Integrand size = 128, antiderivative size = 33 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {1}{4} \log \left (-5 (1-x)+x-\left (-e^3+x-x \left (e^x+x\right )\right )^2\right ) \] Output:
1/4*ln(6*x-5-(x-(exp(x)+x)*x-exp(3))^2)
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.85 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {1}{4} \log \left (5+e^6-6 x+2 e^{3+x} x+2 e^3 (-1+x) x+x^2+e^{2 x} x^2+2 e^x (-1+x) x^2-2 x^3+x^4\right ) \] Input:
Integrate[(-3 + x - 3*x^2 + 2*x^3 + E^3*(-1 + 2*x) + E^(2*x)*(x + x^2) + E ^x*(-2*x + 2*x^2 + x^3 + E^3*(1 + x)))/(10 + 2*E^6 - 12*x + 2*x^2 + 2*E^(2 *x)*x^2 - 4*x^3 + 2*x^4 + E^3*(-4*x + 4*x^2) + E^x*(4*E^3*x - 4*x^2 + 4*x^ 3)),x]
Output:
Log[5 + E^6 - 6*x + 2*E^(3 + x)*x + 2*E^3*(-1 + x)*x + x^2 + E^(2*x)*x^2 + 2*E^x*(-1 + x)*x^2 - 2*x^3 + x^4]/4
Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).
Time = 1.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {7292, 27, 25, 7235}
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.
\(\displaystyle \int \frac {2 x^3-3 x^2+e^{2 x} \left (x^2+x\right )+e^x \left (x^3+2 x^2-2 x+e^3 (x+1)\right )+x+e^3 (2 x-1)-3}{2 x^4-4 x^3+2 e^{2 x} x^2+2 x^2+e^3 \left (4 x^2-4 x\right )+e^x \left (4 x^3-4 x^2+4 e^3 x\right )-12 x+2 e^6+10} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^3-3 x^2+e^{2 x} \left (x^2+x\right )+e^x \left (x^3+2 x^2-2 x+e^3 (x+1)\right )+x+e^3 (2 x-1)-3}{2 \left (x^4+2 e^x x^3-2 x^3-2 e^x x^2+e^{2 x} x^2+\left (1+2 e^3\right ) x^2+2 e^{x+3} x-6 \left (1+\frac {e^3}{3}\right ) x+5 \left (1+\frac {e^6}{5}\right )\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {-2 x^3+3 x^2-x+e^3 (1-2 x)-e^{2 x} \left (x^2+x\right )+e^x \left (-x^3-2 x^2+2 x-e^3 (x+1)\right )+3}{x^4+2 e^x x^3-2 x^3-2 e^x x^2+e^{2 x} x^2+\left (1+2 e^3\right ) x^2+2 e^{x+3} x-2 \left (3+e^3\right ) x+e^6+5}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {-2 x^3+3 x^2-x+e^3 (1-2 x)-e^{2 x} \left (x^2+x\right )+e^x \left (-x^3-2 x^2+2 x-e^3 (x+1)\right )+3}{x^4+2 e^x x^3-2 x^3-2 e^x x^2+e^{2 x} x^2+\left (1+2 e^3\right ) x^2+2 e^{x+3} x-2 \left (3+e^3\right ) x+e^6+5}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle \frac {1}{4} \log \left (x^4+2 e^x x^3-2 x^3-2 e^x x^2+e^{2 x} x^2+\left (1+2 e^3\right ) x^2+2 e^{x+3} x-2 \left (3+e^3\right ) x+e^6+5\right )\) |
Input:
Int[(-3 + x - 3*x^2 + 2*x^3 + E^3*(-1 + 2*x) + E^(2*x)*(x + x^2) + E^x*(-2 *x + 2*x^2 + x^3 + E^3*(1 + x)))/(10 + 2*E^6 - 12*x + 2*x^2 + 2*E^(2*x)*x^ 2 - 4*x^3 + 2*x^4 + E^3*(-4*x + 4*x^2) + E^x*(4*E^3*x - 4*x^2 + 4*x^3)),x]
Output:
Log[5 + E^6 + 2*E^(3 + x)*x - 2*(3 + E^3)*x - 2*E^x*x^2 + E^(2*x)*x^2 + (1 + 2*E^3)*x^2 - 2*x^3 + 2*E^x*x^3 + x^4]/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(25)=50\).
