Integrand size = 219, antiderivative size = 23 \[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=1+\frac {1}{-4+\left ((3-x) x \left (-5+e^x+2 x\right )\right )^x} \]
Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=\frac {1}{-4+\left (-\left ((-3+x) x \left (-5+e^x+2 x\right )\right )\right )^x} \]
Integrate[((-15*x + 11*x^2 - 2*x^3 + E^x*(3*x - x^2))^x*(-15 + 22*x - 6*x^ 2 + E^x*(3 + x - x^2) + (-15 + E^x*(3 - x) + 11*x - 2*x^2)*Log[-15*x + 11* x^2 - 2*x^3 + E^x*(3*x - x^2)]))/(240 - 176*x + 32*x^2 + E^x*(-48 + 16*x) + (-120 + E^x*(24 - 8*x) + 88*x - 16*x^2)*(-15*x + 11*x^2 - 2*x^3 + E^x*(3 *x - x^2))^x + (15 + E^x*(-3 + x) - 11*x + 2*x^2)*(-15*x + 11*x^2 - 2*x^3 + E^x*(3*x - x^2))^(2*x)),x]
Time = 1.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.014, Rules used = {7239, 7259, 17}
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.
| \(\displaystyle \int \frac {\left (-2 x^3+11 x^2+e^x \left (3 x-x^2\right )-15 x\right )^x \left (-6 x^2+e^x \left (-x^2+x+3\right )+\left (-2 x^2+11 x+e^x (3-x)-15\right ) \log \left (-2 x^3+11 x^2+e^x \left (3 x-x^2\right )-15 x\right )+22 x-15\right )}{32 x^2+\left (-16 x^2+88 x+e^x (24-8 x)-120\right ) \left (-2 x^3+11 x^2+e^x \left (3 x-x^2\right )-15 x\right )^x+\left (2 x^2-11 x+e^x (x-3)+15\right ) \left (-2 x^3+11 x^2+e^x \left (3 x-x^2\right )-15 x\right )^{2 x}-176 x+e^x (16 x-48)+240} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\left ((x-3) x \left (2 x+e^x-5\right )\right )\right )^{x-1} \left (6 x^2+e^x \left (x^2-x-3\right )-22 x+(x-3) \left (2 x+e^x-5\right ) \log \left (-\left ((x-3) x \left (2 x+e^x-5\right )\right )\right )+15\right )}{\left (4-\left (-\left ((x-3) x \left (2 x+e^x-5\right )\right )\right )^x\right )^2}dx\) |
| \(\Big \downarrow \) 7259 |
| \(\displaystyle \int \frac {1}{\left (4-\left (-\left (\left (-2 x-e^x+5\right ) (3-x) x\right )\right )^x\right )^2}d\left (-\left (-\left (\left (-2 x-e^x+5\right ) (3-x) x\right )\right )^x\right )\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {1}{4-\left (-\left (\left (-2 x-e^x+5\right ) (3-x) x\right )\right )^x}\) |
Int[((-15*x + 11*x^2 - 2*x^3 + E^x*(3*x - x^2))^x*(-15 + 22*x - 6*x^2 + E^ x*(3 + x - x^2) + (-15 + E^x*(3 - x) + 11*x - 2*x^2)*Log[-15*x + 11*x^2 - 2*x^3 + E^x*(3*x - x^2)]))/(240 - 176*x + 32*x^2 + E^x*(-48 + 16*x) + (-12 0 + E^x*(24 - 8*x) + 88*x - 16*x^2)*(-15*x + 11*x^2 - 2*x^3 + E^x*(3*x - x ^2))^x + (15 + E^x*(-3 + x) - 11*x + 2*x^2)*(-15*x + 11*x^2 - 2*x^3 + E^x* (3*x - x^2))^(2*x)),x]
3.1.39.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)*((a_) + (b_.)*(v_)^(p_.)*(w_)^(p_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(w*D[v, x] + v*D[w, x])]}, Simp[c Subst[Int[(a + b*x^p)^m, x] , x, v*w], x] /; FreeQ[c, x]] /; FreeQ[{a, b, m, p}, x] && IntegerQ[p]
Time = 14.13 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52
| method | result | size |
