Integrand size = 137, antiderivative size = 31 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=x+\frac {e^{3 \left (5+\frac {x}{\log (x)}\right )} x^2}{2 \left (e^x+\log ^2(x)\right )} \]
Time = 0.21 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {1}{2} \left (2 x+\frac {e^{15+\frac {3 x}{\log (x)}} x^2}{e^x+\log ^2(x)}\right ) \]
Integrate[(2*E^(2*x)*Log[x]^2 + 4*E^x*Log[x]^4 + 2*Log[x]^6 + E^((3*x + 15 *Log[x])/Log[x])*(-3*E^x*x^2 + 3*E^x*x^2*Log[x] + (-3*x^2 + E^x*(2*x - x^2 ))*Log[x]^2 + (-2*x + 3*x^2)*Log[x]^3 + 2*x*Log[x]^4))/(2*E^(2*x)*Log[x]^2 + 4*E^x*Log[x]^4 + 2*Log[x]^6),x]
Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(31)=62\).
Time = 3.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.42, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {7292, 27, 7293, 2009}
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.
\(\displaystyle \int \frac {e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+\left (3 x^2-2 x\right ) \log ^3(x)+\left (e^x \left (2 x-x^2\right )-3 x^2\right ) \log ^2(x)+3 e^x x^2 \log (x)+2 x \log ^4(x)\right )+2 \log ^6(x)+4 e^x \log ^4(x)+2 e^{2 x} \log ^2(x)}{2 \log ^6(x)+4 e^x \log ^4(x)+2 e^{2 x} \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+\left (3 x^2-2 x\right ) \log ^3(x)+\left (e^x \left (2 x-x^2\right )-3 x^2\right ) \log ^2(x)+3 e^x x^2 \log (x)+2 x \log ^4(x)\right )+2 \log ^6(x)+4 e^x \log ^4(x)+2 e^{2 x} \log ^2(x)}{2 \log ^2(x) \left (e^x+\log ^2(x)\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {-e^{\frac {3 x}{\log (x)}} \left (-2 x \log ^4(x)+\left (2 x-3 x^2\right ) \log ^3(x)+\left (3 x^2-e^x \left (2 x-x^2\right )\right ) \log ^2(x)-3 e^x x^2 \log (x)+3 e^x x^2\right ) x^{\frac {15}{\log (x)}}+2 \log ^6(x)+4 e^x \log ^4(x)+2 e^{2 x} \log ^2(x)}{\log ^2(x) \left (\log ^2(x)+e^x\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {e^{\frac {3 x}{\log (x)}+15} x \left (2 \log ^4(x)+3 x \log ^3(x)-2 \log ^3(x)+2 e^x \log ^2(x)-e^x x \log ^2(x)-3 x \log ^2(x)+3 e^x x \log (x)-3 e^x x\right )}{\log ^2(x) \left (\log ^2(x)+e^x\right )^2}+2\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (2 x+\frac {x e^{\frac {3 x}{\log (x)}+15} \left (e^x x-x \log ^3(x)+x \log ^2(x)-e^x x \log (x)\right )}{\left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x) \left (e^x+\log ^2(x)\right )^2}\right )\) |
Int[(2*E^(2*x)*Log[x]^2 + 4*E^x*Log[x]^4 + 2*Log[x]^6 + E^((3*x + 15*Log[x ])/Log[x])*(-3*E^x*x^2 + 3*E^x*x^2*Log[x] + (-3*x^2 + E^x*(2*x - x^2))*Log [x]^2 + (-2*x + 3*x^2)*Log[x]^3 + 2*x*Log[x]^4))/(2*E^(2*x)*Log[x]^2 + 4*E ^x*Log[x]^4 + 2*Log[x]^6),x]
(2*x + (E^(15 + (3*x)/Log[x])*x*(E^x*x - E^x*x*Log[x] + x*Log[x]^2 - x*Log [x]^3))/((Log[x]^(-2) - Log[x]^(-1))*Log[x]^2*(E^x + Log[x]^2)^2))/2
3.24.28.3.1 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]]
Time = 3.74 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
risch | \(x +\frac {x^{2} {\mathrm e}^{\frac {15 \ln \left (x \right )+3 x}{\ln \left (x \right )}}}{2 \ln \left (x \right )^{2}+2 \,{\mathrm e}^{x}}\) | \(30\) |
parallelrisch | \(-\frac {-x^{2} {\mathrm e}^{\frac {15 \ln \left (x \right )+3 x}{\ln \left (x \right )}}-2 x \ln \left (x \right )^{2}-2 \,{\mathrm e}^{x} x}{2 \left (\ln \left (x \right )^{2}+{\mathrm e}^{x}\right )}\) | \(43\) |
