0)*Log[E^5 + x])/(E^5*(3 + E^5))
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])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
| method | result | size |
| norman | \(\frac {\left (-2 \,{\mathrm e}^{5}+1\right ) \ln \left (x \right )-2 x \ln \left (x \right )+{\mathrm e}^{5}}{{\mathrm e}^{5}+x}-\ln \left (-3+x \right )\) | \(32\) |
| parallelrisch | \(\frac {-\ln \left (-3+x \right ) {\mathrm e}^{5}-2 \,{\mathrm e}^{5} \ln \left (x \right )-\ln \left (-3+x \right ) x -2 x \ln \left (x \right )+{\mathrm e}^{5}+\ln \left (x \right )}{{\mathrm e}^{5}+x}\) | \(39\) |
| risch | \(\frac {\ln \left (x \right )}{{\mathrm e}^{5}+x}-\frac {2 \ln \left (-x \right ) {\mathrm e}^{5}+\ln \left (-3+x \right ) {\mathrm e}^{5}+2 \ln \left (-x \right ) x +\ln \left (-3+x \right ) x -{\mathrm e}^{5}}{{\mathrm e}^{5}+x}\) | \(52\) |
| default | \(-\frac {\ln \left (x \right ) \left (\ln \left (\frac {{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}-\frac {\operatorname {dilog}\left (\frac {{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}+x}-\frac {\ln \left ({\mathrm e}^{5}+x \right )}{{\mathrm e}^{5}}-2 \ln \left (x \right )+\frac {\ln \left (x \right )}{{\mathrm e}^{5}}-\ln \left (-3+x \right )\) | \(213\) |
| parts | \(-\frac {\ln \left (x \right ) \left (\ln \left (\frac {{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\ln \left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}-\frac {\operatorname {dilog}\left (\frac {{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}-\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )-\operatorname {dilog}\left (\frac {{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}+x}{{\mathrm e}^{5}+\sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}\right )}{2 \sqrt {\left ({\mathrm e}^{5}\right )^{2}-{\mathrm e}^{10}}}+\frac {{\mathrm e}^{5}}{{\mathrm e}^{5}+x}-\frac {\ln \left ({\mathrm e}^{5}+x \right )}{{\mathrm e}^{5}}-2 \ln \left (x \right )+\frac {\ln \left (x \right )}{{\mathrm e}^{5}}-\ln \left (-3+x \right )\) | \(213\) |
Input:
int(((-x^2+3*x)*ln(x)+(-3*x+6)*exp(5)^2+(-7*x^2+16*x-3)*exp(5)-3*x^3+7*x^2 -3*x)/((x^2-3*x)*exp(5)^2+(2*x^3-6*x^2)*exp(5)+x^4-3*x^3),x,method=_RETURN VERBOSE)
Output:
((-2*exp(5)+1)*ln(x)-2*x*ln(x)+exp(5))/(exp(5)+x)-ln(-3+x)
Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=-\frac {{\left (x + e^{5}\right )} \log \left (x - 3\right ) + {\left (2 \, x + 2 \, e^{5} - 1\right )} \log \left (x\right ) - e^{5}}{x + e^{5}} \] Input:
integrate(((-x^2+3*x)*log(x)+(-3*x+6)*exp(5)^2+(-7*x^2+16*x-3)*exp(5)-3*x^ 3+7*x^2-3*x)/((x^2-3*x)*exp(5)^2+(2*x^3-6*x^2)*exp(5)+x^4-3*x^3),x, algori thm="fricas")
Output:
-((x + e^5)*log(x - 3) + (2*x + 2*e^5 - 1)*log(x) - e^5)/(x + e^5)
Time = 0.51 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=- 2 \log {\left (x \right )} - \log {\left (x - 3 \right )} + \frac {\log {\left (x \right )}}{x + e^{5}} + \frac {e^{5}}{x + e^{5}} \] Input:
integrate(((-x**2+3*x)*ln(x)+(-3*x+6)*exp(5)**2+(-7*x**2+16*x-3)*exp(5)-3* x**3+7*x**2-3*x)/((x**2-3*x)*exp(5)**2+(2*x**3-6*x**2)*exp(5)+x**4-3*x**3) ,x)
Output:
-2*log(x) - log(x - 3) + log(x)/(x + exp(5)) + exp(5)/(x + exp(5))
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (28) = 56\).