Time = 0.64 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.94
method | result | size |
risch | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 \left (x^{2}+{\mathrm e}^{3}-x \right ) {\mathrm e}^{x}}{x}+\frac {x^{4}+2 x^{2} {\mathrm e}^{3}-2 x^{3}+{\mathrm e}^{6}-2 x \,{\mathrm e}^{3}+x^{2}-6 x +5}{x^{2}}\right )}{4}\) | \(64\) |
parallelrisch | \(\frac {\ln \left (2 \,{\mathrm e}^{x} x^{3}+x^{4}+2 x \,{\mathrm e}^{3} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{2}+{\mathrm e}^{2 x} x^{2}+2 x^{2} {\mathrm e}^{3}-2 x^{3}-2 x \,{\mathrm e}^{3}+x^{2}+{\mathrm e}^{6}-6 x +5\right )}{4}\) | \(65\) |
norman | \(\frac {\ln \left (4 \,{\mathrm e}^{x} x^{3}+2 x^{4}+4 x \,{\mathrm e}^{3} {\mathrm e}^{x}-4 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{2 x} x^{2}+4 x^{2} {\mathrm e}^{3}-4 x^{3}-4 x \,{\mathrm e}^{3}+2 x^{2}+2 \,{\mathrm e}^{6}-12 x +10\right )}{4}\) | \(72\) |
Input:
int(((x^2+x)*exp(x)^2+((1+x)*exp(3)+x^3+2*x^2-2*x)*exp(x)+(-1+2*x)*exp(3)+ 2*x^3-3*x^2+x-3)/(2*exp(x)^2*x^2+(4*x*exp(3)+4*x^3-4*x^2)*exp(x)+2*exp(3)^ 2+(4*x^2-4*x)*exp(3)+2*x^4-4*x^3+2*x^2-12*x+10),x,method=_RETURNVERBOSE)
Output:
1/2*ln(x)+1/4*ln(exp(2*x)+2/x*(x^2+exp(3)-x)*exp(x)+(x^4+2*x^2*exp(3)-2*x^ 3+exp(6)-2*x*exp(3)+x^2-6*x+5)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {1}{2} \, \log \left (x\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - 2 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + x^{2} + 2 \, {\left (x^{2} - x\right )} e^{3} + 2 \, {\left (x^{3} - x^{2} + x e^{3}\right )} e^{x} - 6 \, x + e^{6} + 5}{x^{2}}\right ) \] Input:
integrate(((x^2+x)*exp(x)^2+((1+x)*exp(3)+x^3+2*x^2-2*x)*exp(x)+(-1+2*x)*e xp(3)+2*x^3-3*x^2+x-3)/(2*exp(x)^2*x^2+(4*x*exp(3)+4*x^3-4*x^2)*exp(x)+2*e xp(3)^2+(4*x^2-4*x)*exp(3)+2*x^4-4*x^3+2*x^2-12*x+10),x, algorithm="fricas ")
Output:
1/2*log(x) + 1/4*log((x^4 - 2*x^3 + x^2*e^(2*x) + x^2 + 2*(x^2 - x)*e^3 + 2*(x^3 - x^2 + x*e^3)*e^x - 6*x + e^6 + 5)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {\log {\left (x \right )}}{2} + \frac {\log {\left (e^{2 x} + \frac {\left (2 x^{2} - 2 x + 2 e^{3}\right ) e^{x}}{x} + \frac {x^{4} - 2 x^{3} + x^{2} + 2 x^{2} e^{3} - 2 x e^{3} - 6 x + 5 + e^{6}}{x^{2}} \right )}}{4} \] Input:
integrate(((x**2+x)*exp(x)**2+((1+x)*exp(3)+x**3+2*x**2-2*x)*exp(x)+(-1+2* x)*exp(3)+2*x**3-3*x**2+x-3)/(2*exp(x)**2*x**2+(4*x*exp(3)+4*x**3-4*x**2)* exp(x)+2*exp(3)**2+(4*x**2-4*x)*exp(3)+2*x**4-4*x**3+2*x**2-12*x+10),x)
Output:
log(x)/2 + log(exp(2*x) + (2*x**2 - 2*x + 2*exp(3))*exp(x)/x + (x**4 - 2*x **3 + x**2 + 2*x**2*exp(3) - 2*x*exp(3) - 6*x + 5 + exp(6))/x**2)/4
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.00 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {1}{2} \, \log \left (x\right ) + \frac {1}{4} \, \log \left (\frac {x^{4} - 2 \, x^{3} + x^{2} {\left (2 \, e^{3} + 1\right )} + x^{2} e^{\left (2 \, x\right )} - 2 \, x {\left (e^{3} + 3\right )} + 2 \, {\left (x^{3} - x^{2} + x e^{3}\right )} e^{x} + e^{6} + 5}{x^{2}}\right ) \] Input:
integrate(((x^2+x)*exp(x)^2+((1+x)*exp(3)+x^3+2*x^2-2*x)*exp(x)+(-1+2*x)*e xp(3)+2*x^3-3*x^2+x-3)/(2*exp(x)^2*x^2+(4*x*exp(3)+4*x^3-4*x^2)*exp(x)+2*e xp(3)^2+(4*x^2-4*x)*exp(3)+2*x^4-4*x^3+2*x^2-12*x+10),x, algorithm="maxima ")
Output:
1/2*log(x) + 1/4*log((x^4 - 2*x^3 + x^2*(2*e^3 + 1) + x^2*e^(2*x) - 2*x*(e ^3 + 3) + 2*(x^3 - x^2 + x*e^3)*e^x + e^6 + 5)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (24) = 48\).
