Optimal. Leaf size=22 \[ -\frac {4 e^{2 x} x^2 \log (2)}{3-x}+\log (x) \]
________________________________________________________________________________________
Rubi [A] time = 0.53, antiderivative size = 38, normalized size of antiderivative = 1.73, number of steps used = 12, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {1594, 27, 6742, 2199, 2194, 2177, 2178, 2176} \begin {gather*} 4 e^{2 x} x \log (2)+\log (x)+12 e^{2 x} \log (2)-\frac {36 e^{2 x} \log (2)}{3-x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 27
Rule 1594
Rule 2176
Rule 2177
Rule 2178
Rule 2194
Rule 2199
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {9-6 x+x^2+e^{2 x} \left (-24 x^2-20 x^3+8 x^4\right ) \log (2)}{x \left (9-6 x+x^2\right )} \, dx\\ &=\int \frac {9-6 x+x^2+e^{2 x} \left (-24 x^2-20 x^3+8 x^4\right ) \log (2)}{(-3+x)^2 x} \, dx\\ &=\int \left (\frac {1}{x}+\frac {4 e^{2 x} x \left (-6-5 x+2 x^2\right ) \log (2)}{(-3+x)^2}\right ) \, dx\\ &=\log (x)+(4 \log (2)) \int \frac {e^{2 x} x \left (-6-5 x+2 x^2\right )}{(-3+x)^2} \, dx\\ &=\log (x)+(4 \log (2)) \int \left (7 e^{2 x}-\frac {9 e^{2 x}}{(-3+x)^2}+\frac {18 e^{2 x}}{-3+x}+2 e^{2 x} x\right ) \, dx\\ &=\log (x)+(8 \log (2)) \int e^{2 x} x \, dx+(28 \log (2)) \int e^{2 x} \, dx-(36 \log (2)) \int \frac {e^{2 x}}{(-3+x)^2} \, dx+(72 \log (2)) \int \frac {e^{2 x}}{-3+x} \, dx\\ &=14 e^{2 x} \log (2)-\frac {36 e^{2 x} \log (2)}{3-x}+4 e^{2 x} x \log (2)+72 e^6 \text {Ei}(-2 (3-x)) \log (2)+\log (x)-(4 \log (2)) \int e^{2 x} \, dx-(72 \log (2)) \int \frac {e^{2 x}}{-3+x} \, dx\\ &=12 e^{2 x} \log (2)-\frac {36 e^{2 x} \log (2)}{3-x}+4 e^{2 x} x \log (2)+\log (x)\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 19, normalized size = 0.86 \begin {gather*} \frac {e^{2 x} x^2 \log (16)}{-3+x}+\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 24, normalized size = 1.09 \begin {gather*} \frac {4 \, x^{2} e^{\left (2 \, x\right )} \log \relax (2) + {\left (x - 3\right )} \log \relax (x)}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 26, normalized size = 1.18 \begin {gather*} \frac {4 \, x^{2} e^{\left (2 \, x\right )} \log \relax (2) + x \log \relax (x) - 3 \, \log \relax (x)}{x - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 20, normalized size = 0.91
method | result | size |
norman | \(\frac {4 x^{2} \ln \relax (2) {\mathrm e}^{2 x}}{x -3}+\ln \relax (x )\) | \(20\) |
risch | \(\frac {4 x^{2} \ln \relax (2) {\mathrm e}^{2 x}}{x -3}+\ln \relax (x )\) | \(20\) |
default | \(\ln \relax (x )+\frac {36 \ln \relax (2) {\mathrm e}^{2 x}}{x -3}+12 \ln \relax (2) {\mathrm e}^{2 x}+4 \ln \relax (2) {\mathrm e}^{2 x} x\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.48, size = 19, normalized size = 0.86 \begin {gather*} \frac {4 \, x^{2} e^{\left (2 \, x\right )} \log \relax (2)}{x - 3} + \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 6.55, size = 19, normalized size = 0.86 \begin {gather*} \ln \relax (x)+\frac {4\,x^2\,{\mathrm {e}}^{2\,x}\,\ln \relax (2)}{x-3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.15, size = 19, normalized size = 0.86 \begin {gather*} \frac {4 x^{2} e^{2 x} \log {\relax (2 )}}{x - 3} + \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________