Integrand size = 124, antiderivative size = 27 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx=e^{4-\frac {\left (-1+\frac {2}{e^3}+\log (5)\right )^2}{x^2}} (4-x) x \]
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx=-5^{\frac {2 \left (-2+e^3\right )}{e^3 x^2}} e^{\frac {-4+4 e^3+e^6 \left (-1+4 x^2-\log ^2(5)\right )}{e^6 x^2}} (-4+x) x \]
Integrate[(E^(-6 + (-4 + 4*E^3 + E^6*(-1 + 4*x^2) + (-4*E^3 + 2*E^6)*Log[5 ] - E^6*Log[5]^2)/(E^6*x^2))*(32 - 8*x + E^3*(-32 + 8*x) + E^6*(8 - 2*x + 4*x^2 - 2*x^3) + (E^3*(32 - 8*x) + E^6*(-16 + 4*x))*Log[5] + E^6*(8 - 2*x) *Log[5]^2))/x^2,x]
-(5^((2*(-2 + E^3))/(E^3*x^2))*E^((-4 + 4*E^3 + E^6*(-1 + 4*x^2 - Log[5]^2 ))/(E^6*x^2))*(-4 + x)*x)
Time = 2.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {7292, 2726}
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.
\(\displaystyle \int \frac {\left (e^6 \left (-2 x^3+4 x^2-2 x+8\right )-8 x+e^3 (8 x-32)+e^6 (8-2 x) \log ^2(5)+\left (e^3 (32-8 x)+e^6 (4 x-16)\right ) \log (5)+32\right ) \exp \left (\frac {e^6 \left (4 x^2-1\right )+4 e^3-4-e^6 \log ^2(5)+\left (2 e^6-4 e^3\right ) \log (5)}{e^6 x^2}-6\right )}{x^2} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-2 e^6 x^3+4 e^6 x^2-2 x \left (2-e^3 (1-\log (5))\right )^2+8 \left (2-e^3 (1-\log (5))\right )^2\right ) \exp \left (\frac {-4+4 e^3-e^6-e^6 \log ^2(5)-4 e^3 \log (5)+2 e^6 \log (5)}{e^6 x^2}-2\right )}{x^2}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle (4-x) x e^{4-\frac {\left (2-e^3 (1-\log (5))\right )^2}{e^6 x^2}}\) |
Int[(E^(-6 + (-4 + 4*E^3 + E^6*(-1 + 4*x^2) + (-4*E^3 + 2*E^6)*Log[5] - E^ 6*Log[5]^2)/(E^6*x^2))*(32 - 8*x + E^3*(-32 + 8*x) + E^6*(8 - 2*x + 4*x^2 - 2*x^3) + (E^3*(32 - 8*x) + E^6*(-16 + 4*x))*Log[5] + E^6*(8 - 2*x)*Log[5 ]^2))/x^2,x]
3.7.8.3.1 Defintions of rubi rules used
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Time = 0.79 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00
method | result | size |
risch | \(25^{\frac {1}{x^{2}}} \left (\frac {1}{625}\right )^{\frac {{\mathrm e}^{-3}}{x^{2}}} \left (-x^{2} {\mathrm e}^{6}+4 x \,{\mathrm e}^{6}\right ) {\mathrm e}^{\frac {-\ln \left (5\right )^{2}-2 x^{2}-4 \,{\mathrm e}^{-6}+4 \,{\mathrm e}^{-3}-1}{x^{2}}}\) | \(54\) |
gosper | \(-{\mathrm e}^{-\frac {\left ({\mathrm e}^{6} \ln \left (5\right )^{2}-4 x^{2} {\mathrm e}^{6}-2 \ln \left (5\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3} \ln \left (5\right )+{\mathrm e}^{6}-4 \,{\mathrm e}^{3}+4\right ) {\mathrm e}^{-6}}{x^{2}}} \left (x -4\right ) x\) | \(59\) |
parallelrisch | \({\mathrm e}^{-6} \left (-{\mathrm e}^{6} x^{2} {\mathrm e}^{\frac {\left (-{\mathrm e}^{6} \ln \left (5\right )^{2}+\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \left (5\right )+\left (4 x^{2}-1\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3}-4\right ) {\mathrm e}^{-6}}{x^{2}}}+4 \,{\mathrm e}^{6} x \,{\mathrm e}^{\frac {\left (-{\mathrm e}^{6} \ln \left (5\right )^{2}+\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \left (5\right )+\left (4 x^{2}-1\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3}-4\right ) {\mathrm e}^{-6}}{x^{2}}}\right )\) | \(125\) |
norman | \(\frac {\left (4 x^{2} {\mathrm e}^{3} {\mathrm e}^{\frac {\left (-{\mathrm e}^{6} \ln \left (5\right )^{2}+\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \left (5\right )+\left (4 x^{2}-1\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3}-4\right ) {\mathrm e}^{-6}}{x^{2}}}-x^{3} {\mathrm e}^{3} {\mathrm e}^{\frac {\left (-{\mathrm e}^{6} \ln \left (5\right )^{2}+\left (2 \,{\mathrm e}^{6}-4 \,{\mathrm e}^{3}\right ) \ln \left (5\right )+\left (4 x^{2}-1\right ) {\mathrm e}^{6}+4 \,{\mathrm e}^{3}-4\right ) {\mathrm e}^{-6}}{x^{2}}}\right ) {\mathrm e}^{-3}}{x}\) | \(126\) |
