Integrand size = 46, antiderivative size = 25 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=x+\left (1+e^x\right ) \left (-2+2 e^{-x} x \left (3-5 x^2\right )\right ) \] Output:
x+(1+exp(x))*(exp(ln(2*x)-x)*(-5*x^2+3)-2)
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=-2 e^x+7 x-10 x^3+2 e^{-x} \left (3 x-5 x^3\right ) \] Input:
Integrate[(x - 2*E^x*x + (2*x*(3 - 3*x - 15*x^2 + 5*x^3 + E^x*(3 - 15*x^2) ))/E^x)/x,x]
Output:
-2*E^x + 7*x - 10*x^3 + (2*(3*x - 5*x^3))/E^x
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2010, 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.
\(\displaystyle \int \frac {2 e^{-x} \left (5 x^3-15 x^2+e^x \left (3-15 x^2\right )-3 x+3\right ) x-2 e^x x+x}{x} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-30 x^2+2 e^{-x} \left (5 x^3-15 x^2-3 x+3\right )-2 e^x+7\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -10 e^{-x} x^3-10 x^3+6 e^{-x} x+7 x-2 e^x\) |
Input:
Int[(x - 2*E^x*x + (2*x*(3 - 3*x - 15*x^2 + 5*x^3 + E^x*(3 - 15*x^2)))/E^x )/x,x]
Output:
-2*E^x + 7*x + (6*x)/E^x - 10*x^3 - (10*x^3)/E^x
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Time = 1.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12
method | result | size |
risch | \(-10 x^{3}+7 x -2 \,{\mathrm e}^{x}+\left (-10 x^{3}+6 x \right ) {\mathrm e}^{-x}\) | \(28\) |
norman | \(\left (6 x -10 x^{3}-2 \,{\mathrm e}^{2 x}+7 \,{\mathrm e}^{x} x -10 \,{\mathrm e}^{x} x^{3}\right ) {\mathrm e}^{-x}\) | \(33\) |
default | \(7 x -10 x^{3}-2 \,{\mathrm e}^{x}-5 \,{\mathrm e}^{\ln \left (2 x \right )-x} x^{2}+3 \,{\mathrm e}^{\ln \left (2 x \right )-x}\) | \(39\) |
parts | \(7 x -10 x^{3}-2 \,{\mathrm e}^{x}-5 \,{\mathrm e}^{\ln \left (2 x \right )-x} x^{2}+3 \,{\mathrm e}^{\ln \left (2 x \right )-x}\) | \(39\) |
parallelrisch | \(-\frac {5 \,{\mathrm e}^{\ln \left (2 x \right )-x} x^{4} {\mathrm e}^{x}+5 \,{\mathrm e}^{\ln \left (2 x \right )-x} x^{4}-3 \,{\mathrm e}^{x} {\mathrm e}^{\ln \left (2 x \right )-x} x^{2}-x^{3}+2 \,{\mathrm e}^{x} x^{2}-3 \,{\mathrm e}^{\ln \left (2 x \right )-x} x^{2}}{x^{2}}\) | \(79\) |
orering | \(\frac {\left (200 x^{6}-1500 x^{5}+940 x^{4}+5340 x^{3}-2001 x^{2}-10104 x +495\right ) \left (\left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) {\mathrm e}^{\ln \left (2 x \right )-x}-2 \,{\mathrm e}^{x} x +x \right )}{2 \left (300 x^{5}-2250 x^{4}+4550 x^{3}-2625 x^{2}-798 x -435\right ) x}+\frac {\left (600 x^{5}-2250 x^{4}+1880 x^{3}+3249 x -1005\right ) \left (\frac {\left (-30 \,{\mathrm e}^{x} x +\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+15 x^{2}-30 x -3\right ) {\mathrm e}^{\ln \left (2 x \right )-x}+\left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) \left (-1+\frac {1}{x}\right ) {\mathrm e}^{\ln \left (2 x \right )-x}-2 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x}+1}{x}-\frac {\left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) {\mathrm e}^{\ln \left (2 x \right )-x}-2 \,{\mathrm e}^{x} x +x}{x^{2}}\right )}{300 x^{5}-2250 x^{4}+4550 x^{3}-2625 x^{2}-798 x -435}-\frac {\left (200 x^{6}-900 x^{5}+940 x^{4}+3249 x^{2}-2010 x -645\right ) \left (\frac {\left (-60 \,{\mathrm e}^{x} x -30 \,{\mathrm e}^{x}+\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+30 x -30\right ) {\mathrm e}^{\ln \left (2 x \right )-x}+2 \left (-30 \,{\mathrm e}^{x} x +\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+15 x^{2}-30 x -3\right ) \left (-1+\frac {1}{x}\right ) {\mathrm e}^{\ln \left (2 x \right )-x}-\frac {\left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) {\mathrm e}^{\ln \left (2 x \right )-x}}{x^{2}}+\left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) \left (-1+\frac {1}{x}\right )^{2} {\mathrm e}^{\ln \left (2 x \right )-x}-2 \,{\mathrm e}^{x} x -4 \,{\mathrm e}^{x}}{x}-\frac {2 \left (\left (-30 \,{\mathrm e}^{x} x +\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+15 x^{2}-30 x -3\right ) {\mathrm e}^{\ln \left (2 x \right )-x}+\left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) \left (-1+\frac {1}{x}\right ) {\mathrm e}^{\ln \left (2 x \right )-x}-2 \,{\mathrm e}^{x} x -2 \,{\mathrm e}^{x}+1\right )}{x^{2}}+\frac {2 \left (\left (-15 x^{2}+3\right ) {\mathrm e}^{x}+5 x^{3}-15 x^{2}-3 x +3\right ) {\mathrm e}^{\ln \left (2 x \right )-x}-4 \,{\mathrm e}^{x} x +2 x}{x^{3}}\right )}{2 \left (300 x^{5}-2250 x^{4}+4550 x^{3}-2625 x^{2}-798 x -435\right )}\) | \(655\) |
Input:
int((((-15*x^2+3)*exp(x)+5*x^3-15*x^2-3*x+3)*exp(ln(2*x)-x)-2*exp(x)*x+x)/ x,x,method=_RETURNVERBOSE)
Output:
-10*x^3+7*x-2*exp(x)+(-10*x^3+6*x)/exp(x)
Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=-{\left ({\left (10 \, x^{3} - 7 \, x\right )} e^{\left (-x + \log \left (2 \, x\right )\right )} + {\left (5 \, x^{2} - 3\right )} e^{\left (-2 \, x + 2 \, \log \left (2 \, x\right )\right )} + 4 \, x\right )} e^{\left (x - \log \left (2 \, x\right )\right )} \] Input:
integrate((((-15*x^2+3)*exp(x)+5*x^3-15*x^2-3*x+3)*exp(log(2*x)-x)-2*exp(x )*x+x)/x,x, algorithm="fricas")
Output:
-((10*x^3 - 7*x)*e^(-x + log(2*x)) + (5*x^2 - 3)*e^(-2*x + 2*log(2*x)) + 4 *x)*e^(x - log(2*x))
Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=- 10 x^{3} + 7 x + \left (- 10 x^{3} + 6 x\right ) e^{- x} - 2 e^{x} \] Input:
integrate((((-15*x**2+3)*exp(x)+5*x**3-15*x**2-3*x+3)*exp(ln(2*x)-x)-2*exp (x)*x+x)/x,x)
Output:
-10*x**3 + 7*x + (-10*x**3 + 6*x)*exp(-x) - 2*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (27) = 54\).
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=-10 \, x^{3} - 10 \, {\left (x^{3} + 3 \, x^{2} + 6 \, x + 6\right )} e^{\left (-x\right )} + 30 \, {\left (x^{2} + 2 \, x + 2\right )} e^{\left (-x\right )} + 6 \, {\left (x + 1\right )} e^{\left (-x\right )} + 7 \, x - 6 \, e^{\left (-x\right )} - 2 \, e^{x} \] Input:
integrate((((-15*x^2+3)*exp(x)+5*x^3-15*x^2-3*x+3)*exp(log(2*x)-x)-2*exp(x )*x+x)/x,x, algorithm="maxima")
Output:
-10*x^3 - 10*(x^3 + 3*x^2 + 6*x + 6)*e^(-x) + 30*(x^2 + 2*x + 2)*e^(-x) + 6*(x + 1)*e^(-x) + 7*x - 6*e^(-x) - 2*e^x
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=-10 \, x^{3} - 2 \, {\left (5 \, x^{3} - 3 \, x\right )} e^{\left (-x\right )} + 7 \, x - 2 \, e^{x} \] Input:
integrate((((-15*x^2+3)*exp(x)+5*x^3-15*x^2-3*x+3)*exp(log(2*x)-x)-2*exp(x )*x+x)/x,x, algorithm="giac")
Output:
-10*x^3 - 2*(5*x^3 - 3*x)*e^(-x) + 7*x - 2*e^x
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=7\,x-2\,{\mathrm {e}}^x+6\,x\,{\mathrm {e}}^{-x}-10\,x^3\,{\mathrm {e}}^{-x}-10\,x^3 \] Input:
int(-(2*x*exp(x) - x + exp(log(2*x) - x)*(3*x + exp(x)*(15*x^2 - 3) + 15*x ^2 - 5*x^3 - 3))/x,x)
Output:
7*x - 2*exp(x) + 6*x*exp(-x) - 10*x^3*exp(-x) - 10*x^3
Time = 0.16 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {x-2 e^x x+2 e^{-x} x \left (3-3 x-15 x^2+5 x^3+e^x \left (3-15 x^2\right )\right )}{x} \, dx=\frac {-2 e^{2 x}-10 e^{x} x^{3}+7 e^{x} x -10 x^{3}+6 x}{e^{x}} \] Input:
int((((-15*x^2+3)*exp(x)+5*x^3-15*x^2-3*x+3)*exp(log(2*x)-x)-2*exp(x)*x+x) /x,x)
Output:
( - 2*e**(2*x) - 10*e**x*x**3 + 7*e**x*x - 10*x**3 + 6*x)/e**x