Integrand size = 94, antiderivative size = 27 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=2 \log \left (10+\frac {20 x}{e^{4+\frac {4}{e^3}}-x^2+\log (4)}\right ) \]
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 173, normalized size of antiderivative = 6.41 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=-4 \text {RootSum}\left [e^{8+\frac {8}{e^3}}+2 e^{4+\frac {4}{e^3}} \log (4)+\log ^2(4)+2 e^{4+\frac {4}{e^3}} \text {$\#$1}+\log (16) \text {$\#$1}-2 e^{4+\frac {4}{e^3}} \text {$\#$1}^2-2 \log (4) \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {e^{4+\frac {4}{e^3}} \log (x-\text {$\#$1})+\log (4) \log (x-\text {$\#$1})+\log (x-\text {$\#$1}) \text {$\#$1}^2}{-2 e^{4+\frac {4}{e^3}}-\log (16)+4 e^{4+\frac {4}{e^3}} \text {$\#$1}+4 \log (4) \text {$\#$1}+6 \text {$\#$1}^2-4 \text {$\#$1}^3}\&\right ] \]
Integrate[(4*E^((4 + 4*E^3)/E^3) + 4*x^2 + 4*Log[4])/(E^((2*(4 + 4*E^3))/E ^3) - 2*x^3 + x^4 + (2*x - 2*x^2)*Log[4] + Log[4]^2 + E^((4 + 4*E^3)/E^3)* (2*x - 2*x^2 + 2*Log[4])),x]
-4*RootSum[E^(8 + 8/E^3) + 2*E^(4 + 4/E^3)*Log[4] + Log[4]^2 + 2*E^(4 + 4/ E^3)*#1 + Log[16]*#1 - 2*E^(4 + 4/E^3)*#1^2 - 2*Log[4]*#1^2 - 2*#1^3 + #1^ 4 & , (E^(4 + 4/E^3)*Log[x - #1] + Log[4]*Log[x - #1] + Log[x - #1]*#1^2)/ (-2*E^(4 + 4/E^3) - Log[16] + 4*E^(4 + 4/E^3)*#1 + 4*Log[4]*#1 + 6*#1^2 - 4*#1^3) & ]
Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2462, 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.
\(\displaystyle \int \frac {4 x^2+4 e^{\frac {4+4 e^3}{e^3}}+4 \log (4)}{x^4-2 x^3+e^{\frac {4+4 e^3}{e^3}} \left (-2 x^2+2 x+2 \log (4)\right )+\left (2 x-2 x^2\right ) \log (4)+e^{\frac {2 \left (4+4 e^3\right )}{e^3}}+\log ^2(4)} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {4 (x-1)}{x^2-2 x-e^{4+\frac {4}{e^3}}-\log (4)}-\frac {4 x}{x^2-e^{4+\frac {4}{e^3}}-\log (4)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \log \left (-x^2+2 x+e^{4+\frac {4}{e^3}}+\log (4)\right )-2 \log \left (-x^2+e^{4+\frac {4}{e^3}}+\log (4)\right )\) |
Int[(4*E^((4 + 4*E^3)/E^3) + 4*x^2 + 4*Log[4])/(E^((2*(4 + 4*E^3))/E^3) - 2*x^3 + x^4 + (2*x - 2*x^2)*Log[4] + Log[4]^2 + E^((4 + 4*E^3)/E^3)*(2*x - 2*x^2 + 2*Log[4])),x]
3.3.32.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 1.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.67
method | result | size |
risch | \(2 \ln \left ({\mathrm e}^{4 \,{\mathrm e}^{-3}+4}-x^{2}+2 \ln \left (2\right )+2 x \right )-2 \ln \left ({\mathrm e}^{4 \,{\mathrm e}^{-3}+4}-x^{2}+2 \ln \left (2\right )\right )\) | \(45\) |
parallelrisch | \(-2 \ln \left (x^{2}-{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}-2 \ln \left (2\right )\right )+2 \ln \left (x^{2}-{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}-2 \ln \left (2\right )-2 x \right )\) | \(53\) |
