Integrand size = 125, antiderivative size = 29 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=-2+\frac {1}{3} \left (5+e^x x\right ) \log \left (2+x+\left (6-e^2\right ) x (3+x)\right ) \] Output:
1/3*ln(x+2+(3+x)*x*(6-exp(2)))*(exp(x)*x+5)-2
Time = 0.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {1}{3} \left (5+e^x x\right ) \log \left (2+\left (19-3 e^2\right ) x-\left (-6+e^2\right ) x^2\right ) \] Input:
Integrate[(-95 - 60*x + E^2*(15 + 10*x) + E^x*(-19*x - 12*x^2 + E^2*(3*x + 2*x^2)) + E^x*(-2 - 21*x - 25*x^2 - 6*x^3 + E^2*(3*x + 4*x^2 + x^3))*Log[ 2 + 19*x + 6*x^2 + E^2*(-3*x - x^2)])/(-6 - 57*x - 18*x^2 + E^2*(9*x + 3*x ^2)),x]
Output:
((5 + E^x*x)*Log[2 + (19 - 3*E^2)*x - (-6 + E^2)*x^2])/3
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(29)=58\).
Time = 2.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.14, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {7292, 7279, 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.
\(\displaystyle \int \frac {e^x \left (-12 x^2+e^2 \left (2 x^2+3 x\right )-19 x\right )+e^x \left (-6 x^3-25 x^2+e^2 \left (x^3+4 x^2+3 x\right )-21 x-2\right ) \log \left (6 x^2+e^2 \left (-x^2-3 x\right )+19 x+2\right )-60 x+e^2 (10 x+15)-95}{-18 x^2+e^2 \left (3 x^2+9 x\right )-57 x-6} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-e^x \left (-12 x^2+e^2 \left (2 x^2+3 x\right )-19 x\right )-e^x \left (-6 x^3-25 x^2+e^2 \left (x^3+4 x^2+3 x\right )-21 x-2\right ) \log \left (6 x^2+e^2 \left (-x^2-3 x\right )+19 x+2\right )+60 x-e^2 (10 x+15)+95}{3 \left (6-e^2\right ) x^2+3 \left (19-3 e^2\right ) x+6}dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {5 \left (2 \left (6-e^2\right ) x-3 e^2+19\right )}{3 \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )}+\frac {e^x \left (12 \left (1-\frac {e^2}{6}\right ) x^2+25 \left (1-\frac {4 e^2}{25}\right ) x^2 \log \left (6 x^2-e^2 (x+3) x+19 x+2\right )+21 \left (1-\frac {e^2}{7}\right ) x \log \left (6 x^2-e^2 (x+3) x+19 x+2\right )+2 \log \left (6 x^2-e^2 (x+3) x+19 x+2\right )+6 \left (1-\frac {e^2}{6}\right ) x^3 \log \left (6 x^2-e^2 (x+3) x+19 x+2\right )+19 \left (1-\frac {3 e^2}{19}\right ) x\right )}{3 \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{3} e^x \log \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )+\frac {1}{3} e^x (x+1) \log \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )+\frac {5}{3} \log \left (\left (6-e^2\right ) x^2+\left (19-3 e^2\right ) x+2\right )\) |
Input:
Int[(-95 - 60*x + E^2*(15 + 10*x) + E^x*(-19*x - 12*x^2 + E^2*(3*x + 2*x^2 )) + E^x*(-2 - 21*x - 25*x^2 - 6*x^3 + E^2*(3*x + 4*x^2 + x^3))*Log[2 + 19 *x + 6*x^2 + E^2*(-3*x - x^2)])/(-6 - 57*x - 18*x^2 + E^2*(9*x + 3*x^2)),x ]
Output:
(5*Log[2 + (19 - 3*E^2)*x + (6 - E^2)*x^2])/3 - (E^x*Log[2 + (19 - 3*E^2)* x + (6 - E^2)*x^2])/3 + (E^x*(1 + x)*Log[2 + (19 - 3*E^2)*x + (6 - E^2)*x^ 2])/3
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 5.41 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76
method | result | size |
risch | \(\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {5 \ln \left (\left ({\mathrm e}^{2}-6\right ) x^{2}+\left (3 \,{\mathrm e}^{2}-19\right ) x -2\right )}{3}\) | \(51\) |
default | \(\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {5 \ln \left (x^{2} {\mathrm e}^{2}+3 \,{\mathrm e}^{2} x -6 x^{2}-19 x -2\right )}{3}\) | \(54\) |