default | \(\text {Expression too large to display}\) | \(826\) |
derivativedivides | \(\text {Expression too large to display}\) | \(827\) |
meijerg | \(\text {Expression too large to display}\) | \(1041\) |
int(((-2*x+8)*exp(3)^2*ln(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3))*ln(5)+ (-2*x^3+4*x^2-2*x+8)*exp(3)^2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3)^2*ln(5) ^2+(2*exp(3)^2-4*exp(3))*ln(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/x^2/exp(3)^2 )/x^2/exp(3)^2,x,method=_RETURNVERBOSE)
25^(1/x^2)*(1/625)^(1/x^2*exp(-3))*(-x^2*exp(6)+4*x*exp(6))*exp((-ln(5)^2- 2*x^2-4*exp(-6)+4*exp(-3)-1)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx=-{\left (x^{2} - 4 \, x\right )} e^{\left (-\frac {{\left (e^{6} \log \left (5\right )^{2} + {\left (2 \, x^{2} + 1\right )} e^{6} - 2 \, {\left (e^{6} - 2 \, e^{3}\right )} \log \left (5\right ) - 4 \, e^{3} + 4\right )} e^{\left (-6\right )}}{x^{2}} + 6\right )} \]
integrate(((-2*x+8)*exp(3)^2*log(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3)) *log(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3) ^2*log(5)^2+(2*exp(3)^2-4*exp(3))*log(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/x^ 2/exp(3)^2)/x^2/exp(3)^2,x, algorithm=\
-(x^2 - 4*x)*e^(-(e^6*log(5)^2 + (2*x^2 + 1)*e^6 - 2*(e^6 - 2*e^3)*log(5) - 4*e^3 + 4)*e^(-6)/x^2 + 6)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).
Time = 0.42 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.96 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx=\left (- x^{2} + 4 x\right ) e^{\frac {\left (4 x^{2} - 1\right ) e^{6} - e^{6} \log {\left (5 \right )}^{2} - 4 + 4 e^{3} + \left (- 4 e^{3} + 2 e^{6}\right ) \log {\left (5 \right )}}{x^{2} e^{6}}} \]
integrate(((-2*x+8)*exp(3)**2*ln(5)**2+((4*x-16)*exp(3)**2+(-8*x+32)*exp(3 ))*ln(5)+(-2*x**3+4*x**2-2*x+8)*exp(3)**2+(8*x-32)*exp(3)-8*x+32)*exp((-ex p(3)**2*ln(5)**2+(2*exp(3)**2-4*exp(3))*ln(5)+(4*x**2-1)*exp(3)**2+4*exp(3 )-4)/x**2/exp(3)**2)/x**2/exp(3)**2,x)
(-x**2 + 4*x)*exp(((4*x**2 - 1)*exp(6) - exp(6)*log(5)**2 - 4 + 4*exp(3) + (-4*exp(3) + 2*exp(6))*log(5))*exp(-6)/x**2)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.60 (sec) , antiderivative size = 450, normalized size of antiderivative = 16.67 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx =\text {Too large to display} \]
integrate(((-2*x+8)*exp(3)^2*log(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3)) *log(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3) ^2*log(5)^2+(2*exp(3)^2-4*exp(3))*log(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/x^ 2/exp(3)^2)/x^2/exp(3)^2,x, algorithm=\
2*x*sqrt(x^(-2))*abs(e^3*log(5) - e^3 + 2)*e*gamma(-1/2, (e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2) - (e^3*log(5) - e^3 + 2)^2*e^(-2)*gamma(-1, (e^3*log(5 ) - e^3 + 2)^2*e^(-6)/x^2) + Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^4* log(5)^2 - 4*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^7*log(5)^2/(( log(5) - 1)*e^3 + 2) - 2*Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^4*log( 5) + 4*Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e*log(5) + 8*sqrt(pi)*erf( ((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^7*log(5)/((log(5) - 1)*e^3 + 2) - 16*sq rt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^4*log(5)/((log(5) - 1)*e^3 + 2) + Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^4 - 4*Ei(-(e^3*log(5) - e ^3 + 2)^2*e^(-6)/x^2)*e + 4*Ei(-(e^3*log(5) - e^3 + 2)^2*e^(-6)/x^2)*e^(-2 ) - 4*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^7/((log(5) - 1)*e^3 + 2) + 16*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e^4/((log(5) - 1)* e^3 + 2) - 16*sqrt(pi)*erf(((log(5) - 1)*e^3 + 2)*e^(-3)/x)*e/((log(5) - 1 )*e^3 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (24) = 48\).