norman | \(-2 \ln \left (-x^{2}+{\mathrm e}^{\left (4 \,{\mathrm e}^{3}+4\right ) {\mathrm e}^{-3}}+2 \ln \left (2\right )\right )+2 \ln \left (-x^{2}+{\mathrm e}^{\left (4 \,{\mathrm e}^{3}+4\right ) {\mathrm e}^{-3}}+2 \ln \left (2\right )+2 x \right )\) | \(55\) |
default | \(2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}+\left (-4 \ln \left (2\right )-2 \,{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}\right ) \textit {\_Z}^{2}+\left (4 \ln \left (2\right )+2 \,{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}\right ) \textit {\_Z} +4 \ln \left (2\right )^{2}+4 \ln \left (2\right ) {\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}+{\mathrm e}^{8 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}\right )}{\sum }\frac {\left (\textit {\_R}^{2}+2 \ln \left (2\right )+{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}-4 \textit {\_R} \ln \left (2\right )-2 \,{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}} \textit {\_R} -3 \textit {\_R}^{2}+2 \ln \left (2\right )+{\mathrm e}^{4 \left ({\mathrm e}^{3}+1\right ) {\mathrm e}^{-3}}}\right )\) | \(149\) |
int((4*exp((4*exp(3)+4)/exp(3))+8*ln(2)+4*x^2)/(exp((4*exp(3)+4)/exp(3))^2 +(4*ln(2)-2*x^2+2*x)*exp((4*exp(3)+4)/exp(3))+4*ln(2)^2+2*(-2*x^2+2*x)*ln( 2)+x^4-2*x^3),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=2 \, \log \left (x^{2} - 2 \, x - e^{\left (4 \, {\left (e^{3} + 1\right )} e^{\left (-3\right )}\right )} - 2 \, \log \left (2\right )\right ) - 2 \, \log \left (x^{2} - e^{\left (4 \, {\left (e^{3} + 1\right )} e^{\left (-3\right )}\right )} - 2 \, \log \left (2\right )\right ) \]
integrate((4*exp((4*exp(3)+4)/exp(3))+8*log(2)+4*x^2)/(exp((4*exp(3)+4)/ex p(3))^2+(4*log(2)-2*x^2+2*x)*exp((4*exp(3)+4)/exp(3))+4*log(2)^2+2*(-2*x^2 +2*x)*log(2)+x^4-2*x^3),x, algorithm=\
2*log(x^2 - 2*x - e^(4*(e^3 + 1)*e^(-3)) - 2*log(2)) - 2*log(x^2 - e^(4*(e ^3 + 1)*e^(-3)) - 2*log(2))
Time = 1.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=- 2 \log {\left (x^{2} - e^{4} e^{\frac {4}{e^{3}}} - 2 \log {\left (2 \right )} \right )} + 2 \log {\left (x^{2} - 2 x - e^{4} e^{\frac {4}{e^{3}}} - 2 \log {\left (2 \right )} \right )} \]
integrate((4*exp((4*exp(3)+4)/exp(3))+8*ln(2)+4*x**2)/(exp((4*exp(3)+4)/ex p(3))**2+(4*ln(2)-2*x**2+2*x)*exp((4*exp(3)+4)/exp(3))+4*ln(2)**2+2*(-2*x* *2+2*x)*ln(2)+x**4-2*x**3),x)
-2*log(x**2 - exp(4)*exp(4*exp(-3)) - 2*log(2)) + 2*log(x**2 - 2*x - exp(4 )*exp(4*exp(-3)) - 2*log(2))
Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.78 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=2 \, \log \left (x^{2} - 2 \, x - e^{\left (4 \, {\left (e^{3} + 1\right )} e^{\left (-3\right )}\right )} - 2 \, \log \left (2\right )\right ) - 2 \, \log \left (x^{2} - e^{\left (4 \, {\left (e^{3} + 1\right )} e^{\left (-3\right )}\right )} - 2 \, \log \left (2\right )\right ) \]
integrate((4*exp((4*exp(3)+4)/exp(3))+8*log(2)+4*x^2)/(exp((4*exp(3)+4)/ex p(3))^2+(4*log(2)-2*x^2+2*x)*exp((4*exp(3)+4)/exp(3))+4*log(2)^2+2*(-2*x^2 +2*x)*log(2)+x^4-2*x^3),x, algorithm=\