| parallelrisch | \(\frac {1}{{\mathrm e}^{x \ln \left (\left (-x^{2}+3 x \right ) {\mathrm e}^{x}-2 x^{3}+11 x^{2}-15 x \right )}-4}\) | \(35\) |
| risch | \(\frac {1}{\left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )^{x} x^{x} {\mathrm e}^{-\frac {i x \pi \left (-\operatorname {csgn}\left (i \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right ) {\operatorname {csgn}\left (i x \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right )}^{2}+\operatorname {csgn}\left (i \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right ) \operatorname {csgn}\left (i x \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right ) \operatorname {csgn}\left (i x \right )-{\operatorname {csgn}\left (i x \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right )}^{3}-{\operatorname {csgn}\left (i x \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right )}^{2} \operatorname {csgn}\left (i x \right )+2 {\operatorname {csgn}\left (i x \left (x^{2}+\left (\frac {{\mathrm e}^{x}}{2}-\frac {11}{2}\right ) x -\frac {3 \,{\mathrm e}^{x}}{2}+\frac {15}{2}\right )\right )}^{2}-2\right )}{2}}-4}\) | \(215\) |
int((((-x+3)*exp(x)-2*x^2+11*x-15)*ln((-x^2+3*x)*exp(x)-2*x^3+11*x^2-15*x) +(-x^2+x+3)*exp(x)-6*x^2+22*x-15)*exp(x*ln((-x^2+3*x)*exp(x)-2*x^3+11*x^2- 15*x))/(((-3+x)*exp(x)+2*x^2-11*x+15)*exp(x*ln((-x^2+3*x)*exp(x)-2*x^3+11* x^2-15*x))^2+((-8*x+24)*exp(x)-16*x^2+88*x-120)*exp(x*ln((-x^2+3*x)*exp(x) -2*x^3+11*x^2-15*x))+(16*x-48)*exp(x)+32*x^2-176*x+240),x,method=_RETURNVE RBOSE)
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=\frac {1}{{\left (-2 \, x^{3} + 11 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x} - 15 \, x\right )}^{x} - 4} \]
integrate((((-x+3)*exp(x)-2*x^2+11*x-15)*log((-x^2+3*x)*exp(x)-2*x^3+11*x^ 2-15*x)+(-x^2+x+3)*exp(x)-6*x^2+22*x-15)*exp(x*log((-x^2+3*x)*exp(x)-2*x^3 +11*x^2-15*x))/(((-3+x)*exp(x)+2*x^2-11*x+15)*exp(x*log((-x^2+3*x)*exp(x)- 2*x^3+11*x^2-15*x))^2+((-8*x+24)*exp(x)-16*x^2+88*x-120)*exp(x*log((-x^2+3 *x)*exp(x)-2*x^3+11*x^2-15*x))+(16*x-48)*exp(x)+32*x^2-176*x+240),x, algor ithm=\
Timed out. \[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=\text {Timed out} \]
integrate((((-x+3)*exp(x)-2*x**2+11*x-15)*ln((-x**2+3*x)*exp(x)-2*x**3+11* x**2-15*x)+(-x**2+x+3)*exp(x)-6*x**2+22*x-15)*exp(x*ln((-x**2+3*x)*exp(x)- 2*x**3+11*x**2-15*x))/(((-3+x)*exp(x)+2*x**2-11*x+15)*exp(x*ln((-x**2+3*x) *exp(x)-2*x**3+11*x**2-15*x))**2+((-8*x+24)*exp(x)-16*x**2+88*x-120)*exp(x *ln((-x**2+3*x)*exp(x)-2*x**3+11*x**2-15*x))+(16*x-48)*exp(x)+32*x**2-176* x+240),x)
Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=\frac {1}{e^{\left (x \log \left (x - 3\right ) + x \log \left (x\right ) + x \log \left (-2 \, x - e^{x} + 5\right )\right )} - 4} \]