Time = 0.17 (sec) , antiderivative size = 443, normalized size of antiderivative = 15.82 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=-2 \, {\left (e^{\left (-10\right )} \log \left (x\right ) - \frac {3 \, {\left (2 \, e^{5} + 3\right )} \log \left (x + e^{5}\right )}{e^{20} + 6 \, e^{15} + 9 \, e^{10}} - \frac {\log \left (x - 3\right )}{e^{10} + 6 \, e^{5} + 9} + \frac {3}{x {\left (e^{10} + 3 \, e^{5}\right )} + e^{15} + 3 \, e^{10}}\right )} e^{10} + 3 \, {\left (\frac {\log \left (x + e^{5}\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {\log \left (x - 3\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {1}{x {\left (e^{5} + 3\right )} + e^{10} + 3 \, e^{5}}\right )} e^{10} + {\left (e^{\left (-10\right )} \log \left (x\right ) - \frac {3 \, {\left (2 \, e^{5} + 3\right )} \log \left (x + e^{5}\right )}{e^{20} + 6 \, e^{15} + 9 \, e^{10}} - \frac {\log \left (x - 3\right )}{e^{10} + 6 \, e^{5} + 9} + \frac {3}{x {\left (e^{10} + 3 \, e^{5}\right )} + e^{15} + 3 \, e^{10}}\right )} e^{5} + 7 \, {\left (\frac {e^{5}}{x {\left (e^{5} + 3\right )} + e^{10} + 3 \, e^{5}} + \frac {3 \, \log \left (x + e^{5}\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {3 \, \log \left (x - 3\right )}{e^{10} + 6 \, e^{5} + 9}\right )} e^{5} - 16 \, {\left (\frac {\log \left (x + e^{5}\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {\log \left (x - 3\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {1}{x {\left (e^{5} + 3\right )} + e^{10} + 3 \, e^{5}}\right )} e^{5} + e^{\left (-5\right )} \log \left (x + e^{5}\right ) - e^{\left (-5\right )} \log \left (x\right ) - \frac {3 \, {\left (e^{10} + 6 \, e^{5}\right )} \log \left (x + e^{5}\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {3 \, e^{10}}{x {\left (e^{5} + 3\right )} + e^{10} + 3 \, e^{5}} - \frac {7 \, e^{5}}{x {\left (e^{5} + 3\right )} + e^{10} + 3 \, e^{5}} - \frac {18 \, \log \left (x + e^{5}\right )}{e^{10} + 6 \, e^{5} + 9} - \frac {9 \, \log \left (x - 3\right )}{e^{10} + 6 \, e^{5} + 9} + \frac {\log \left (x\right )}{x + e^{5}} - \frac {3}{x {\left (e^{5} + 3\right )} + e^{10} + 3 \, e^{5}} \] Input:
integrate(((-x^2+3*x)*log(x)+(-3*x+6)*exp(5)^2+(-7*x^2+16*x-3)*exp(5)-3*x^ 3+7*x^2-3*x)/((x^2-3*x)*exp(5)^2+(2*x^3-6*x^2)*exp(5)+x^4-3*x^3),x, algori thm="maxima")
Output:
