Integrand size = 120, antiderivative size = 26 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=2+x+\frac {x}{\left (x+\frac {e^{-e} x}{\log (x)}\right )^2}+\frac {3 \log (x)}{2} \]
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {1}{2} \left (3 \log (x)+2 \left (x-\frac {2}{x+e^e x \log (x)}+\frac {1+\frac {1}{\left (1+e^e \log (x)\right )^2}}{x}\right )\right ) \]
Integrate[(3*x + 2*x^2 + (4*E^(2*E) + E^E*(9*x + 6*x^2))*Log[x] + E^(2*E)* (-2 + 9*x + 6*x^2)*Log[x]^2 + E^(3*E)*(-2 + 3*x + 2*x^2)*Log[x]^3)/(2*x^2 + 6*E^E*x^2*Log[x] + 6*E^(2*E)*x^2*Log[x]^2 + 2*E^(3*E)*x^2*Log[x]^3),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.54 (sec) , antiderivative size = 173, normalized size of antiderivative = 6.65, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, 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.
\(\displaystyle \int \frac {2 x^2+e^{3 e} \left (2 x^2+3 x-2\right ) \log ^3(x)+e^{2 e} \left (6 x^2+9 x-2\right ) \log ^2(x)+\left (e^e \left (6 x^2+9 x\right )+4 e^{2 e}\right ) \log (x)+3 x}{2 x^2+2 e^{3 e} x^2 \log ^3(x)+6 e^{2 e} x^2 \log ^2(x)+6 e^e x^2 \log (x)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 x^2+e^{3 e} \left (2 x^2+3 x-2\right ) \log ^3(x)+e^{2 e} \left (6 x^2+9 x-2\right ) \log ^2(x)+\left (e^e \left (6 x^2+9 x\right )+4 e^{2 e}\right ) \log (x)+3 x}{2 x^2 \left (e^e \log (x)+1\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int \frac {-e^{3 e} \left (-2 x^2-3 x+2\right ) \log ^3(x)-e^{2 e} \left (-6 x^2-9 x+2\right ) \log ^2(x)+\left (3 e^e \left (2 x^2+3 x\right )+4 e^{2 e}\right ) \log (x)+2 x^2+3 x}{x^2 \left (e^e \log (x)+1\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{2} \int \left (\frac {2 x^2+3 x-2}{x^2}+\frac {4}{x^2 \left (e^e \log (x)+1\right )}+\frac {2 \left (-1+2 e^e\right )}{x^2 \left (e^e \log (x)+1\right )^2}-\frac {4 e^e}{x^2 \left (e^e \log (x)+1\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (2 e^{e^{-e}-2 e} \left (1-2 e^e\right ) \operatorname {ExpIntegralEi}\left (-e^{-e} \left (e^e \log (x)+1\right )\right )+4 e^{e^{-e}-e} \operatorname {ExpIntegralEi}\left (-e^{-e} \left (e^e \log (x)+1\right )\right )-2 e^{e^{-e}-2 e} \operatorname {ExpIntegralEi}\left (-e^{-e} \left (e^e \log (x)+1\right )\right )+2 x+\frac {2}{x}+3 \log (x)+\frac {2 e^{-e} \left (1-2 e^e\right )}{x \left (e^e \log (x)+1\right )}-\frac {2 e^{-e}}{x \left (e^e \log (x)+1\right )}+\frac {2}{x \left (e^e \log (x)+1\right )^2}\right )\) |
Int[(3*x + 2*x^2 + (4*E^(2*E) + E^E*(9*x + 6*x^2))*Log[x] + E^(2*E)*(-2 + 9*x + 6*x^2)*Log[x]^2 + E^(3*E)*(-2 + 3*x + 2*x^2)*Log[x]^3)/(2*x^2 + 6*E^ E*x^2*Log[x] + 6*E^(2*E)*x^2*Log[x]^2 + 2*E^(3*E)*x^2*Log[x]^3),x]
(2/x + 2*x - 2*E^(-2*E + E^(-E))*ExpIntegralEi[-((1 + E^E*Log[x])/E^E)] + 4*E^(-E + E^(-E))*ExpIntegralEi[-((1 + E^E*Log[x])/E^E)] + 2*E^(-2*E + E^( -E))*(1 - 2*E^E)*ExpIntegralEi[-((1 + E^E*Log[x])/E^E)] + 3*Log[x] + 2/(x* (1 + E^E*Log[x])^2) - 2/(E^E*x*(1 + E^E*Log[x])) + (2*(1 - 2*E^E))/(E^E*x* (1 + E^E*Log[x])))/2
3.18.27.3.1 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 = 0.36 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65
method | result | size |
risch | \(\frac {3 x \ln \left (x \right )+2 x^{2}+2}{2 x}-\frac {2 \,{\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1\right )^{2}}\) | \(43\) |
norman | \(\frac {x^{2}+{\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+x^{2} {\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}} x \ln \left (x \right )^{3}}{2}+\frac {3 x \ln \left (x \right )}{2}+3 \,{\mathrm e}^{{\mathrm e}} x \ln \left (x \right )^{2}+2 x^{2} {\mathrm e}^{{\mathrm e}} \ln \left (x \right )}{x \left ({\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1\right )^{2}}\) | \(79\) |
