Optimal. Leaf size=30 \[ x+e^5 \left (\log \left (x^2\right )+\log ^2\left (\frac {1}{3} \left (1+\log \left (\left (-x+x^2\right )^2\right )\right )\right )\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.80, antiderivative size = 34, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, integrand size = 106, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {6741, 6742, 43, 6684, 6686} \begin {gather*} e^5 \log ^2\left (\frac {1}{3} \left (\log \left ((1-x)^2 x^2\right )+1\right )\right )+x+2 e^5 \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 6684
Rule 6686
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x-x^2-e^5 (-2+2 x)-\left (-x+x^2+e^5 (-2+2 x)\right ) \log \left (x^2-2 x^3+x^4\right )-e^5 (-4+8 x) \log \left (\frac {1}{3} \left (1+\log \left (x^2-2 x^3+x^4\right )\right )\right )}{(1-x) x \left (1+\log \left ((-1+x)^2 x^2\right )\right )} \, dx\\ &=\int \left (\frac {2 e^5+x}{x}+\frac {4 e^5 (-1+2 x) \log \left (\frac {1}{3} \left (1+\log \left ((-1+x)^2 x^2\right )\right )\right )}{(-1+x) x \left (1+\log \left ((-1+x)^2 x^2\right )\right )}\right ) \, dx\\ &=\left (4 e^5\right ) \int \frac {(-1+2 x) \log \left (\frac {1}{3} \left (1+\log \left ((-1+x)^2 x^2\right )\right )\right )}{(-1+x) x \left (1+\log \left ((-1+x)^2 x^2\right )\right )} \, dx+\int \frac {2 e^5+x}{x} \, dx\\ &=e^5 \log ^2\left (\frac {1}{3} \left (1+\log \left ((1-x)^2 x^2\right )\right )\right )+\int \left (1+\frac {2 e^5}{x}\right ) \, dx\\ &=x+2 e^5 \log (x)+e^5 \log ^2\left (\frac {1}{3} \left (1+\log \left ((1-x)^2 x^2\right )\right )\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 32, normalized size = 1.07 \begin {gather*} x+2 e^5 \log (x)+e^5 \log ^2\left (\frac {1}{3} \left (1+\log \left ((-1+x)^2 x^2\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 31, normalized size = 1.03 \begin {gather*} e^{5} \log \left (\frac {1}{3} \, \log \left (x^{4} - 2 \, x^{3} + x^{2}\right ) + \frac {1}{3}\right )^{2} + 2 \, e^{5} \log \relax (x) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.78, size = 51, normalized size = 1.70 \begin {gather*} -2 \, e^{5} \log \relax (3) \log \left (\log \left (x^{4} - 2 \, x^{3} + x^{2}\right ) + 1\right ) + e^{5} \log \left (\log \left (x^{4} - 2 \, x^{3} + x^{2}\right ) + 1\right )^{2} + 2 \, e^{5} \log \relax (x) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 32, normalized size = 1.07
method | result | size |
norman | \(x +{\mathrm e}^{5} \ln \left (\frac {\ln \left (x^{4}-2 x^{3}+x^{2}\right )}{3}+\frac {1}{3}\right )^{2}+2 \,{\mathrm e}^{5} \ln \relax (x )\) | \(32\) |
default | \(x +2 \,{\mathrm e}^{5} \ln \relax (x )-2 \,{\mathrm e}^{5} \ln \relax (3) \ln \left (\ln \left (x^{4}-2 x^{3}+x^{2}\right )+1\right )+{\mathrm e}^{5} \ln \left (\ln \left (x^{4}-2 x^{3}+x^{2}\right )+1\right )^{2}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.93, size = 45, normalized size = 1.50 \begin {gather*} -2 \, e^{5} \log \relax (3) \log \left (2 \, \log \left (x - 1\right ) + 2 \, \log \relax (x) + 1\right ) + e^{5} \log \left (2 \, \log \left (x - 1\right ) + 2 \, \log \relax (x) + 1\right )^{2} + 2 \, e^{5} \log \relax (x) + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.80, size = 31, normalized size = 1.03 \begin {gather*} {\mathrm {e}}^5\,{\ln \left (\frac {\ln \left (x^4-2\,x^3+x^2\right )}{3}+\frac {1}{3}\right )}^2+x+2\,{\mathrm {e}}^5\,\ln \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.59, size = 34, normalized size = 1.13 \begin {gather*} x + 2 e^{5} \log {\relax (x )} + e^{5} \log {\left (\frac {\log {\left (x^{4} - 2 x^{3} + x^{2} \right )}}{3} + \frac {1}{3} \right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________