Integrand size = 96, antiderivative size = 33 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {1}{3} \left (-\frac {-2+e^{1+x}}{x}+x\right ) \left (-1+\frac {x}{2}+\frac {x^2}{\log (x)}\right ) \] Output:
(x-(exp(1+x)-2)/x)*(1/3*x^2/ln(x)-1/3+1/6*x)
Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {1}{6} \left (e^x \left (-e+\frac {2 e}{x}\right )-\frac {4}{x}-2 x+x^2+\frac {2 x \left (2-e^{1+x}+x^2\right )}{\log (x)}\right ) \] Input:
Integrate[(-4*x^2 + 2*E^(1 + x)*x^2 - 2*x^4 + (4*x^2 + 6*x^4 + E^(1 + x)*( -2*x^2 - 2*x^3))*Log[x] + (4 - 2*x^2 + 2*x^3 + E^(1 + x)*(-2 + 2*x - x^2)) *Log[x]^2)/(6*x^2*Log[x]^2),x]
Output:
(E^x*(-E + (2*E)/x) - 4/x - 2*x + x^2 + (2*x*(2 - E^(1 + x) + x^2))/Log[x] )/6
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(33)=66\).
Time = 1.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {27, 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.
\(\displaystyle \int \frac {-2 x^4+2 e^{x+1} x^2-4 x^2+\left (2 x^3-2 x^2+e^{x+1} \left (-x^2+2 x-2\right )+4\right ) \log ^2(x)+\left (6 x^4+4 x^2+e^{x+1} \left (-2 x^3-2 x^2\right )\right ) \log (x)}{6 x^2 \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{6} \int -\frac {2 x^4-2 e^{x+1} x^2+4 x^2-\left (2 x^3-2 x^2-e^{x+1} \left (x^2-2 x+2\right )+4\right ) \log ^2(x)-2 \left (3 x^4+2 x^2-e^{x+1} \left (x^3+x^2\right )\right ) \log (x)}{x^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{6} \int \frac {2 x^4-2 e^{x+1} x^2+4 x^2-\left (2 x^3-2 x^2-e^{x+1} \left (x^2-2 x+2\right )+4\right ) \log ^2(x)-2 \left (3 x^4+2 x^2-e^{x+1} \left (x^3+x^2\right )\right ) \log (x)}{x^2 \log ^2(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{6} \int \left (\frac {e^{x+1} \left (2 \log (x) x^3+\log ^2(x) x^2+2 \log (x) x^2-2 x^2-2 \log ^2(x) x+2 \log ^2(x)\right )}{x^2 \log ^2(x)}-\frac {2 \left (3 \log (x) x^4-x^4+\log ^2(x) x^3-\log ^2(x) x^2+2 \log (x) x^2-2 x^2+2 \log ^2(x)\right )}{x^2 \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{6} \left (\frac {2 x^3}{\log (x)}+x^2-\frac {e^{x+1} \left (2 x^3 \log (x)+x^2 \log ^2(x)-2 x \log ^2(x)\right )}{x^2 \log ^2(x)}-2 x-\frac {4}{x}+\frac {4 x}{\log (x)}\right )\) |
Input:
Int[(-4*x^2 + 2*E^(1 + x)*x^2 - 2*x^4 + (4*x^2 + 6*x^4 + E^(1 + x)*(-2*x^2 - 2*x^3))*Log[x] + (4 - 2*x^2 + 2*x^3 + E^(1 + x)*(-2 + 2*x - x^2))*Log[x ]^2)/(6*x^2*Log[x]^2),x]
Output:
(-4/x - 2*x + x^2 + (4*x)/Log[x] + (2*x^3)/Log[x] - (E^(1 + x)*(2*x^3*Log[ x] - 2*x*Log[x]^2 + x^2*Log[x]^2))/(x^2*Log[x]^2))/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45
\[\frac {x^{3}-2 x^{2}-x \,{\mathrm e}^{1+x}+2 \,{\mathrm e}^{1+x}-4}{6 x}+\frac {x \left (x^{2}-{\mathrm e}^{1+x}+2\right )}{3 \ln \left (x \right )}\]
Input:
int(1/6*(((-x^2+2*x-2)*exp(1+x)+2*x^3-2*x^2+4)*ln(x)^2+((-2*x^3-2*x^2)*exp (1+x)+6*x^4+4*x^2)*ln(x)+2*x^2*exp(1+x)-2*x^4-4*x^2)/x^2/ln(x)^2,x)
Output:
1/6*(x^3-2*x^2-x*exp(1+x)+2*exp(1+x)-4)/x+1/3*x*(x^2-exp(1+x)+2)/ln(x)
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {2 \, x^{4} - 2 \, x^{2} e^{\left (x + 1\right )} + 4 \, x^{2} + {\left (x^{3} - 2 \, x^{2} - {\left (x - 2\right )} e^{\left (x + 1\right )} - 4\right )} \log \left (x\right )}{6 \, x \log \left (x\right )} \] Input:
integrate(1/6*(((-x^2+2*x-2)*exp(1+x)+2*x^3-2*x^2+4)*log(x)^2+((-2*x^3-2*x ^2)*exp(1+x)+6*x^4+4*x^2)*log(x)+2*x^2*exp(1+x)-2*x^4-4*x^2)/x^2/log(x)^2, x, algorithm="fricas")
Output:
