Integrand size = 47, antiderivative size = 26 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {x^3 \left (e^3+x\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \] Output:
1/2*x^3*(exp(3)+x)*ln(1/12*(exp(4)-1)/x)/ln(2)
Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {e^3 x^3 \log \left (-\frac {1-e^4}{12 x}\right )+x^4 \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \] Input:
Integrate[(-(E^3*x^2) - x^3 + (3*E^3*x^2 + 4*x^3)*Log[(-1 + E^4)/(12*x)])/ Log[4],x]
Output:
(E^3*x^3*Log[-1/12*(1 - E^4)/x] + x^4*Log[(-1 + E^4)/(12*x)])/Log[4]
Time = 0.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {27, 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.
\(\displaystyle \int \frac {-x^3-e^3 x^2+\left (4 x^3+3 e^3 x^2\right ) \log \left (\frac {e^4-1}{12 x}\right )}{\log (4)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (-x^3-e^3 x^2+\left (4 x^3+3 e^3 x^2\right ) \log \left (-\frac {1-e^4}{12 x}\right )\right )dx}{\log (4)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (x^4+e^3 x^3\right ) \log \left (-\frac {1-e^4}{12 x}\right )}{\log (4)}\) |
Input:
Int[(-(E^3*x^2) - x^3 + (3*E^3*x^2 + 4*x^3)*Log[(-1 + E^4)/(12*x)])/Log[4] ,x]
Output:
((E^3*x^3 + x^4)*Log[-1/12*(1 - E^4)/x])/Log[4]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {\left (x^{3} {\mathrm e}^{3}+x^{4}\right ) \ln \left (\frac {{\mathrm e}^{4}-1}{12 x}\right )}{2 \ln \left (2\right )}\) | \(27\) |
parallelrisch | \(\frac {{\mathrm e}^{3} x^{3} \ln \left (\frac {{\mathrm e}^{4}-1}{12 x}\right )+x^{4} \ln \left (\frac {{\mathrm e}^{4}-1}{12 x}\right )}{2 \ln \left (2\right )}\) | \(38\) |
norman | \(\frac {x^{4} \ln \left (\frac {{\mathrm e}^{4}-1}{12 x}\right )}{2 \ln \left (2\right )}+\frac {{\mathrm e}^{3} x^{3} \ln \left (\frac {{\mathrm e}^{4}-1}{12 x}\right )}{2 \ln \left (2\right )}\) | \(42\) |
parts | \(-\frac {x^{4}}{8 \ln \left (2\right )}-\frac {{\mathrm e}^{3} x^{3}}{6 \ln \left (2\right )}-\frac {\left ({\mathrm e}^{4}-1\right )^{3} \left (3 \,{\mathrm e}^{3} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )+\frac {{\mathrm e}^{4} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{3}+\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{12 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}+\frac {x^{4}}{48 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{3456 \ln \left (2\right )}\) | \(156\) |
default | \(\frac {-\left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right ) \left (\frac {{\mathrm e}^{3} {\mathrm e}^{8} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )}{48}+\frac {{\mathrm e}^{12} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{432}-\frac {{\mathrm e}^{8} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{144}-\frac {{\mathrm e}^{3} {\mathrm e}^{4} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )}{24}+\frac {{\mathrm e}^{4} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{144}+\frac {{\mathrm e}^{3} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )}{48}+\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{1728 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}+\frac {x^{4}}{6912 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )-\frac {x^{4}}{4}-\frac {x^{3} {\mathrm e}^{3}}{3}}{2 \ln \left (2\right )}\) | \(328\) |
derivativedivides | \(-\frac {\left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right ) \left (\frac {{\mathrm e}^{3} {\mathrm e}^{8} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )}{48}+\frac {{\mathrm e}^{12} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{432}+\frac {{\mathrm e}^{3} {\mathrm e}^{8} x^{3}}{432 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {{\mathrm e}^{8} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{144}+\frac {{\mathrm e}^{12} x^{4}}{6912 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {{\mathrm e}^{8} x^{4}}{2304 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {{\mathrm e}^{3} {\mathrm e}^{4} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )}{24}-\frac {{\mathrm e}^{3} {\mathrm e}^{4} x^{3}}{216 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}+\frac {{\mathrm e}^{4} \left (-\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{4 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}-\frac {x^{4}}{16 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{144}+\frac {{\mathrm e}^{4} x^{4}}{2304 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}+\frac {{\mathrm e}^{3} \left (-\frac {x^{3} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{3 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}-\frac {x^{3}}{9 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}\right )}{48}+\frac {{\mathrm e}^{3} x^{3}}{432 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{3}}+\frac {x^{4} \ln \left (\frac {\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}}{x}\right )}{1728 \left (\frac {{\mathrm e}^{4}}{12}-\frac {1}{12}\right )^{4}}\right )}{2 \ln \left (2\right )}\) | \(400\) |
Input:
int(1/2*((3*x^2*exp(3)+4*x^3)*ln(1/12*(exp(4)-1)/x)-x^2*exp(3)-x^3)/ln(2), x,method=_RETURNVERBOSE)
Output:
1/2/ln(2)*(x^3*exp(3)+x^4)*ln(1/12*(exp(4)-1)/x)
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {{\left (x^{4} + x^{3} e^{3}\right )} \log \left (\frac {e^{4} - 1}{12 \, x}\right )}{2 \, \log \left (2\right )} \] Input:
integrate(1/2*((3*x^2*exp(3)+4*x^3)*log(1/12*(exp(4)-1)/x)-x^2*exp(3)-x^3) /log(2),x, algorithm="fricas")
Output:
1/2*(x^4 + x^3*e^3)*log(1/12*(e^4 - 1)/x)/log(2)
Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {\left (x^{4} + x^{3} e^{3}\right ) \log {\left (\frac {- \frac {1}{12} + \frac {e^{4}}{12}}{x} \right )}}{2 \log {\left (2 \right )}} \] Input:
integrate(1/2*((3*x**2*exp(3)+4*x**3)*ln(1/12*(exp(4)-1)/x)-x**2*exp(3)-x* *3)/ln(2),x)
Output:
(x**4 + x**3*exp(3))*log((-1/12 + exp(4)/12)/x)/(2*log(2))
Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {{\left (x^{4} + x^{3} e^{3}\right )} \log \left (\frac {e^{4} - 1}{12 \, x}\right )}{2 \, \log \left (2\right )} \] Input:
integrate(1/2*((3*x^2*exp(3)+4*x^3)*log(1/12*(exp(4)-1)/x)-x^2*exp(3)-x^3) /log(2),x, algorithm="maxima")
Output:
1/2*(x^4 + x^3*e^3)*log(1/12*(e^4 - 1)/x)/log(2)
Leaf count of result is larger than twice the leaf count of optimal. 156 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 6.00 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=-\frac {3 \, x^{4} + 4 \, x^{3} e^{3} - \frac {12 \, {\left (\frac {x^{4}}{e^{4} - 1} + \frac {x^{3} e^{3}}{e^{4} - 1}\right )} {\left (e^{4} - 1\right )}^{2} \log \left (\frac {e^{4} - 1}{12 \, x}\right ) + \frac {x^{4} {\left (\frac {4 \, {\left (e^{4} - 1\right )} e^{19}}{x} - \frac {16 \, {\left (e^{4} - 1\right )} e^{15}}{x} + \frac {24 \, {\left (e^{4} - 1\right )} e^{11}}{x} - \frac {16 \, {\left (e^{4} - 1\right )} e^{7}}{x} + \frac {4 \, {\left (e^{4} - 1\right )} e^{3}}{x} + 3 \, e^{20} - 15 \, e^{16} + 30 \, e^{12} - 30 \, e^{8} + 15 \, e^{4} - 3\right )}}{{\left (e^{4} - 1\right )}^{4}}}{e^{4} - 1}}{24 \, \log \left (2\right )} \] Input:
integrate(1/2*((3*x^2*exp(3)+4*x^3)*log(1/12*(exp(4)-1)/x)-x^2*exp(3)-x^3) /log(2),x, algorithm="giac")
Output:
-1/24*(3*x^4 + 4*x^3*e^3 - (12*(x^4/(e^4 - 1) + x^3*e^3/(e^4 - 1))*(e^4 - 1)^2*log(1/12*(e^4 - 1)/x) + x^4*(4*(e^4 - 1)*e^19/x - 16*(e^4 - 1)*e^15/x + 24*(e^4 - 1)*e^11/x - 16*(e^4 - 1)*e^7/x + 4*(e^4 - 1)*e^3/x + 3*e^20 - 15*e^16 + 30*e^12 - 30*e^8 + 15*e^4 - 3)/(e^4 - 1)^4)/(e^4 - 1))/log(2)
Time = 2.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {x^3\,\left (\ln \left (\frac {1}{x}\right )+\ln \left (\frac {{\mathrm {e}}^4}{12}-\frac {1}{12}\right )\right )\,\left (x+{\mathrm {e}}^3\right )}{2\,\ln \left (2\right )} \] Input:
int(-((x^2*exp(3))/2 - (log((exp(4)/12 - 1/12)/x)*(3*x^2*exp(3) + 4*x^3))/ 2 + x^3/2)/log(2),x)
Output:
(x^3*(log(1/x) + log(exp(4)/12 - 1/12))*(x + exp(3)))/(2*log(2))
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-e^3 x^2-x^3+\left (3 e^3 x^2+4 x^3\right ) \log \left (\frac {-1+e^4}{12 x}\right )}{\log (4)} \, dx=\frac {\mathrm {log}\left (\frac {e^{4}-1}{12 x}\right ) x^{3} \left (e^{3}+x \right )}{2 \,\mathrm {log}\left (2\right )} \] Input:
int(1/2*((3*x^2*exp(3)+4*x^3)*log(1/12*(exp(4)-1)/x)-x^2*exp(3)-x^3)/log(2 ),x)
Output:
(log((e**4 - 1)/(12*x))*x**3*(e**3 + x))/(2*log(2))