int(((2*x*ln(x)^4+(3*x^2-2*x)*ln(x)^3+((-x^2+2*x)*exp(x)-3*x^2)*ln(x)^2+3* x^2*exp(x)*ln(x)-3*exp(x)*x^2)*exp((15*ln(x)+3*x)/ln(x))+2*ln(x)^6+4*exp(x )*ln(x)^4+2*exp(x)^2*ln(x)^2)/(2*ln(x)^6+4*exp(x)*ln(x)^4+2*exp(x)^2*ln(x) ^2),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {x^{2} e^{\left (\frac {3 \, {\left (x + 5 \, \log \left (x\right )\right )}}{\log \left (x\right )}\right )} + 2 \, x \log \left (x\right )^{2} + 2 \, x e^{x}}{2 \, {\left (\log \left (x\right )^{2} + e^{x}\right )}} \]
integrate(((2*x*log(x)^4+(3*x^2-2*x)*log(x)^3+((-x^2+2*x)*exp(x)-3*x^2)*lo g(x)^2+3*x^2*exp(x)*log(x)-3*exp(x)*x^2)*exp((15*log(x)+3*x)/log(x))+2*log (x)^6+4*exp(x)*log(x)^4+2*exp(x)^2*log(x)^2)/(2*log(x)^6+4*exp(x)*log(x)^4 +2*exp(x)^2*log(x)^2),x, algorithm=\
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {x^{2} e^{\frac {3 x + 15 \log {\left (x \right )}}{\log {\left (x \right )}}}}{2 e^{x} + 2 \log {\left (x \right )}^{2}} + x \]
integrate(((2*x*ln(x)**4+(3*x**2-2*x)*ln(x)**3+((-x**2+2*x)*exp(x)-3*x**2) *ln(x)**2+3*x**2*exp(x)*ln(x)-3*exp(x)*x**2)*exp((15*ln(x)+3*x)/ln(x))+2*l n(x)**6+4*exp(x)*ln(x)**4+2*exp(x)**2*ln(x)**2)/(2*ln(x)**6+4*exp(x)*ln(x) **4+2*exp(x)**2*ln(x)**2),x)
Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {x^{2} e^{\left (\frac {3 \, x}{\log \left (x\right )} + 15\right )} + 2 \, x \log \left (x\right )^{2} + 2 \, x e^{x}}{2 \, {\left (\log \left (x\right )^{2} + e^{x}\right )}} \]
integrate(((2*x*log(x)^4+(3*x^2-2*x)*log(x)^3+((-x^2+2*x)*exp(x)-3*x^2)*lo g(x)^2+3*x^2*exp(x)*log(x)-3*exp(x)*x^2)*exp((15*log(x)+3*x)/log(x))+2*log (x)^6+4*exp(x)*log(x)^4+2*exp(x)^2*log(x)^2)/(2*log(x)^6+4*exp(x)*log(x)^4 +2*exp(x)^2*log(x)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 119 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.84 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=\frac {2 \, x^{3} e^{x} \log \left (x\right )^{2} + x^{4} e^{\left (x + \frac {3 \, {\left (x + 5 \, \log \left (x\right )\right )}}{\log \left (x\right )}\right )} + 2 \, x^{3} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x^{2} e^{\left (\frac {3 \, {\left (x + 5 \, \log \left (x\right )\right )}}{\log \left (x\right )}\right )} - 4 \, x e^{x} \log \left (x\right ) + 8 \, x \log \left (x\right )^{2} + 8 \, x e^{x}}{2 \, {\left (x^{2} e^{x} \log \left (x\right )^{2} + x^{2} e^{\left (2 \, x\right )} + 4 \, \log \left (x\right )^{2} + 4 \, e^{x}\right )}} \]
integrate(((2*x*log(x)^4+(3*x^2-2*x)*log(x)^3+((-x^2+2*x)*exp(x)-3*x^2)*lo g(x)^2+3*x^2*exp(x)*log(x)-3*exp(x)*x^2)*exp((15*log(x)+3*x)/log(x))+2*log (x)^6+4*exp(x)*log(x)^4+2*exp(x)^2*log(x)^2)/(2*log(x)^6+4*exp(x)*log(x)^4 +2*exp(x)^2*log(x)^2),x, algorithm=\
1/2*(2*x^3*e^x*log(x)^2 + x^4*e^(x + 3*(x + 5*log(x))/log(x)) + 2*x^3*e^(2 *x) + 2*x^2*e^(2*x) + 4*x^2*e^(3*(x + 5*log(x))/log(x)) - 4*x*e^x*log(x) + 8*x*log(x)^2 + 8*x*e^x)/(x^2*e^x*log(x)^2 + x^2*e^(2*x) + 4*log(x)^2 + 4* e^x)
Time = 14.73 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94 \[ \int \frac {2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)+e^{\frac {3 x+15 \log (x)}{\log (x)}} \left (-3 e^x x^2+3 e^x x^2 \log (x)+\left (-3 x^2+e^x \left (2 x-x^2\right )\right ) \log ^2(x)+\left (-2 x+3 x^2\right ) \log ^3(x)+2 x \log ^4(x)\right )}{2 e^{2 x} \log ^2(x)+4 e^x \log ^4(x)+2 \log ^6(x)} \, dx=x+\frac {x^2\,{\mathrm {e}}^{15}\,{\mathrm {e}}^{\frac {3\,x}{\ln \left (x\right )}}}{2\,\left ({\ln \left (x\right )}^2+{\mathrm {e}}^x\right )} \]