Integrand size = 153, antiderivative size = 39 \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=\frac {x^2}{\log ^2(2) \log ^2\left (\frac {\left (-e^{1-x}+2 x\right ) \left (\frac {2}{x}-x^2\right )}{x^2}\right )} \]
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=\frac {x^2}{\log ^2(2) \log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (-2+x^3\right )}{x^3}\right )} \]
Integrate[(16*x^2 + 4*x^5 + E^(1 - x)*(-12*x - 4*x^2 + 2*x^5) + (8*x^2 - 4 *x^5 + E^(1 - x)*(-4*x + 2*x^4))*Log[(4*x - 2*x^4 + E^(1 - x)*(-2 + x^3))/ x^3])/((E^(1 - x)*(-2 + x^3)*Log[2]^2 + (4*x - 2*x^4)*Log[2]^2)*Log[(4*x - 2*x^4 + E^(1 - x)*(-2 + x^3))/x^3]^3),x]
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.
| \(\displaystyle \int \frac {4 x^5+16 x^2+e^{1-x} \left (2 x^5-4 x^2-12 x\right )+\left (-4 x^5+e^{1-x} \left (2 x^4-4 x\right )+8 x^2\right ) \log \left (\frac {-2 x^4+e^{1-x} \left (x^3-2\right )+4 x}{x^3}\right )}{\left (\left (4 x-2 x^4\right ) \log ^2(2)+e^{1-x} \left (x^3-2\right ) \log ^2(2)\right ) \log ^3\left (\frac {-2 x^4+e^{1-x} \left (x^3-2\right )+4 x}{x^3}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^x \left (-4 x^5-16 x^2-e^{1-x} \left (2 x^5-4 x^2-12 x\right )-\left (-4 x^5+e^{1-x} \left (2 x^4-4 x\right )+8 x^2\right ) \log \left (\frac {-2 x^4+e^{1-x} \left (x^3-2\right )+4 x}{x^3}\right )\right )}{\left (e-2 e^x x\right ) \left (2-x^3\right ) \log ^2(2) \log ^3\left (-\frac {e^{-x} \left (2 e^x x-e\right ) \left (x^3-2\right )}{x^3}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {2 e^x \left (2 x^5+8 x^2-e^{1-x} \left (-x^5+2 x^2+6 x\right )+\left (-2 x^5+4 x^2-e^{1-x} \left (2 x-x^4\right )\right ) \log \left (\frac {-2 x^4+4 x-e^{1-x} \left (2-x^3\right )}{x^3}\right )\right )}{\left (e-2 e^x x\right ) \left (2-x^3\right ) \log ^3\left (-\frac {e^{-x} \left (e-2 e^x x\right ) \left (2-x^3\right )}{x^3}\right )}dx}{\log ^2(2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \int \frac {e^x \left (2 x^5+8 x^2-e^{1-x} \left (-x^5+2 x^2+6 x\right )+\left (-2 x^5+4 x^2-e^{1-x} \left (2 x-x^4\right )\right ) \log \left (\frac {-2 x^4+4 x-e^{1-x} \left (2-x^3\right )}{x^3}\right )\right )}{\left (e-2 e^x x\right ) \left (2-x^3\right ) \log ^3\left (-\frac {e^{-x} \left (e-2 e^x x\right ) \left (2-x^3\right )}{x^3}\right )}dx}{\log ^2(2)}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \int \left (\frac {2 e^x x^2 (x+1)}{\left (2 e^x x-e\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}-\frac {x \left (x^4+\log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right ) x^3-2 x-2 \log \left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )-6\right )}{\left (x^3-2\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}\right )dx}{\log ^2(2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (-2^{2/3} \int \frac {1}{\left (\sqrt [3]{2}-x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx-(-2)^{2/3} \int \frac {1}{\left (\sqrt [3]{-1} x+\sqrt [3]{2}\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx+\sqrt [3]{-1} 2^{2/3} \int \frac {1}{\left (\sqrt [3]{2}-(-1)^{2/3} x\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx+2 \int \frac {e^x x^3}{\left (2 e^x x-e\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx-\int \frac {x}{\log ^2\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx-\int \frac {x^2}{\log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx+2 \int \frac {e^x x^2}{\left (2 e^x x-e\right ) \log ^3\left (\frac {e^{-x} \left (e-2 e^x x\right ) \left (x^3-2\right )}{x^3}\right )}dx\right )}{\log ^2(2)}\) |