-2*(e^(-10)*log(x) - 3*(2*e^5 + 3)*log(x + e^5)/(e^20 + 6*e^15 + 9*e^10) - log(x - 3)/(e^10 + 6*e^5 + 9) + 3/(x*(e^10 + 3*e^5) + e^15 + 3*e^10))*e^1 0 + 3*(log(x + e^5)/(e^10 + 6*e^5 + 9) - log(x - 3)/(e^10 + 6*e^5 + 9) - 1 /(x*(e^5 + 3) + e^10 + 3*e^5))*e^10 + (e^(-10)*log(x) - 3*(2*e^5 + 3)*log( x + e^5)/(e^20 + 6*e^15 + 9*e^10) - log(x - 3)/(e^10 + 6*e^5 + 9) + 3/(x*( e^10 + 3*e^5) + e^15 + 3*e^10))*e^5 + 7*(e^5/(x*(e^5 + 3) + e^10 + 3*e^5) + 3*log(x + e^5)/(e^10 + 6*e^5 + 9) - 3*log(x - 3)/(e^10 + 6*e^5 + 9))*e^5 - 16*(log(x + e^5)/(e^10 + 6*e^5 + 9) - log(x - 3)/(e^10 + 6*e^5 + 9) - 1 /(x*(e^5 + 3) + e^10 + 3*e^5))*e^5 + e^(-5)*log(x + e^5) - e^(-5)*log(x) - 3*(e^10 + 6*e^5)*log(x + e^5)/(e^10 + 6*e^5 + 9) - 3*e^10/(x*(e^5 + 3) + e^10 + 3*e^5) - 7*e^5/(x*(e^5 + 3) + e^10 + 3*e^5) - 18*log(x + e^5)/(e^10 + 6*e^5 + 9) - 9*log(x - 3)/(e^10 + 6*e^5 + 9) + log(x)/(x + e^5) - 3/(x* (e^5 + 3) + e^10 + 3*e^5)
Time = 0.12 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=-\frac {x \log \left (x - 3\right ) + e^{5} \log \left (x - 3\right ) + 2 \, x \log \left (x\right ) + 2 \, e^{5} \log \left (x\right ) - e^{5} - \log \left (x\right )}{x + e^{5}} \] Input:
integrate(((-x^2+3*x)*log(x)+(-3*x+6)*exp(5)^2+(-7*x^2+16*x-3)*exp(5)-3*x^ 3+7*x^2-3*x)/((x^2-3*x)*exp(5)^2+(2*x^3-6*x^2)*exp(5)+x^4-3*x^3),x, algori thm="giac")
Output:
-(x*log(x - 3) + e^5*log(x - 3) + 2*x*log(x) + 2*e^5*log(x) - e^5 - log(x) )/(x + e^5)
Time = 0.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=\frac {{\mathrm {e}}^5}{x+{\mathrm {e}}^5}-2\,\ln \left (x\right )-\ln \left (x-3\right )+\frac {\ln \left (x\right )}{x+{\mathrm {e}}^5} \] Input:
int((3*x + exp(5)*(7*x^2 - 16*x + 3) - log(x)*(3*x - x^2) - 7*x^2 + 3*x^3 + exp(10)*(3*x - 6))/(exp(10)*(3*x - x^2) + exp(5)*(6*x^2 - 2*x^3) + 3*x^3 - x^4),x)
Output:
exp(5)/(x + exp(5)) - 2*log(x) - log(x - 3) + log(x)/(x + exp(5))
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.50 \[ \int \frac {e^{10} (6-3 x)-3 x+7 x^2-3 x^3+e^5 \left (-3+16 x-7 x^2\right )+\left (3 x-x^2\right ) \log (x)}{-3 x^3+x^4+e^{10} \left (-3 x+x^2\right )+e^5 \left (-6 x^2+2 x^3\right )} \, dx=\frac {-\mathrm {log}\left (x -3\right ) e^{5}-\mathrm {log}\left (x -3\right ) x -2 \,\mathrm {log}\left (x \right ) e^{5}-2 \,\mathrm {log}\left (x \right ) x +\mathrm {log}\left (x \right )-x}{e^{5}+x} \] Input:
int(((-x^2+3*x)*log(x)+(-3*x+6)*exp(5)^2+(-7*x^2+16*x-3)*exp(5)-3*x^3+7*x^ 2-3*x)/((x^2-3*x)*exp(5)^2+(2*x^3-6*x^2)*exp(5)+x^4-3*x^3),x)
Output:
( - log(x - 3)*e**5 - log(x - 3)*x - 2*log(x)*e**5 - 2*log(x)*x + log(x) - x)/(e**5 + x)