Integrand size = 84, antiderivative size = 22 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x \left (x^2-\left (1+\log \left (-\frac {2}{5}+e^3\right )\right )^2\right )^2 \]
Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x \left (x^2-\left (1+\log \left (-\frac {2}{5}+e^3\right )\right )^2\right )^2 \]
Integrate[1 - 6*x^2 + 5*x^4 + (4 - 12*x^2)*Log[(-2 + 5*E^3)/5] + (6 - 6*x^ 2)*Log[(-2 + 5*E^3)/5]^2 + 4*Log[(-2 + 5*E^3)/5]^3 + Log[(-2 + 5*E^3)/5]^4 ,x]
Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(22)=44\).
Time = 0.22 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {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.
\(\displaystyle \int \left (5 x^4-6 x^2+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (5 e^3-2\right )\right )+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (5 e^3-2\right )\right )+1+\log ^4\left (\frac {1}{5} \left (5 e^3-2\right )\right )+4 \log ^3\left (\frac {1}{5} \left (5 e^3-2\right )\right )\right ) \, dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^5-2 x^3-2 x^3 \log ^2\left (e^3-\frac {2}{5}\right )-4 x^3 \log \left (e^3-\frac {2}{5}\right )+6 x \log ^2\left (e^3-\frac {2}{5}\right )+x \left (1+\log ^4\left (e^3-\frac {2}{5}\right )+4 \log ^3\left (e^3-\frac {2}{5}\right )\right )+4 x \log \left (e^3-\frac {2}{5}\right )\) |
Int[1 - 6*x^2 + 5*x^4 + (4 - 12*x^2)*Log[(-2 + 5*E^3)/5] + (6 - 6*x^2)*Log [(-2 + 5*E^3)/5]^2 + 4*Log[(-2 + 5*E^3)/5]^3 + Log[(-2 + 5*E^3)/5]^4,x]
-2*x^3 + x^5 + 4*x*Log[-2/5 + E^3] - 4*x^3*Log[-2/5 + E^3] + 6*x*Log[-2/5 + E^3]^2 - 2*x^3*Log[-2/5 + E^3]^2 + x*(1 + 4*Log[-2/5 + E^3]^3 + Log[-2/5 + E^3]^4)
3.13.24.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(69\) vs. \(2(19)=38\).
Time = 0.90 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.18
method | result | size |
parallelrisch | \(-2 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2} x^{3}+x^{5}-4 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right ) x^{3}+6 x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2}-2 x^{3}+4 x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )+\left (\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{4}+4 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{3}+1\right ) x\) | \(70\) |
parts | \(x +x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{4}-2 x^{3}+x^{5}+6 x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2}+4 x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{3}-4 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right ) x^{3}-2 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2} x^{3}+4 x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )\) | \(70\) |
default | \(x^{5}+\frac {\left (-\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2}-2 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )-1+\left (-5 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )-5\right ) \left (\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )+1\right )\right ) x^{3}}{3}+x \left (\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2}+2 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )+1\right ) {\left (\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )+1\right )}^{2}\) | \(72\) |