parts | \(\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {5 \ln \left (x^{2} {\mathrm e}^{2}+3 \,{\mathrm e}^{2} x -6 x^{2}-19 x -2\right )}{3}\) | \(54\) |
norman | \(\frac {5 \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}+\frac {{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3}\) | \(55\) |
parallelrisch | \(\frac {{\mathrm e}^{4} x \,{\mathrm e}^{x} \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )-12 \,{\mathrm e}^{2} {\mathrm e}^{x} \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right ) x +5 \,{\mathrm e}^{4} \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )+36 \,{\mathrm e}^{x} x \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )-60 \,{\mathrm e}^{2} \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )+180 \ln \left (\left (-x^{2}-3 x \right ) {\mathrm e}^{2}+6 x^{2}+19 x +2\right )}{3 \left ({\mathrm e}^{2}-6\right )^{2}}\) | \(180\) |
Input:
int((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*ln((-x^2-3*x)*exp (2)+6*x^2+19*x+2)+((2*x^2+3*x)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15)*exp(2) -60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x,method=_RETURNVERBOSE)
Output:
1/3*exp(x)*x*ln((-x^2-3*x)*exp(2)+6*x^2+19*x+2)+5/3*ln((exp(2)-6)*x^2+(3*e xp(2)-19)*x-2)
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {1}{3} \, {\left (x e^{x} + 5\right )} \log \left (6 \, x^{2} - {\left (x^{2} + 3 \, x\right )} e^{2} + 19 \, x + 2\right ) \] Input:
integrate((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3 *x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15) *exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x, algorithm="fricas")
Output:
1/3*(x*e^x + 5)*log(6*x^2 - (x^2 + 3*x)*e^2 + 19*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 2.38 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {x e^{x} \log {\left (6 x^{2} + 19 x + \left (- x^{2} - 3 x\right ) e^{2} + 2 \right )}}{3} + \frac {5 \log {\left (x^{2} \left (-6 + e^{2}\right ) + x \left (-19 + 3 e^{2}\right ) - 2 \right )}}{3} \] Input:
integrate((((x**3+4*x**2+3*x)*exp(2)-6*x**3-25*x**2-21*x-2)*exp(x)*ln((-x* *2-3*x)*exp(2)+6*x**2+19*x+2)+((2*x**2+3*x)*exp(2)-12*x**2-19*x)*exp(x)+(1 0*x+15)*exp(2)-60*x-95)/((3*x**2+9*x)*exp(2)-18*x**2-57*x-6),x)
Output:
x*exp(x)*log(6*x**2 + 19*x + (-x**2 - 3*x)*exp(2) + 2)/3 + 5*log(x**2*(-6 + exp(2)) + x*(-19 + 3*exp(2)) - 2)/3
Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 391, normalized size of antiderivative = 13.48 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {1}{3} \, x e^{x} \log \left (-x^{2} {\left (e^{2} - 6\right )} - x {\left (3 \, e^{2} - 19\right )} + 2\right ) - \frac {5}{3} \, {\left (\frac {{\left (3 \, e^{2} - 19\right )} \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{\sqrt {9 \, e^{4} - 106 \, e^{2} + 313} {\left (e^{2} - 6\right )}} - \frac {\log \left (x^{2} {\left (e^{2} - 6\right )} + x {\left (3 \, e^{2} - 19\right )} - 2\right )}{e^{2} - 6}\right )} e^{2} + \frac {5 \, e^{2} \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{\sqrt {9 \, e^{4} - 106 \, e^{2} + 313}} - \frac {95 \, \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{3 \, \sqrt {9 \, e^{4} - 106 \, e^{2} + 313}} + \frac {10 \, {\left (3 \, e^{2} - 19\right )} \log \left (\frac {2 \, x {\left (e^{2} - 6\right )} - \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}{2 \, x {\left (e^{2} - 6\right )} + \sqrt {9 \, e^{4} - 106 \, e^{2} + 313} + 3 \, e^{2} - 19}\right )}{\sqrt {9 \, e^{4} - 106 \, e^{2} + 313} {\left (e^{2} - 6\right )}} - \frac {10 \, \log \left (x^{2} {\left (e^{2} - 6\right )} + x {\left (3 \, e^{2} - 19\right )} - 2\right )}{e^{2} - 6} \] Input:
