Integrand size = 173, antiderivative size = 28 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=5+e^{-2-2 x} \left (3+\frac {25}{(x-(2-x) x+\log (x))^2}\right ) \]
Time = 0.07 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.82 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=-e^{-2 (1+x)} \left (-3-\frac {25}{((-1+x) x+\log (x))^2}\right ) \]
Integrate[(-50 + 50*x - 50*x^2 - 50*x^3 + 6*x^4 - 18*x^5 + 18*x^6 - 6*x^7 + (-50*x - 18*x^3 + 36*x^4 - 18*x^5)*Log[x] + (18*x^2 - 18*x^3)*Log[x]^2 - 6*x*Log[x]^3)/(E^(2 + 2*x)*(-x^4 + 3*x^5 - 3*x^6 + x^7) + E^(2 + 2*x)*(3* x^3 - 6*x^4 + 3*x^5)*Log[x] + E^(2 + 2*x)*(-3*x^2 + 3*x^3)*Log[x]^2 + E^(2 + 2*x)*x*Log[x]^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.
\(\displaystyle \int \frac {-6 x^7+18 x^6-18 x^5+6 x^4-50 x^3-50 x^2+\left (18 x^2-18 x^3\right ) \log ^2(x)+\left (-18 x^5+36 x^4-18 x^3-50 x\right ) \log (x)+50 x-6 x \log ^3(x)-50}{e^{2 x+2} \left (3 x^3-3 x^2\right ) \log ^2(x)+e^{2 x+2} \left (3 x^5-6 x^4+3 x^3\right ) \log (x)+e^{2 x+2} \left (x^7-3 x^6+3 x^5-x^4\right )+e^{2 x+2} x \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 e^{-2 x-2} \left (-3 x^7+9 x^6-9 x^5+3 x^4-25 x^3-25 x^2-9 (x-1) x^2 \log ^2(x)-\left (9 x^4-18 x^3+9 x^2+25\right ) x \log (x)+25 x-3 x \log ^3(x)-25\right )}{x ((x-1) x+\log (x))^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {e^{-2 x-2} \left (3 x^7-9 x^6+9 x^5-3 x^4+25 x^3-9 (1-x) \log ^2(x) x^2+25 x^2+3 \log ^3(x) x+\left (9 x^4-18 x^3+9 x^2+25\right ) \log (x) x-25 x+25\right )}{x ((1-x) x-\log (x))^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (-\frac {25 e^{-2 x-2} \left (2 x^2-x+1\right )}{x \left (x^2-x+\log (x)\right )^3}-3 e^{-2 x-2}-\frac {25 e^{-2 x-2}}{\left (x^2-x+\log (x)\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (25 \int \frac {e^{-2 x-2}}{\left (x^2-x+\log (x)\right )^3}dx-25 \int \frac {e^{-2 x-2}}{x \left (x^2-x+\log (x)\right )^3}dx-50 \int \frac {e^{-2 x-2} x}{\left (x^2-x+\log (x)\right )^3}dx-25 \int \frac {e^{-2 x-2}}{\left (x^2-x+\log (x)\right )^2}dx+\frac {3}{2} e^{-2 x-2}\right )\) |
Int[(-50 + 50*x - 50*x^2 - 50*x^3 + 6*x^4 - 18*x^5 + 18*x^6 - 6*x^7 + (-50 *x - 18*x^3 + 36*x^4 - 18*x^5)*Log[x] + (18*x^2 - 18*x^3)*Log[x]^2 - 6*x*L og[x]^3)/(E^(2 + 2*x)*(-x^4 + 3*x^5 - 3*x^6 + x^7) + E^(2 + 2*x)*(3*x^3 - 6*x^4 + 3*x^5)*Log[x] + E^(2 + 2*x)*(-3*x^2 + 3*x^3)*Log[x]^2 + E^(2 + 2*x )*x*Log[x]^3),x]
3.29.33.3.1 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_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04
method | result | size |
risch | \(3 \,{\mathrm e}^{-2-2 x}+\frac {25 \,{\mathrm e}^{-2-2 x}}{\left (x^{2}+\ln \left (x \right )-x \right )^{2}}\) | \(29\) |
parallelrisch | \(\frac {\left (50-12 x \ln \left (x \right )+6 \ln \left (x \right )^{2}+6 x^{4}-12 x^{3}+6 x^{2}+12 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{-2-2 x}}{2 x^{4}-4 x^{3}+4 x^{2} \ln \left (x \right )+2 x^{2}-4 x \ln \left (x \right )+2 \ln \left (x \right )^{2}}\) | \(74\) |
int((-6*x*ln(x)^3+(-18*x^3+18*x^2)*ln(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)*ln (x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^2*ln(x)^3 +(3*x^3-3*x^2)*exp(1+x)^2*ln(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^2*ln(x)+(x^ 7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (24) = 48\).
Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.04 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {3 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + 25}{2 \, {\left (x^{2} - x\right )} e^{\left (2 \, x + 2\right )} \log \left (x\right ) + e^{\left (2 \, x + 2\right )} \log \left (x\right )^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x + 2\right )}} \]
integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3- 50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^ 2*log(x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^ 2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x, algorithm=\
(3*x^4 - 6*x^3 + 3*x^2 + 6*(x^2 - x)*log(x) + 3*log(x)^2 + 25)/(2*(x^2 - x )*e^(2*x + 2)*log(x) + e^(2*x + 2)*log(x)^2 + (x^4 - 2*x^3 + x^2)*e^(2*x + 2))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (24) = 48\).
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.79 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {\left (3 x^{4} - 6 x^{3} + 6 x^{2} \log {\left (x \right )} + 3 x^{2} - 6 x \log {\left (x \right )} + 3 \log {\left (x \right )}^{2} + 25\right ) e^{- 2 x - 2}}{x^{4} - 2 x^{3} + 2 x^{2} \log {\left (x \right )} + x^{2} - 2 x \log {\left (x \right )} + \log {\left (x \right )}^{2}} \]
integrate((-6*x*ln(x)**3+(-18*x**3+18*x**2)*ln(x)**2+(-18*x**5+36*x**4-18* x**3-50*x)*ln(x)-6*x**7+18*x**6-18*x**5+6*x**4-50*x**3-50*x**2+50*x-50)/(x *exp(1+x)**2*ln(x)**3+(3*x**3-3*x**2)*exp(1+x)**2*ln(x)**2+(3*x**5-6*x**4+ 3*x**3)*exp(1+x)**2*ln(x)+(x**7-3*x**6+3*x**5-x**4)*exp(1+x)**2),x)
(3*x**4 - 6*x**3 + 6*x**2*log(x) + 3*x**2 - 6*x*log(x) + 3*log(x)**2 + 25) *exp(-2*x - 2)/(x**4 - 2*x**3 + 2*x**2*log(x) + x**2 - 2*x*log(x) + log(x) **2)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00 \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\frac {{\left (3 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2} + 25\right )} e^{\left (-2 \, x\right )}}{x^{4} e^{2} - 2 \, x^{3} e^{2} + x^{2} e^{2} + e^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{2} e^{2} - x e^{2}\right )} \log \left (x\right )} \]
integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3- 50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^ 2*log(x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^ 2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x, algorithm=\
(3*x^4 - 6*x^3 + 3*x^2 + 6*(x^2 - x)*log(x) + 3*log(x)^2 + 25)*e^(-2*x)/(x ^4*e^2 - 2*x^3*e^2 + x^2*e^2 + e^2*log(x)^2 + 2*(x^2*e^2 - x*e^2)*log(x))