parallelrisch | \(\frac {3 \,{\mathrm e}^{2 \,{\mathrm e}} x \ln \left (x \right )^{3}+2 x^{2} {\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+2 \,{\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+6 \,{\mathrm e}^{{\mathrm e}} x \ln \left (x \right )^{2}+4 x^{2} {\mathrm e}^{{\mathrm e}} \ln \left (x \right )+3 x \ln \left (x \right )+2 x^{2}}{2 x \left ({\mathrm e}^{2 \,{\mathrm e}} \ln \left (x \right )^{2}+2 \,{\mathrm e}^{{\mathrm e}} \ln \left (x \right )+1\right )}\) | \(95\) |
default | \(\text {Expression too large to display}\) | \(960\) |
int(((2*x^2+3*x-2)*exp(exp(1))^3*ln(x)^3+(6*x^2+9*x-2)*exp(exp(1))^2*ln(x) ^2+(4*exp(exp(1))^2+(6*x^2+9*x)*exp(exp(1)))*ln(x)+2*x^2+3*x)/(2*x^2*exp(e xp(1))^3*ln(x)^3+6*x^2*exp(exp(1))^2*ln(x)^2+6*x^2*exp(exp(1))*ln(x)+2*x^2 ),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {3 \, x e^{\left (2 \, e\right )} \log \left (x\right )^{3} + 2 \, {\left ({\left (x^{2} + 1\right )} e^{\left (2 \, e\right )} + 3 \, x e^{e}\right )} \log \left (x\right )^{2} + 2 \, x^{2} + {\left (4 \, x^{2} e^{e} + 3 \, x\right )} \log \left (x\right )}{2 \, {\left (x e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 2 \, x e^{e} \log \left (x\right ) + x\right )}} \]
integrate(((2*x^2+3*x-2)*exp(exp(1))^3*log(x)^3+(6*x^2+9*x-2)*exp(exp(1))^ 2*log(x)^2+(4*exp(exp(1))^2+(6*x^2+9*x)*exp(exp(1)))*log(x)+2*x^2+3*x)/(2* x^2*exp(exp(1))^3*log(x)^3+6*x^2*exp(exp(1))^2*log(x)^2+6*x^2*exp(exp(1))* log(x)+2*x^2),x, algorithm=\
1/2*(3*x*e^(2*e)*log(x)^3 + 2*((x^2 + 1)*e^(2*e) + 3*x*e^e)*log(x)^2 + 2*x ^2 + (4*x^2*e^e + 3*x)*log(x))/(x*e^(2*e)*log(x)^2 + 2*x*e^e*log(x) + x)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=x + \frac {- 2 e^{e} \log {\left (x \right )} - 1}{x e^{2 e} \log {\left (x \right )}^{2} + 2 x e^{e} \log {\left (x \right )} + x} + \frac {3 \log {\left (x \right )}}{2} + \frac {1}{x} \]
integrate(((2*x**2+3*x-2)*exp(exp(1))**3*ln(x)**3+(6*x**2+9*x-2)*exp(exp(1 ))**2*ln(x)**2+(4*exp(exp(1))**2+(6*x**2+9*x)*exp(exp(1)))*ln(x)+2*x**2+3* x)/(2*x**2*exp(exp(1))**3*ln(x)**3+6*x**2*exp(exp(1))**2*ln(x)**2+6*x**2*e xp(exp(1))*ln(x)+2*x**2),x)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (24) = 48\).
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 4.15 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {8 \, x^{2} e^{\left (2 \, e\right )} \log \left (x\right ) + 4 \, x^{2} e^{e} + 4 \, {\left (x^{2} e^{\left (3 \, e\right )} + e^{\left (3 \, e\right )}\right )} \log \left (x\right )^{2} + 3 \, x}{4 \, {\left (x e^{\left (3 \, e\right )} \log \left (x\right )^{2} + 2 \, x e^{\left (2 \, e\right )} \log \left (x\right ) + x e^{e}\right )}} - \frac {3}{4 \, {\left (e^{\left (3 \, e\right )} \log \left (x\right )^{2} + 2 \, e^{\left (2 \, e\right )} \log \left (x\right ) + e^{e}\right )}} + \frac {3}{2} \, \log \left (x\right ) \]
integrate(((2*x^2+3*x-2)*exp(exp(1))^3*log(x)^3+(6*x^2+9*x-2)*exp(exp(1))^ 2*log(x)^2+(4*exp(exp(1))^2+(6*x^2+9*x)*exp(exp(1)))*log(x)+2*x^2+3*x)/(2* x^2*exp(exp(1))^3*log(x)^3+6*x^2*exp(exp(1))^2*log(x)^2+6*x^2*exp(exp(1))* log(x)+2*x^2),x, algorithm=\
1/4*(8*x^2*e^(2*e)*log(x) + 4*x^2*e^e + 4*(x^2*e^(3*e) + e^(3*e))*log(x)^2 + 3*x)/(x*e^(3*e)*log(x)^2 + 2*x*e^(2*e)*log(x) + x*e^e) - 3/4/(e^(3*e)*l og(x)^2 + 2*e^(2*e)*log(x) + e^e) + 3/2*log(x)
Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (24) = 48\).