gosper | \(\left (\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )+1-x \right ) x \left (\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{3}+x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2}-\ln \left ({\mathrm e}^{3}-\frac {2}{5}\right ) x^{2}-x^{3}+3 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )^{2}+2 x \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )-x^{2}+3 \ln \left ({\mathrm e}^{3}-\frac {2}{5}\right )+x +1\right )\) | \(76\) |
norman | \(x^{5}+\left (-2 \ln \left (5\right )^{2}+4 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right )-2 \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2}+4 \ln \left (5\right )-4 \ln \left (5 \,{\mathrm e}^{3}-2\right )-2\right ) x^{3}+\left (1+4 \ln \left (5 \,{\mathrm e}^{3}-2\right )^{3}+\ln \left (5 \,{\mathrm e}^{3}-2\right )^{4}+\ln \left (5\right )^{4}-4 \ln \left (5\right )^{3}-4 \ln \left (5\right )+6 \ln \left (5\right )^{2}+4 \ln \left (5 \,{\mathrm e}^{3}-2\right )-12 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right )+6 \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2}+12 \ln \left (5\right )^{2} \ln \left (5 \,{\mathrm e}^{3}-2\right )-12 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2}-4 \ln \left (5\right )^{3} \ln \left (5 \,{\mathrm e}^{3}-2\right )+6 \ln \left (5\right )^{2} \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2}-4 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right )^{3}\right ) x\) | \(194\) |
risch | \(x \ln \left (5\right )^{4}-4 \ln \left (5\right )^{3} \ln \left (5 \,{\mathrm e}^{3}-2\right ) x +6 \ln \left (5\right )^{2} \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2} x -2 x^{3} \ln \left (5\right )^{2}-4 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right )^{3} x +4 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right ) x^{3}+\ln \left (5 \,{\mathrm e}^{3}-2\right )^{4} x -2 \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2} x^{3}+x^{5}-4 \ln \left (5\right )^{3} x +12 \ln \left (5\right )^{2} \ln \left (5 \,{\mathrm e}^{3}-2\right ) x -12 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2} x +4 x^{3} \ln \left (5\right )+4 \ln \left (5 \,{\mathrm e}^{3}-2\right )^{3} x -4 x^{3} \ln \left (5 \,{\mathrm e}^{3}-2\right )+6 x \ln \left (5\right )^{2}-12 \ln \left (5\right ) \ln \left (5 \,{\mathrm e}^{3}-2\right ) x +6 \ln \left (5 \,{\mathrm e}^{3}-2\right )^{2} x -2 x^{3}-4 x \ln \left (5\right )+4 x \ln \left (5 \,{\mathrm e}^{3}-2\right )+x\) | \(221\) |
int(ln(exp(3)-2/5)^4+4*ln(exp(3)-2/5)^3+(-6*x^2+6)*ln(exp(3)-2/5)^2+(-12*x ^2+4)*ln(exp(3)-2/5)+5*x^4-6*x^2+1,x,method=_RETURNVERBOSE)
-2*ln(exp(3)-2/5)^2*x^3+x^5-4*ln(exp(3)-2/5)*x^3+6*x*ln(exp(3)-2/5)^2-2*x^ 3+4*x*ln(exp(3)-2/5)+(ln(exp(3)-2/5)^4+4*ln(exp(3)-2/5)^3+1)*x
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (19) = 38\).
Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x^{5} + x \log \left (e^{3} - \frac {2}{5}\right )^{4} + 4 \, x \log \left (e^{3} - \frac {2}{5}\right )^{3} - 2 \, x^{3} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (e^{3} - \frac {2}{5}\right )^{2} - 4 \, {\left (x^{3} - x\right )} \log \left (e^{3} - \frac {2}{5}\right ) + x \]
integrate(log(exp(3)-2/5)^4+4*log(exp(3)-2/5)^3+(-6*x^2+6)*log(exp(3)-2/5) ^2+(-12*x^2+4)*log(exp(3)-2/5)+5*x^4-6*x^2+1,x, algorithm=\
x^5 + x*log(e^3 - 2/5)^4 + 4*x*log(e^3 - 2/5)^3 - 2*x^3 - 2*(x^3 - 3*x)*lo g(e^3 - 2/5)^2 - 4*(x^3 - x)*log(e^3 - 2/5) + x
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (17) = 34\).