Time = 0.19 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {1}{4} \, \log \left (x^{4} + 2 \, x^{3} e^{x} - 2 \, x^{3} + 2 \, x^{2} e^{3} + x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2} - 2 \, x e^{3} + 2 \, x e^{\left (x + 3\right )} - 6 \, x + e^{6} + 5\right ) \] Input:
integrate(((x^2+x)*exp(x)^2+((1+x)*exp(3)+x^3+2*x^2-2*x)*exp(x)+(-1+2*x)*e xp(3)+2*x^3-3*x^2+x-3)/(2*exp(x)^2*x^2+(4*x*exp(3)+4*x^3-4*x^2)*exp(x)+2*e xp(3)^2+(4*x^2-4*x)*exp(3)+2*x^4-4*x^3+2*x^2-12*x+10),x, algorithm="giac")
Output:
1/4*log(x^4 + 2*x^3*e^x - 2*x^3 + 2*x^2*e^3 + x^2*e^(2*x) - 2*x^2*e^x + x^ 2 - 2*x*e^3 + 2*x*e^(x + 3) - 6*x + e^6 + 5)
Time = 3.69 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx=\frac {\ln \left ({\mathrm {e}}^6-6\,x+2\,x\,{\mathrm {e}}^{x+3}-2\,x^2\,{\mathrm {e}}^x+2\,x^3\,{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^3+x^2\,{\mathrm {e}}^{2\,x}+2\,x^2\,{\mathrm {e}}^3+x^2-2\,x^3+x^4+5\right )}{4} \] Input:
int((x + exp(x)*(exp(3)*(x + 1) - 2*x + 2*x^2 + x^3) + exp(2*x)*(x + x^2) - 3*x^2 + 2*x^3 + exp(3)*(2*x - 1) - 3)/(2*exp(6) - 12*x - exp(3)*(4*x - 4 *x^2) + exp(x)*(4*x*exp(3) - 4*x^2 + 4*x^3) + 2*x^2*exp(2*x) + 2*x^2 - 4*x ^3 + 2*x^4 + 10),x)
Output:
log(exp(6) - 6*x + 2*x*exp(x + 3) - 2*x^2*exp(x) + 2*x^3*exp(x) - 2*x*exp( 3) + x^2*exp(2*x) + 2*x^2*exp(3) + x^2 - 2*x^3 + x^4 + 5)/4
\[ \int \frac {-3+x-3 x^2+2 x^3+e^3 (-1+2 x)+e^{2 x} \left (x+x^2\right )+e^x \left (-2 x+2 x^2+x^3+e^3 (1+x)\right )}{10+2 e^6-12 x+2 x^2+2 e^{2 x} x^2-4 x^3+2 x^4+e^3 \left (-4 x+4 x^2\right )+e^x \left (4 e^3 x-4 x^2+4 x^3\right )} \, dx =\text {Too large to display} \] Input:
int(((x^2+x)*exp(x)^2+((1+x)*exp(3)+x^3+2*x^2-2*x)*exp(x)+(-1+2*x)*exp(3)+ 2*x^3-3*x^2+x-3)/(2*exp(x)^2*x^2+(4*x*exp(3)+4*x^3-4*x^2)*exp(x)+2*exp(3)^ 2+(4*x^2-4*x)*exp(3)+2*x^4-4*x^3+2*x^2-12*x+10),x)
Output:
(int(e**x/(e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e** 6 + 2*e**3*x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x)*e**3 + 2*i nt(x**3/(e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e**3*x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) - 3*int(x**2 /(e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e** 3*x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) + int((e**(2*x)*x** 2)/(e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e **3*x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) + int((e**(2*x)*x )/(e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e* *3*x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) + int((e**x*x**3)/ (e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e**3 *x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) + 2*int((e**x*x**2)/ (e**(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e**3 *x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) + int((e**x*x)/(e**( 2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e**3*x**2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x)*e**3 - 2*int((e**x*x)/(e* *(2*x)*x**2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e**3*x* *2 - 2*e**3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x) + 2*int(x/(e**(2*x)*x** 2 + 2*e**x*e**3*x + 2*e**x*x**3 - 2*e**x*x**2 + e**6 + 2*e**3*x**2 - 2*e** 3*x + x**4 - 2*x**3 + x**2 - 6*x + 5),x)*e**3 + int(x/(e**(2*x)*x**2 + ...