integrate((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3 *x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15) *exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x, algorithm="maxima")
Output:
1/3*x*e^x*log(-x^2*(e^2 - 6) - x*(3*e^2 - 19) + 2) - 5/3*((3*e^2 - 19)*log ((2*x*(e^2 - 6) - sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19)/(2*x*(e^2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19))/(sqrt(9*e^4 - 106*e^2 + 313) *(e^2 - 6)) - log(x^2*(e^2 - 6) + x*(3*e^2 - 19) - 2)/(e^2 - 6))*e^2 + 5*e ^2*log((2*x*(e^2 - 6) - sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19)/(2*x*(e^ 2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19))/sqrt(9*e^4 - 106*e^2 + 313) - 95/3*log((2*x*(e^2 - 6) - sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19 )/(2*x*(e^2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^2 - 19))/sqrt(9*e^4 - 106*e^2 + 313) + 10*(3*e^2 - 19)*log((2*x*(e^2 - 6) - sqrt(9*e^4 - 106*e^ 2 + 313) + 3*e^2 - 19)/(2*x*(e^2 - 6) + sqrt(9*e^4 - 106*e^2 + 313) + 3*e^ 2 - 19))/(sqrt(9*e^4 - 106*e^2 + 313)*(e^2 - 6)) - 10*log(x^2*(e^2 - 6) + x*(3*e^2 - 19) - 2)/(e^2 - 6)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {1}{3} \, x e^{x} \log \left (-x^{2} e^{2} + 6 \, x^{2} - 3 \, x e^{2} + 19 \, x + 2\right ) + \frac {5}{3} \, \log \left (x^{2} e^{2} - 6 \, x^{2} + 3 \, x e^{2} - 19 \, x - 2\right ) \] Input:
integrate((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3 *x)*exp(2)+6*x^2+19*x+2)+((2*x^2+3*x)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15) *exp(2)-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x, algorithm="giac")
Output:
1/3*x*e^x*log(-x^2*e^2 + 6*x^2 - 3*x*e^2 + 19*x + 2) + 5/3*log(x^2*e^2 - 6 *x^2 + 3*x*e^2 - 19*x - 2)
Time = 8.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {5\,\ln \left (\left ({\mathrm {e}}^2-6\right )\,x^2+\left (3\,{\mathrm {e}}^2-19\right )\,x-2\right )}{3}+\frac {x\,{\mathrm {e}}^x\,\ln \left (19\,x-{\mathrm {e}}^2\,\left (x^2+3\,x\right )+6\,x^2+2\right )}{3} \] Input:
int((60*x + exp(x)*(19*x - exp(2)*(3*x + 2*x^2) + 12*x^2) - exp(2)*(10*x + 15) + exp(x)*log(19*x - exp(2)*(3*x + x^2) + 6*x^2 + 2)*(21*x - exp(2)*(3 *x + 4*x^2 + x^3) + 25*x^2 + 6*x^3 + 2) + 95)/(57*x - exp(2)*(9*x + 3*x^2) + 18*x^2 + 6),x)
Output:
(5*log(x*(3*exp(2) - 19) + x^2*(exp(2) - 6) - 2))/3 + (x*exp(x)*log(19*x - exp(2)*(3*x + x^2) + 6*x^2 + 2))/3
Time = 0.17 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {-95-60 x+e^2 (15+10 x)+e^x \left (-19 x-12 x^2+e^2 \left (3 x+2 x^2\right )\right )+e^x \left (-2-21 x-25 x^2-6 x^3+e^2 \left (3 x+4 x^2+x^3\right )\right ) \log \left (2+19 x+6 x^2+e^2 \left (-3 x-x^2\right )\right )}{-6-57 x-18 x^2+e^2 \left (9 x+3 x^2\right )} \, dx=\frac {e^{x} \mathrm {log}\left (-e^{2} x^{2}-3 e^{2} x +6 x^{2}+19 x +2\right ) x}{3}+\frac {5 \,\mathrm {log}\left (e^{2} x^{2}+3 e^{2} x -6 x^{2}-19 x -2\right )}{3} \] Input:
int((((x^3+4*x^2+3*x)*exp(2)-6*x^3-25*x^2-21*x-2)*exp(x)*log((-x^2-3*x)*ex p(2)+6*x^2+19*x+2)+((2*x^2+3*x)*exp(2)-12*x^2-19*x)*exp(x)+(10*x+15)*exp(2 )-60*x-95)/((3*x^2+9*x)*exp(2)-18*x^2-57*x-6),x)
Output:
(e**x*log( - e**2*x**2 - 3*e**2*x + 6*x**2 + 19*x + 2)*x + 5*log(e**2*x**2 + 3*e**2*x - 6*x**2 - 19*x - 2))/3