2*log(x^2 - 2*x - e^(4*(e^3 + 1)*e^(-3)) - 2*log(2)) - 2*log(x^2 - e^(4*(e ^3 + 1)*e^(-3)) - 2*log(2))
Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=-2.00000000000000 \, \log \left (x + 8.24717230730000\right ) + 2.00000000000000 \, \log \left (x + 7.30757793020000\right ) - 2.00000000000000 \, \log \left (x - 8.24717230730000\right ) + 2.00000000000000 \, \log \left (x - 9.30757793020000\right ) \]
integrate((4*exp((4*exp(3)+4)/exp(3))+8*log(2)+4*x^2)/(exp((4*exp(3)+4)/ex p(3))^2+(4*log(2)-2*x^2+2*x)*exp((4*exp(3)+4)/exp(3))+4*log(2)^2+2*(-2*x^2 +2*x)*log(2)+x^4-2*x^3),x, algorithm=\
-2.00000000000000*log(x + 8.24717230730000) + 2.00000000000000*log(x + 7.3 0757793020000) - 2.00000000000000*log(x - 8.24717230730000) + 2.0000000000 0000*log(x - 9.30757793020000)
Time = 16.30 (sec) , antiderivative size = 11837, normalized size of antiderivative = 438.41 \[ \int \frac {4 e^{\frac {4+4 e^3}{e^3}}+4 x^2+4 \log (4)}{e^{\frac {2 \left (4+4 e^3\right )}{e^3}}-2 x^3+x^4+\left (2 x-2 x^2\right ) \log (4)+\log ^2(4)+e^{\frac {4+4 e^3}{e^3}} \left (2 x-2 x^2+2 \log (4)\right )} \, dx=\text {Too large to display} \]
int((4*exp(exp(-3)*(4*exp(3) + 4)) + 8*log(2) + 4*x^2)/(exp(2*exp(-3)*(4*e xp(3) + 4)) + 2*log(2)*(2*x - 2*x^2) + 4*log(2)^2 - 2*x^3 + x^4 + exp(exp( -3)*(4*exp(3) + 4))*(2*x + 4*log(2) - 2*x^2)),x)
symsum(log(- 256*exp(8*exp(-3) + 8) - 16*log(16)*log(256) - 256*exp(4*exp( -3) + 4)*log(2) - 64*exp(4*exp(-3) + 4)*log(16) - 64*exp(4*exp(-3) + 4)*lo g(256) - x*(64*log(16) + 32*log(256) + 256*exp(4*exp(-3) + 4)) - 256*log(2 )^2 - 4*log(256)^2 - root(4096*z^4*exp(4*exp(-3)*(exp(3) + 1))*log(2)^3*lo g(16)^2 + 1536*z^4*exp(8*exp(-3)*(exp(3) + 1))*log(2)^2*log(16)^2 - 768*z^ 4*exp(4*exp(-3)*(exp(3) + 1))*log(2)^2*log(16)^2 - 512*z^4*exp(4*exp(-3)*( exp(3) + 1))*log(2)^2*log(16)^3 + 16384*z^4*exp(8*exp(-3)*(exp(3) + 1))*lo g(2)^3*log(16) + 10240*z^4*exp(12*exp(-3)*(exp(3) + 1))*log(2)^2*log(16) + 8192*z^4*exp(4*exp(-3)*(exp(3) + 1))*log(2)^4*log(16) + 6144*z^4*exp(4*ex p(-3)*(exp(3) + 1))*log(2)^3*log(16) + 3072*z^4*exp(8*exp(-3)*(exp(3) + 1) )*log(2)^2*log(16) - 1920*z^4*exp(4*exp(-3)*(exp(3) + 1))*log(2)^2*log(16) - 768*z^4*exp(8*exp(-3)*(exp(3) + 1))*log(2)*log(16)^2 - 512*z^4*exp(12*e xp(-3)*(exp(3) + 1))*log(2)*log(16)^2 - 512*z^4*exp(8*exp(-3)*(exp(3) + 1) )*log(2)*log(16)^3 - 480*z^4*exp(4*exp(-3)*(exp(3) + 1))*log(2)*log(16)^2 - 128*z^4*exp(4*exp(-3)*(exp(3) + 1))*log(2)*log(16)^3 - 64*z^4*exp(4*exp( -3)*(exp(3) + 1))*log(2)*log(16)^4 + 2048*z^4*exp(16*exp(-3)*(exp(3) + 1)) *log(2)*log(16) - 1920*z^4*exp(8*exp(-3)*(exp(3) + 1))*log(2)*log(16) - 53 248*z^4*exp(8*exp(-3)*(exp(3) + 1))*log(2)^4 - 49152*z^4*exp(4*exp(-3)*(ex p(3) + 1))*log(2)^5 - 24576*z^4*exp(12*exp(-3)*(exp(3) + 1))*log(2)^3 + 13 824*z^4*exp(4*exp(-3)*(exp(3) + 1))*log(2)^3 + 12288*z^4*exp(8*exp(-3)*...