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.58 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=\frac {2 \, x^{2} e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 3 \, x e^{\left (2 \, e\right )} \log \left (x\right )^{3} + 4 \, x^{2} e^{e} \log \left (x\right ) + 6 \, x e^{e} \log \left (x\right )^{2} + 2 \, e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 2 \, x^{2} + 3 \, x \log \left (x\right )}{2 \, {\left (x e^{\left (2 \, e\right )} \log \left (x\right )^{2} + 2 \, x e^{e} \log \left (x\right ) + x\right )}} \]
integrate(((2*x^2+3*x-2)*exp(exp(1))^3*log(x)^3+(6*x^2+9*x-2)*exp(exp(1))^ 2*log(x)^2+(4*exp(exp(1))^2+(6*x^2+9*x)*exp(exp(1)))*log(x)+2*x^2+3*x)/(2* x^2*exp(exp(1))^3*log(x)^3+6*x^2*exp(exp(1))^2*log(x)^2+6*x^2*exp(exp(1))* log(x)+2*x^2),x, algorithm=\
1/2*(2*x^2*e^(2*e)*log(x)^2 + 3*x*e^(2*e)*log(x)^3 + 4*x^2*e^e*log(x) + 6* x*e^e*log(x)^2 + 2*e^(2*e)*log(x)^2 + 2*x^2 + 3*x*log(x))/(x*e^(2*e)*log(x )^2 + 2*x*e^e*log(x) + x)
Time = 10.89 (sec) , antiderivative size = 178, normalized size of antiderivative = 6.85 \[ \int \frac {3 x+2 x^2+\left (4 e^{2 e}+e^e \left (9 x+6 x^2\right )\right ) \log (x)+e^{2 e} \left (-2+9 x+6 x^2\right ) \log ^2(x)+e^{3 e} \left (-2+3 x+2 x^2\right ) \log ^3(x)}{2 x^2+6 e^e x^2 \log (x)+6 e^{2 e} x^2 \log ^2(x)+2 e^{3 e} x^2 \log ^3(x)} \, dx=x+\frac {3\,\ln \left (x\right )}{2}-\frac {\frac {{\mathrm {e}}^{-3\,\mathrm {e}}}{2\,x}+\frac {{\mathrm {e}}^{-\mathrm {e}}\,{\ln \left (x\right )}^2}{x}+\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{\mathrm {e}}+3\right )}{2\,x}}{{\ln \left (x\right )}^2+2\,{\mathrm {e}}^{-\mathrm {e}}\,\ln \left (x\right )+{\mathrm {e}}^{-2\,\mathrm {e}}}-\frac {\frac {{\mathrm {e}}^{-3\,\mathrm {e}}\,\left (2\,{\mathrm {e}}^{2\,\mathrm {e}}+3\,{\mathrm {e}}^{\mathrm {e}}-1\right )}{2\,x}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,{\ln \left (x\right )}^2}{x}+\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\ln \left (x\right )\,\left (2\,{\mathrm {e}}^{\mathrm {e}}-3\right )}{2\,x}}{{\mathrm {e}}^{-\mathrm {e}}+\ln \left (x\right )}-\frac {{\mathrm {e}}^{-\mathrm {e}}\,\ln \left (x\right )}{x}+\frac {{\mathrm {e}}^{-2\,\mathrm {e}}\,\left (2\,{\mathrm {e}}^{2\,\mathrm {e}}+4\,{\mathrm {e}}^{\mathrm {e}}-1\right )}{2\,x} \]
int((3*x + log(x)*(4*exp(2*exp(1)) + exp(exp(1))*(9*x + 6*x^2)) + 2*x^2 + exp(3*exp(1))*log(x)^3*(3*x + 2*x^2 - 2) + exp(2*exp(1))*log(x)^2*(9*x + 6 *x^2 - 2))/(2*x^2 + 6*x^2*exp(2*exp(1))*log(x)^2 + 2*x^2*exp(3*exp(1))*log (x)^3 + 6*x^2*exp(exp(1))*log(x)),x)
x + (3*log(x))/2 - (exp(-3*exp(1))/(2*x) + (exp(-exp(1))*log(x)^2)/x + (ex p(-2*exp(1))*log(x)*(2*exp(exp(1)) + 3))/(2*x))/(exp(-2*exp(1)) + log(x)^2 + 2*exp(-exp(1))*log(x)) - ((exp(-3*exp(1))*(2*exp(2*exp(1)) + 3*exp(exp( 1)) - 1))/(2*x) - (exp(-exp(1))*log(x)^2)/x + (exp(-2*exp(1))*log(x)*(2*ex p(exp(1)) - 3))/(2*x))/(exp(-exp(1)) + log(x)) - (exp(-exp(1))*log(x))/x + (exp(-2*exp(1))*(2*exp(2*exp(1)) + 4*exp(exp(1)) - 1))/(2*x)