Integrand size = 118, antiderivative size = 28 \[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=\frac {e^{-x+\frac {x}{3+3 (4-x) x^2}} x}{\log (3)} \]
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=\frac {e^{x \left (-1+\frac {1}{3+12 x^2-3 x^3}\right )} x}{\log (3)} \]
Integrate[(E^((2*x + 12*x^3 - 3*x^4 + (-3 - 12*x^2 + 3*x^3)*Log[x/Log[3]]) /(-3 - 12*x^2 + 3*x^3))*(3 - 2*x + 24*x^2 - 34*x^3 + 56*x^4 - 72*x^5 + 27* x^6 - 3*x^7))/(3*x + 24*x^3 - 6*x^4 + 48*x^5 - 24*x^6 + 3*x^7),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.
| \(\displaystyle \int \frac {\left (-3 x^7+27 x^6-72 x^5+56 x^4-34 x^3+24 x^2-2 x+3\right ) \exp \left (\frac {-3 x^4+12 x^3+\left (3 x^3-12 x^2-3\right ) \log \left (\frac {x}{\log (3)}\right )+2 x}{3 x^3-12 x^2-3}\right )}{3 x^7-24 x^6+48 x^5-6 x^4+24 x^3+3 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-3 x^7+27 x^6-72 x^5+56 x^4-34 x^3+24 x^2-2 x+3\right ) \exp \left (\frac {-3 x^4+12 x^3+\left (3 x^3-12 x^2-3\right ) \log \left (\frac {x}{\log (3)}\right )+2 x}{3 x^3-12 x^2-3}\right )}{x \left (3 x^6-24 x^5+48 x^4-6 x^3+24 x^2+3\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \frac {\left (-3 x^7+27 x^6-72 x^5+56 x^4-34 x^3+24 x^2-2 x+3\right ) \exp \left (\frac {-3 x^4+12 x^3+\left (3 x^3-12 x^2-3\right ) \log \left (\frac {x}{\log (3)}\right )+2 x}{3 x^3-12 x^2-3}\right )}{3 x \left (x^3-4 x^2-1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int \frac {\exp \left (-\frac {-3 x^4+12 x^3+2 x}{3 \left (-x^3+4 x^2+1\right )}\right ) \left (-3 x^7+27 x^6-72 x^5+56 x^4-34 x^3+24 x^2-2 x+3\right )}{\left (-x^3+4 x^2+1\right )^2 \log (3)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\exp \left (-\frac {-3 x^4+12 x^3+2 x}{3 \left (-x^3+4 x^2+1\right )}\right ) \left (-3 x^7+27 x^6-72 x^5+56 x^4-34 x^3+24 x^2-2 x+3\right )}{\left (-x^3+4 x^2+1\right )^2}dx}{3 \log (3)}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {\int \frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) \left (-3 x^7+27 x^6-72 x^5+56 x^4-34 x^3+24 x^2-2 x+3\right )}{\left (-x^3+4 x^2+1\right )^2}dx}{3 \log (3)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (-3 \exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) x+3 \exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right )+\frac {2 \exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) (x+2)}{x^3-4 x^2-1}+\frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) \left (16 x^2+3 x+4\right )}{\left (x^3-4 x^2-1\right )^2}\right )dx}{3 \log (3)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \int \exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right )dx-3 \int \exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) xdx+4 \int \frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right )}{\left (x^3-4 x^2-1\right )^2}dx+3 \int \frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) x}{\left (x^3-4 x^2-1\right )^2}dx+16 \int \frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) x^2}{\left (x^3-4 x^2-1\right )^2}dx+4 \int \frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right )}{x^3-4 x^2-1}dx+2 \int \frac {\exp \left (-\frac {x \left (-3 x^3+12 x^2+2\right )}{3 \left (-x^3+4 x^2+1\right )}\right ) x}{x^3-4 x^2-1}dx}{3 \log (3)}\) |
Int[(E^((2*x + 12*x^3 - 3*x^4 + (-3 - 12*x^2 + 3*x^3)*Log[x/Log[3]])/(-3 - 12*x^2 + 3*x^3))*(3 - 2*x + 24*x^2 - 34*x^3 + 56*x^4 - 72*x^5 + 27*x^6 - 3*x^7))/(3*x + 24*x^3 - 6*x^4 + 48*x^5 - 24*x^6 + 3*x^7),x]
3.19.77.3.1 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[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 3.53 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \left (3\right )}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}\) | \(50\) |
| gosper | \({\mathrm e}^{\frac {3 \ln \left (\frac {x}{\ln \left (3\right )}\right ) x^{3}-3 x^{4}-12 \ln \left (\frac {x}{\ln \left (3\right )}\right ) x^{2}+12 x^{3}-3 \ln \left (\frac {x}{\ln \left (3\right )}\right )+2 x}{3 x^{3}-12 x^{2}-3}}\) | \(63\) |
| risch | \({\mathrm e}^{-\frac {-3 \ln \left (\frac {x}{\ln \left (3\right )}\right ) x^{3}+3 x^{4}+12 \ln \left (\frac {x}{\ln \left (3\right )}\right ) x^{2}-12 x^{3}+3 \ln \left (\frac {x}{\ln \left (3\right )}\right )-2 x}{3 \left (x^{3}-4 x^{2}-1\right )}}\) | \(63\) |