Int[(16*x^2 + 4*x^5 + E^(1 - x)*(-12*x - 4*x^2 + 2*x^5) + (8*x^2 - 4*x^5 + E^(1 - x)*(-4*x + 2*x^4))*Log[(4*x - 2*x^4 + E^(1 - x)*(-2 + x^3))/x^3])/ ((E^(1 - x)*(-2 + x^3)*Log[2]^2 + (4*x - 2*x^4)*Log[2]^2)*Log[(4*x - 2*x^4 + E^(1 - x)*(-2 + x^3))/x^3]^3),x]
3.23.22.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 8.44 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95
| method | result | size |
| parallelrisch | \(\frac {x^{2}}{{\ln \left (\frac {\left (x^{3}-2\right ) {\mathrm e}^{1-x}-2 x^{4}+4 x}{x^{3}}\right )}^{2} \ln \left (2\right )^{2}}\) | \(37\) |
| risch | \(-\frac {4 x^{2}}{{\left (\pi \operatorname {csgn}\left (i x^{3}\right )^{3}-\pi \operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x \right )-\pi \operatorname {csgn}\left (i x^{3}\right )^{2} \operatorname {csgn}\left (i x^{2}\right )+\pi \,\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )-2 \pi {\operatorname {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )}^{2}+\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}-\pi \,\operatorname {csgn}\left (\frac {i}{x^{3}}\right ) \operatorname {csgn}\left (i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{x^{3}}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )}^{2}+\pi \,\operatorname {csgn}\left (i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )}^{2}+\pi {\operatorname {csgn}\left (\frac {i \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )}{x^{3}}\right )}^{3}+2 \pi -2 i \ln \left (2\right )+6 i \ln \left (x \right )-2 i \ln \left (x^{4}-\frac {{\mathrm e}^{1-x} x^{3}}{2}-2 x +{\mathrm e}^{1-x}\right )\right )}^{2} \ln \left (2\right )^{2}}\) | \(411\) |
int((((2*x^4-4*x)*exp(1-x)-4*x^5+8*x^2)*ln(((x^3-2)*exp(1-x)-2*x^4+4*x)/x^ 3)+(2*x^5-4*x^2-12*x)*exp(1-x)+4*x^5+16*x^2)/((x^3-2)*ln(2)^2*exp(1-x)+(-2 *x^4+4*x)*ln(2)^2)/ln(((x^3-2)*exp(1-x)-2*x^4+4*x)/x^3)^3,x,method=_RETURN VERBOSE)
Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=\frac {x^{2}}{\log \left (2\right )^{2} \log \left (-\frac {2 \, x^{4} - {\left (x^{3} - 2\right )} e^{\left (-x + 1\right )} - 4 \, x}{x^{3}}\right )^{2}} \]
integrate((((2*x^4-4*x)*exp(1-x)-4*x^5+8*x^2)*log(((x^3-2)*exp(1-x)-2*x^4+ 4*x)/x^3)+(2*x^5-4*x^2-12*x)*exp(1-x)+4*x^5+16*x^2)/((x^3-2)*log(2)^2*exp( 1-x)+(-2*x^4+4*x)*log(2)^2)/log(((x^3-2)*exp(1-x)-2*x^4+4*x)/x^3)^3,x, alg orithm=\
Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.82 \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=\frac {x^{2}}{\log {\left (2 \right )}^{2} \log {\left (\frac {- 2 x^{4} + 4 x + \left (x^{3} - 2\right ) e^{1 - x}}{x^{3}} \right )}^{2}} \]
integrate((((2*x**4-4*x)*exp(1-x)-4*x**5+8*x**2)*ln(((x**3-2)*exp(1-x)-2*x **4+4*x)/x**3)+(2*x**5-4*x**2-12*x)*exp(1-x)+4*x**5+16*x**2)/((x**3-2)*ln( 2)**2*exp(1-x)+(-2*x**4+4*x)*ln(2)**2)/ln(((x**3-2)*exp(1-x)-2*x**4+4*x)/x **3)**3,x)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (37) = 74\).