integrate((((-x+3)*exp(x)-2*x^2+11*x-15)*log((-x^2+3*x)*exp(x)-2*x^3+11*x^ 2-15*x)+(-x^2+x+3)*exp(x)-6*x^2+22*x-15)*exp(x*log((-x^2+3*x)*exp(x)-2*x^3 +11*x^2-15*x))/(((-3+x)*exp(x)+2*x^2-11*x+15)*exp(x*log((-x^2+3*x)*exp(x)- 2*x^3+11*x^2-15*x))^2+((-8*x+24)*exp(x)-16*x^2+88*x-120)*exp(x*log((-x^2+3 *x)*exp(x)-2*x^3+11*x^2-15*x))+(16*x-48)*exp(x)+32*x^2-176*x+240),x, algor ithm=\
\[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=\int { -\frac {{\left (6 \, x^{2} + {\left (x^{2} - x - 3\right )} e^{x} + {\left (2 \, x^{2} + {\left (x - 3\right )} e^{x} - 11 \, x + 15\right )} \log \left (-2 \, x^{3} + 11 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x} - 15 \, x\right ) - 22 \, x + 15\right )} {\left (-2 \, x^{3} + 11 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x} - 15 \, x\right )}^{x}}{{\left (2 \, x^{2} + {\left (x - 3\right )} e^{x} - 11 \, x + 15\right )} {\left (-2 \, x^{3} + 11 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x} - 15 \, x\right )}^{2 \, x} - 8 \, {\left (2 \, x^{2} + {\left (x - 3\right )} e^{x} - 11 \, x + 15\right )} {\left (-2 \, x^{3} + 11 \, x^{2} - {\left (x^{2} - 3 \, x\right )} e^{x} - 15 \, x\right )}^{x} + 32 \, x^{2} + 16 \, {\left (x - 3\right )} e^{x} - 176 \, x + 240} \,d x } \]
integrate((((-x+3)*exp(x)-2*x^2+11*x-15)*log((-x^2+3*x)*exp(x)-2*x^3+11*x^ 2-15*x)+(-x^2+x+3)*exp(x)-6*x^2+22*x-15)*exp(x*log((-x^2+3*x)*exp(x)-2*x^3 +11*x^2-15*x))/(((-3+x)*exp(x)+2*x^2-11*x+15)*exp(x*log((-x^2+3*x)*exp(x)- 2*x^3+11*x^2-15*x))^2+((-8*x+24)*exp(x)-16*x^2+88*x-120)*exp(x*log((-x^2+3 *x)*exp(x)-2*x^3+11*x^2-15*x))+(16*x-48)*exp(x)+32*x^2-176*x+240),x, algor ithm=\
integrate(-(6*x^2 + (x^2 - x - 3)*e^x + (2*x^2 + (x - 3)*e^x - 11*x + 15)* log(-2*x^3 + 11*x^2 - (x^2 - 3*x)*e^x - 15*x) - 22*x + 15)*(-2*x^3 + 11*x^ 2 - (x^2 - 3*x)*e^x - 15*x)^x/((2*x^2 + (x - 3)*e^x - 11*x + 15)*(-2*x^3 + 11*x^2 - (x^2 - 3*x)*e^x - 15*x)^(2*x) - 8*(2*x^2 + (x - 3)*e^x - 11*x + 15)*(-2*x^3 + 11*x^2 - (x^2 - 3*x)*e^x - 15*x)^x + 32*x^2 + 16*(x - 3)*e^x - 176*x + 240), x)
Time = 11.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x \left (-15+22 x-6 x^2+e^x \left (3+x-x^2\right )+\left (-15+e^x (3-x)+11 x-2 x^2\right ) \log \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )\right )}{240-176 x+32 x^2+e^x (-48+16 x)+\left (-120+e^x (24-8 x)+88 x-16 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^x+\left (15+e^x (-3+x)-11 x+2 x^2\right ) \left (-15 x+11 x^2-2 x^3+e^x \left (3 x-x^2\right )\right )^{2 x}} \, dx=\frac {1}{{\left (3\,x\,{\mathrm {e}}^x-x^2\,{\mathrm {e}}^x-15\,x+11\,x^2-2\,x^3\right )}^x-4} \]
int(-(exp(x*log(exp(x)*(3*x - x^2) - 15*x + 11*x^2 - 2*x^3))*(log(exp(x)*( 3*x - x^2) - 15*x + 11*x^2 - 2*x^3)*(exp(x)*(x - 3) - 11*x + 2*x^2 + 15) - exp(x)*(x - x^2 + 3) - 22*x + 6*x^2 + 15))/(exp(x)*(16*x - 48) - exp(x*lo g(exp(x)*(3*x - x^2) - 15*x + 11*x^2 - 2*x^3))*(exp(x)*(8*x - 24) - 88*x + 16*x^2 + 120) - 176*x + 32*x^2 + exp(2*x*log(exp(x)*(3*x - x^2) - 15*x + 11*x^2 - 2*x^3))*(exp(x)*(x - 3) - 11*x + 2*x^2 + 15) + 240),x)