| norman | \(\frac {x^{3} {\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \left (3\right )}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}-4 x^{2} {\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \left (3\right )}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}-{\mathrm e}^{\frac {\left (3 x^{3}-12 x^{2}-3\right ) \ln \left (\frac {x}{\ln \left (3\right )}\right )-3 x^{4}+12 x^{3}+2 x}{3 x^{3}-12 x^{2}-3}}}{x^{3}-4 x^{2}-1}\) | \(176\) |
int((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12*x^2-3 )*ln(x/ln(3))-3*x^4+12*x^3+2*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48*x^5-6*x ^4+24*x^3+3*x),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=e^{\left (-\frac {3 \, x^{4} - 12 \, x^{3} - 3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )} \log \left (\frac {x}{\log \left (3\right )}\right ) - 2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )}}\right )} \]
integrate((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12 *x^2-3)*log(x/log(3))-3*x^4+12*x^3+2*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48 *x^5-6*x^4+24*x^3+3*x),x, algorithm=\
Timed out. \[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=\text {Timed out} \]
integrate((-3*x**7+27*x**6-72*x**5+56*x**4-34*x**3+24*x**2-2*x+3)*exp(((3* x**3-12*x**2-3)*ln(x/ln(3))-3*x**4+12*x**3+2*x)/(3*x**3-12*x**2-3))/(3*x** 7-24*x**6+48*x**5-6*x**4+24*x**3+3*x),x)
\[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=\int { -\frac {{\left (3 \, x^{7} - 27 \, x^{6} + 72 \, x^{5} - 56 \, x^{4} + 34 \, x^{3} - 24 \, x^{2} + 2 \, x - 3\right )} e^{\left (-\frac {3 \, x^{4} - 12 \, x^{3} - 3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )} \log \left (\frac {x}{\log \left (3\right )}\right ) - 2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )}}\right )}}{3 \, {\left (x^{7} - 8 \, x^{6} + 16 \, x^{5} - 2 \, x^{4} + 8 \, x^{3} + x\right )}} \,d x } \]
integrate((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12 *x^2-3)*log(x/log(3))-3*x^4+12*x^3+2*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48 *x^5-6*x^4+24*x^3+3*x),x, algorithm=\
-1/3*integrate((3*x^7 - 27*x^6 + 72*x^5 - 56*x^4 + 34*x^3 - 24*x^2 + 2*x - 3)*e^(-1/3*(3*x^4 - 12*x^3 - 3*(x^3 - 4*x^2 - 1)*log(x/log(3)) - 2*x)/(x^ 3 - 4*x^2 - 1))/(x^7 - 8*x^6 + 16*x^5 - 2*x^4 + 8*x^3 + x), x)
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (26) = 52\).
Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.25 \[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=e^{\left (-\frac {x^{4}}{x^{3} - 4 \, x^{2} - 1} + \frac {x^{3} \log \left (\frac {x}{\log \left (3\right )}\right )}{x^{3} - 4 \, x^{2} - 1} + \frac {4 \, x^{3}}{x^{3} - 4 \, x^{2} - 1} - \frac {4 \, x^{2} \log \left (\frac {x}{\log \left (3\right )}\right )}{x^{3} - 4 \, x^{2} - 1} + \frac {2 \, x}{3 \, {\left (x^{3} - 4 \, x^{2} - 1\right )}} - \frac {\log \left (\frac {x}{\log \left (3\right )}\right )}{x^{3} - 4 \, x^{2} - 1}\right )} \]
integrate((-3*x^7+27*x^6-72*x^5+56*x^4-34*x^3+24*x^2-2*x+3)*exp(((3*x^3-12 *x^2-3)*log(x/log(3))-3*x^4+12*x^3+2*x)/(3*x^3-12*x^2-3))/(3*x^7-24*x^6+48 *x^5-6*x^4+24*x^3+3*x),x, algorithm=\
e^(-x^4/(x^3 - 4*x^2 - 1) + x^3*log(x/log(3))/(x^3 - 4*x^2 - 1) + 4*x^3/(x ^3 - 4*x^2 - 1) - 4*x^2*log(x/log(3))/(x^3 - 4*x^2 - 1) + 2/3*x/(x^3 - 4*x ^2 - 1) - log(x/log(3))/(x^3 - 4*x^2 - 1))
Time = 9.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18 \[ \int \frac {e^{\frac {2 x+12 x^3-3 x^4+\left (-3-12 x^2+3 x^3\right ) \log \left (\frac {x}{\log (3)}\right )}{-3-12 x^2+3 x^3}} \left (3-2 x+24 x^2-34 x^3+56 x^4-72 x^5+27 x^6-3 x^7\right )}{3 x+24 x^3-6 x^4+48 x^5-24 x^6+3 x^7} \, dx=x^{\frac {12\,x^2}{-3\,x^3+12\,x^2+3}-\frac {x^3-1}{-x^3+4\,x^2+1}}\,{\mathrm {e}}^{-\frac {2\,x}{-3\,x^3+12\,x^2+3}}\,{\mathrm {e}}^{\frac {3\,x^4}{-3\,x^3+12\,x^2+3}}\,{\mathrm {e}}^{-\frac {12\,x^3}{-3\,x^3+12\,x^2+3}}\,{\ln \left (3\right )}^{\frac {x^3-1}{-x^3+4\,x^2+1}-\frac {12\,x^2}{-3\,x^3+12\,x^2+3}} \]
int(-(exp(-(2*x - log(x/log(3))*(12*x^2 - 3*x^3 + 3) + 12*x^3 - 3*x^4)/(12 *x^2 - 3*x^3 + 3))*(2*x - 24*x^2 + 34*x^3 - 56*x^4 + 72*x^5 - 27*x^6 + 3*x ^7 - 3))/(3*x + 24*x^3 - 6*x^4 + 48*x^5 - 24*x^6 + 3*x^7),x)