Time = 0.51 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.67 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx=-x^{2} e^{\left (\frac {{\left (x^{2} e^{6} - e^{6} \log \left (5\right )^{2} + 2 \, e^{6} \log \left (5\right ) - 4 \, e^{3} \log \left (5\right ) - e^{6} + 4 \, e^{3} - 4\right )} e^{\left (-6\right )}}{x^{2}} + 3\right )} + 4 \, x e^{\left (\frac {{\left (x^{2} e^{6} - e^{6} \log \left (5\right )^{2} + 2 \, e^{6} \log \left (5\right ) - 4 \, e^{3} \log \left (5\right ) - e^{6} + 4 \, e^{3} - 4\right )} e^{\left (-6\right )}}{x^{2}} + 3\right )} \]
integrate(((-2*x+8)*exp(3)^2*log(5)^2+((4*x-16)*exp(3)^2+(-8*x+32)*exp(3)) *log(5)+(-2*x^3+4*x^2-2*x+8)*exp(3)^2+(8*x-32)*exp(3)-8*x+32)*exp((-exp(3) ^2*log(5)^2+(2*exp(3)^2-4*exp(3))*log(5)+(4*x^2-1)*exp(3)^2+4*exp(3)-4)/x^ 2/exp(3)^2)/x^2/exp(3)^2,x, algorithm=\
-x^2*e^((x^2*e^6 - e^6*log(5)^2 + 2*e^6*log(5) - 4*e^3*log(5) - e^6 + 4*e^ 3 - 4)*e^(-6)/x^2 + 3) + 4*x*e^((x^2*e^6 - e^6*log(5)^2 + 2*e^6*log(5) - 4 *e^3*log(5) - e^6 + 4*e^3 - 4)*e^(-6)/x^2 + 3)
Time = 11.66 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.07 \[ \int \frac {e^{-6+\frac {-4+4 e^3+e^6 \left (-1+4 x^2\right )+\left (-4 e^3+2 e^6\right ) \log (5)-e^6 \log ^2(5)}{e^6 x^2}} \left (32-8 x+e^3 (-32+8 x)+e^6 \left (8-2 x+4 x^2-2 x^3\right )+\left (e^3 (32-8 x)+e^6 (-16+4 x)\right ) \log (5)+e^6 (8-2 x) \log ^2(5)\right )}{x^2} \, dx=-\frac {{25}^{\frac {1}{x^2}}\,x\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^{-3}}{x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{-6}}{x^2}}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-\frac {{\ln \left (5\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {1}{x^2}}\,\left (x-4\right )}{{25}^{\frac {2\,{\mathrm {e}}^{-3}}{x^2}}} \]
int(-(exp(-6)*exp(-(exp(-6)*(exp(6)*log(5)^2 - exp(6)*(4*x^2 - 1) - 4*exp( 3) + log(5)*(4*exp(3) - 2*exp(6)) + 4))/x^2)*(8*x - log(5)*(exp(6)*(4*x - 16) - exp(3)*(8*x - 32)) + exp(6)*(2*x - 4*x^2 + 2*x^3 - 8) - exp(3)*(8*x - 32) + exp(6)*log(5)^2*(2*x - 8) - 32))/x^2,x)