\[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\int { -\frac {2 \, {\left (3 \, x^{7} - 9 \, x^{6} + 9 \, x^{5} - 3 \, x^{4} + 3 \, x \log \left (x\right )^{3} + 25 \, x^{3} + 9 \, {\left (x^{3} - x^{2}\right )} \log \left (x\right )^{2} + 25 \, x^{2} + {\left (9 \, x^{5} - 18 \, x^{4} + 9 \, x^{3} + 25 \, x\right )} \log \left (x\right ) - 25 \, x + 25\right )}}{x e^{\left (2 \, x + 2\right )} \log \left (x\right )^{3} + 3 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x + 2\right )} \log \left (x\right )^{2} + 3 \, {\left (x^{5} - 2 \, x^{4} + x^{3}\right )} e^{\left (2 \, x + 2\right )} \log \left (x\right ) + {\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} e^{\left (2 \, x + 2\right )}} \,d x } \]
integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3- 50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3-50*x^2+50*x-50)/(x*exp(1+x)^ 2*log(x)^3+(3*x^3-3*x^2)*exp(1+x)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(1+x)^ 2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(1+x)^2),x, algorithm=\
integrate(-2*(3*x^7 - 9*x^6 + 9*x^5 - 3*x^4 + 3*x*log(x)^3 + 25*x^3 + 9*(x ^3 - x^2)*log(x)^2 + 25*x^2 + (9*x^5 - 18*x^4 + 9*x^3 + 25*x)*log(x) - 25* x + 25)/(x*e^(2*x + 2)*log(x)^3 + 3*(x^3 - x^2)*e^(2*x + 2)*log(x)^2 + 3*( x^5 - 2*x^4 + x^3)*e^(2*x + 2)*log(x) + (x^7 - 3*x^6 + 3*x^5 - x^4)*e^(2*x + 2)), x)
Timed out. \[ \int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx=\int \frac {6\,x\,{\ln \left (x\right )}^3-50\,x+\ln \left (x\right )\,\left (18\,x^5-36\,x^4+18\,x^3+50\,x\right )-{\ln \left (x\right )}^2\,\left (18\,x^2-18\,x^3\right )+50\,x^2+50\,x^3-6\,x^4+18\,x^5-18\,x^6+6\,x^7+50}{-x\,{\mathrm {e}}^{2\,x+2}\,{\ln \left (x\right )}^3+{\mathrm {e}}^{2\,x+2}\,\left (3\,x^2-3\,x^3\right )\,{\ln \left (x\right )}^2-{\mathrm {e}}^{2\,x+2}\,\left (3\,x^5-6\,x^4+3\,x^3\right )\,\ln \left (x\right )+{\mathrm {e}}^{2\,x+2}\,\left (-x^7+3\,x^6-3\,x^5+x^4\right )} \,d x \]
int((6*x*log(x)^3 - 50*x + log(x)*(50*x + 18*x^3 - 36*x^4 + 18*x^5) - log( x)^2*(18*x^2 - 18*x^3) + 50*x^2 + 50*x^3 - 6*x^4 + 18*x^5 - 18*x^6 + 6*x^7 + 50)/(exp(2*x + 2)*(x^4 - 3*x^5 + 3*x^6 - x^7) + exp(2*x + 2)*log(x)^2*( 3*x^2 - 3*x^3) - x*exp(2*x + 2)*log(x)^3 - exp(2*x + 2)*log(x)*(3*x^3 - 6* x^4 + 3*x^5)),x)
int((6*x*log(x)^3 - 50*x + log(x)*(50*x + 18*x^3 - 36*x^4 + 18*x^5) - log( x)^2*(18*x^2 - 18*x^3) + 50*x^2 + 50*x^3 - 6*x^4 + 18*x^5 - 18*x^6 + 6*x^7 + 50)/(exp(2*x + 2)*(x^4 - 3*x^5 + 3*x^6 - x^7) + exp(2*x + 2)*log(x)^2*( 3*x^2 - 3*x^3) - x*exp(2*x + 2)*log(x)^3 - exp(2*x + 2)*log(x)*(3*x^3 - 6* x^4 + 3*x^5)), x)