Time = 0.53 (sec) , antiderivative size = 124, normalized size of antiderivative = 3.18 \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=\frac {x^{2}}{x^{2} \log \left (2\right )^{2} + \log \left (2\right )^{2} \log \left (x^{3} - 2\right )^{2} + \log \left (2\right )^{2} \log \left (-2 \, x e^{x} + e\right )^{2} + 6 \, x \log \left (2\right )^{2} \log \left (x\right ) + 9 \, \log \left (2\right )^{2} \log \left (x\right )^{2} - 2 \, {\left (x \log \left (2\right )^{2} + 3 \, \log \left (2\right )^{2} \log \left (x\right )\right )} \log \left (x^{3} - 2\right ) - 2 \, {\left (x \log \left (2\right )^{2} - \log \left (2\right )^{2} \log \left (x^{3} - 2\right ) + 3 \, \log \left (2\right )^{2} \log \left (x\right )\right )} \log \left (-2 \, x e^{x} + e\right )} \]
integrate((((2*x^4-4*x)*exp(1-x)-4*x^5+8*x^2)*log(((x^3-2)*exp(1-x)-2*x^4+ 4*x)/x^3)+(2*x^5-4*x^2-12*x)*exp(1-x)+4*x^5+16*x^2)/((x^3-2)*log(2)^2*exp( 1-x)+(-2*x^4+4*x)*log(2)^2)/log(((x^3-2)*exp(1-x)-2*x^4+4*x)/x^3)^3,x, alg orithm=\
x^2/(x^2*log(2)^2 + log(2)^2*log(x^3 - 2)^2 + log(2)^2*log(-2*x*e^x + e)^2 + 6*x*log(2)^2*log(x) + 9*log(2)^2*log(x)^2 - 2*(x*log(2)^2 + 3*log(2)^2* log(x))*log(x^3 - 2) - 2*(x*log(2)^2 - log(2)^2*log(x^3 - 2) + 3*log(2)^2* log(x))*log(-2*x*e^x + e))
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (37) = 74\).
Time = 0.64 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.33 \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=\frac {x^{2}}{\log \left (2\right )^{2} \log \left (-2 \, x^{4} + x^{3} e^{\left (-x + 1\right )} + 4 \, x - 2 \, e^{\left (-x + 1\right )}\right )^{2} - 2 \, \log \left (2\right )^{2} \log \left (-2 \, x^{4} + x^{3} e^{\left (-x + 1\right )} + 4 \, x - 2 \, e^{\left (-x + 1\right )}\right ) \log \left (x^{3}\right ) + \log \left (2\right )^{2} \log \left (x^{3}\right )^{2}} \]
integrate((((2*x^4-4*x)*exp(1-x)-4*x^5+8*x^2)*log(((x^3-2)*exp(1-x)-2*x^4+ 4*x)/x^3)+(2*x^5-4*x^2-12*x)*exp(1-x)+4*x^5+16*x^2)/((x^3-2)*log(2)^2*exp( 1-x)+(-2*x^4+4*x)*log(2)^2)/log(((x^3-2)*exp(1-x)-2*x^4+4*x)/x^3)^3,x, alg orithm=\
x^2/(log(2)^2*log(-2*x^4 + x^3*e^(-x + 1) + 4*x - 2*e^(-x + 1))^2 - 2*log( 2)^2*log(-2*x^4 + x^3*e^(-x + 1) + 4*x - 2*e^(-x + 1))*log(x^3) + log(2)^2 *log(x^3)^2)
Timed out. \[ \int \frac {16 x^2+4 x^5+e^{1-x} \left (-12 x-4 x^2+2 x^5\right )+\left (8 x^2-4 x^5+e^{1-x} \left (-4 x+2 x^4\right )\right ) \log \left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )}{\left (e^{1-x} \left (-2+x^3\right ) \log ^2(2)+\left (4 x-2 x^4\right ) \log ^2(2)\right ) \log ^3\left (\frac {4 x-2 x^4+e^{1-x} \left (-2+x^3\right )}{x^3}\right )} \, dx=-\int \frac {{\mathrm {e}}^{1-x}\,\left (-2\,x^5+4\,x^2+12\,x\right )-16\,x^2-4\,x^5+\ln \left (\frac {4\,x+{\mathrm {e}}^{1-x}\,\left (x^3-2\right )-2\,x^4}{x^3}\right )\,\left ({\mathrm {e}}^{1-x}\,\left (4\,x-2\,x^4\right )-8\,x^2+4\,x^5\right )}{{\ln \left (\frac {4\,x+{\mathrm {e}}^{1-x}\,\left (x^3-2\right )-2\,x^4}{x^3}\right )}^3\,\left ({\ln \left (2\right )}^2\,\left (4\,x-2\,x^4\right )+{\mathrm {e}}^{1-x}\,{\ln \left (2\right )}^2\,\left (x^3-2\right )\right )} \,d x \]
int(-(exp(1 - x)*(12*x + 4*x^2 - 2*x^5) - 16*x^2 - 4*x^5 + log((4*x + exp( 1 - x)*(x^3 - 2) - 2*x^4)/x^3)*(exp(1 - x)*(4*x - 2*x^4) - 8*x^2 + 4*x^5)) /(log((4*x + exp(1 - x)*(x^3 - 2) - 2*x^4)/x^3)^3*(log(2)^2*(4*x - 2*x^4) + exp(1 - x)*log(2)^2*(x^3 - 2))),x)