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.55 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x^{5} + x^{3} \left (- 2 \log {\left (- \frac {2}{5} + e^{3} \right )}^{2} - 4 \log {\left (- \frac {2}{5} + e^{3} \right )} - 2\right ) + x \left (1 + 4 \log {\left (- \frac {2}{5} + e^{3} \right )} + 6 \log {\left (- \frac {2}{5} + e^{3} \right )}^{2} + \log {\left (- \frac {2}{5} + e^{3} \right )}^{4} + 4 \log {\left (- \frac {2}{5} + e^{3} \right )}^{3}\right ) \]
integrate(ln(exp(3)-2/5)**4+4*ln(exp(3)-2/5)**3+(-6*x**2+6)*ln(exp(3)-2/5) **2+(-12*x**2+4)*ln(exp(3)-2/5)+5*x**4-6*x**2+1,x)
x**5 + x**3*(-2*log(-2/5 + exp(3))**2 - 4*log(-2/5 + exp(3)) - 2) + x*(1 + 4*log(-2/5 + exp(3)) + 6*log(-2/5 + exp(3))**2 + log(-2/5 + exp(3))**4 + 4*log(-2/5 + exp(3))**3)
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (19) = 38\).
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x^{5} + x \log \left (e^{3} - \frac {2}{5}\right )^{4} + 4 \, x \log \left (e^{3} - \frac {2}{5}\right )^{3} - 2 \, x^{3} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (e^{3} - \frac {2}{5}\right )^{2} - 4 \, {\left (x^{3} - x\right )} \log \left (e^{3} - \frac {2}{5}\right ) + x \]
integrate(log(exp(3)-2/5)^4+4*log(exp(3)-2/5)^3+(-6*x^2+6)*log(exp(3)-2/5) ^2+(-12*x^2+4)*log(exp(3)-2/5)+5*x^4-6*x^2+1,x, algorithm=\
x^5 + x*log(e^3 - 2/5)^4 + 4*x*log(e^3 - 2/5)^3 - 2*x^3 - 2*(x^3 - 3*x)*lo g(e^3 - 2/5)^2 - 4*(x^3 - x)*log(e^3 - 2/5) + x
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.68 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x^{5} + x \log \left (e^{3} - \frac {2}{5}\right )^{4} + 4 \, x \log \left (e^{3} - \frac {2}{5}\right )^{3} - 2 \, x^{3} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (e^{3} - \frac {2}{5}\right )^{2} - 4 \, {\left (x^{3} - x\right )} \log \left (e^{3} - \frac {2}{5}\right ) + x \]
integrate(log(exp(3)-2/5)^4+4*log(exp(3)-2/5)^3+(-6*x^2+6)*log(exp(3)-2/5) ^2+(-12*x^2+4)*log(exp(3)-2/5)+5*x^4-6*x^2+1,x, algorithm=\
x^5 + x*log(e^3 - 2/5)^4 + 4*x*log(e^3 - 2/5)^3 - 2*x^3 - 2*(x^3 - 3*x)*lo g(e^3 - 2/5)^2 - 4*(x^3 - x)*log(e^3 - 2/5) + x
Time = 11.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \left (1-6 x^2+5 x^4+\left (4-12 x^2\right ) \log \left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\left (6-6 x^2\right ) \log ^2\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+4 \log ^3\left (\frac {1}{5} \left (-2+5 e^3\right )\right )+\log ^4\left (\frac {1}{5} \left (-2+5 e^3\right )\right )\right ) \, dx=x^5+\left (-4\,\ln \left ({\mathrm {e}}^3-\frac {2}{5}\right )-2\,{\ln \left ({\mathrm {e}}^3-\frac {2}{5}\right )}^2-2\right )\,x^3+\left (4\,\ln \left ({\mathrm {e}}^3-\frac {2}{5}\right )+6\,{\ln \left ({\mathrm {e}}^3-\frac {2}{5}\right )}^2+4\,{\ln \left ({\mathrm {e}}^3-\frac {2}{5}\right )}^3+{\ln \left ({\mathrm {e}}^3-\frac {2}{5}\right )}^4+1\right )\,x \]
int(4*log(exp(3) - 2/5)^3 - log(exp(3) - 2/5)^2*(6*x^2 - 6) - log(exp(3) - 2/5)*(12*x^2 - 4) + log(exp(3) - 2/5)^4 - 6*x^2 + 5*x^4 + 1,x)