1/6*(2*x^4 - 2*x^2*e^(x + 1) + 4*x^2 + (x^3 - 2*x^2 - (x - 2)*e^(x + 1) - 4)*log(x))/(x*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {x^{2}}{6} - \frac {x}{3} + \frac {x^{3} + 2 x}{3 \log {\left (x \right )}} + \frac {\left (- 2 x^{2} - x \log {\left (x \right )} + 2 \log {\left (x \right )}\right ) e^{x + 1}}{6 x \log {\left (x \right )}} - \frac {2}{3 x} \] Input:
integrate(1/6*(((-x**2+2*x-2)*exp(1+x)+2*x**3-2*x**2+4)*ln(x)**2+((-2*x**3 -2*x**2)*exp(1+x)+6*x**4+4*x**2)*ln(x)+2*x**2*exp(1+x)-2*x**4-4*x**2)/x**2 /ln(x)**2,x)
Output:
x**2/6 - x/3 + (x**3 + 2*x)/(3*log(x)) + (-2*x**2 - x*log(x) + 2*log(x))*e xp(x + 1)/(6*x*log(x)) - 2/(3*x)
\[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\int { -\frac {2 \, x^{4} - 2 \, x^{2} e^{\left (x + 1\right )} - {\left (2 \, x^{3} - 2 \, x^{2} - {\left (x^{2} - 2 \, x + 2\right )} e^{\left (x + 1\right )} + 4\right )} \log \left (x\right )^{2} + 4 \, x^{2} - 2 \, {\left (3 \, x^{4} + 2 \, x^{2} - {\left (x^{3} + x^{2}\right )} e^{\left (x + 1\right )}\right )} \log \left (x\right )}{6 \, x^{2} \log \left (x\right )^{2}} \,d x } \] Input:
integrate(1/6*(((-x^2+2*x-2)*exp(1+x)+2*x^3-2*x^2+4)*log(x)^2+((-2*x^3-2*x ^2)*exp(1+x)+6*x^4+4*x^2)*log(x)+2*x^2*exp(1+x)-2*x^4-4*x^2)/x^2/log(x)^2, x, algorithm="maxima")
Output:
1/6*x^2 + 1/3*Ei(x)*e - 1/3*e*gamma(-1, -x) - 1/3*x - 1/3*x*e^(x + 1)/log( x) - 2/3/x - 1/6*e^(x + 1) - 2/3*gamma(-1, -log(x)) - gamma(-1, -3*log(x)) + 1/6*integrate(2*(3*x^2 + 2)/log(x), x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {2 \, x^{4} + x^{3} \log \left (x\right ) - 2 \, x^{2} e^{\left (x + 1\right )} - 2 \, x^{2} \log \left (x\right ) - x e^{\left (x + 1\right )} \log \left (x\right ) + 4 \, x^{2} + 2 \, e^{\left (x + 1\right )} \log \left (x\right ) - 4 \, \log \left (x\right )}{6 \, x \log \left (x\right )} \] Input:
integrate(1/6*(((-x^2+2*x-2)*exp(1+x)+2*x^3-2*x^2+4)*log(x)^2+((-2*x^3-2*x ^2)*exp(1+x)+6*x^4+4*x^2)*log(x)+2*x^2*exp(1+x)-2*x^4-4*x^2)/x^2/log(x)^2, x, algorithm="giac")
Output:
1/6*(2*x^4 + x^3*log(x) - 2*x^2*e^(x + 1) - 2*x^2*log(x) - x*e^(x + 1)*log (x) + 4*x^2 + 2*e^(x + 1)*log(x) - 4*log(x))/(x*log(x))
Time = 3.81 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {2\,x}{3\,\ln \left (x\right )}-\frac {{\mathrm {e}}^{x+1}}{6}-\frac {x}{3}+\frac {{\mathrm {e}}^{x+1}}{3\,x}+\frac {x^3}{3\,\ln \left (x\right )}-\frac {2}{3\,x}+\frac {x^2}{6}-\frac {x\,{\mathrm {e}}^{x+1}}{3\,\ln \left (x\right )} \] Input:
int(-((log(x)^2*(exp(x + 1)*(x^2 - 2*x + 2) + 2*x^2 - 2*x^3 - 4))/6 - (x^2 *exp(x + 1))/3 - (log(x)*(4*x^2 - exp(x + 1)*(2*x^2 + 2*x^3) + 6*x^4))/6 + (2*x^2)/3 + x^4/3)/(x^2*log(x)^2),x)
Output:
(2*x)/(3*log(x)) - exp(x + 1)/6 - x/3 + exp(x + 1)/(3*x) + x^3/(3*log(x)) - 2/(3*x) + x^2/6 - (x*exp(x + 1))/(3*log(x))
Time = 0.16 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {-4 x^2+2 e^{1+x} x^2-2 x^4+\left (4 x^2+6 x^4+e^{1+x} \left (-2 x^2-2 x^3\right )\right ) \log (x)+\left (4-2 x^2+2 x^3+e^{1+x} \left (-2+2 x-x^2\right )\right ) \log ^2(x)}{6 x^2 \log ^2(x)} \, dx=\frac {-e^{x} \mathrm {log}\left (x \right ) e x +2 e^{x} \mathrm {log}\left (x \right ) e -2 e^{x} e \,x^{2}+\mathrm {log}\left (x \right ) x^{3}-2 \,\mathrm {log}\left (x \right ) x^{2}-4 \,\mathrm {log}\left (x \right )+2 x^{4}+4 x^{2}}{6 \,\mathrm {log}\left (x \right ) x} \] Input:
int(1/6*(((-x^2+2*x-2)*exp(1+x)+2*x^3-2*x^2+4)*log(x)^2+((-2*x^3-2*x^2)*ex p(1+x)+6*x^4+4*x^2)*log(x)+2*x^2*exp(1+x)-2*x^4-4*x^2)/x^2/log(x)^2,x)
Output:
( - e**x*log(x)*e*x + 2*e**x*log(x)*e - 2*e**x*e*x**2 + log(x)*x**3 - 2*lo g(x)*x**2 - 4*log(x) + 2*x**4 + 4*x**2)/(6*log(x)*x)