Optimal. Leaf size=24 \[ 2 x \left (1-\log (x)+\frac {e^x \log (-5+x) \log \left (x^2\right )}{x}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.79, antiderivative size = 22, normalized size of antiderivative = 0.92, number of steps used = 14, number of rules used = 10, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 6688, 2295, 2178, 2554, 12, 2288, 6742, 2194, 2557} \begin {gather*} 2 e^x \log (x-5) \log \left (x^2\right )+2 x-2 x \log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 1593
Rule 2178
Rule 2194
Rule 2288
Rule 2295
Rule 2554
Rule 2557
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x (-20+4 x) \log (-5+x)+\left (10 x-2 x^2\right ) \log (x)+\left (2 e^x x+e^x \left (-10 x+2 x^2\right ) \log (-5+x)\right ) \log \left (x^2\right )}{(-5+x) x} \, dx\\ &=\int \left (-2 \log (x)+\frac {2 e^x \log \left (x^2\right )}{-5+x}+\frac {2 e^x \log (-5+x) \left (2+x \log \left (x^2\right )\right )}{x}\right ) \, dx\\ &=-(2 \int \log (x) \, dx)+2 \int \frac {e^x \log \left (x^2\right )}{-5+x} \, dx+2 \int \frac {e^x \log (-5+x) \left (2+x \log \left (x^2\right )\right )}{x} \, dx\\ &=2 x-2 x \log (x)+2 e^5 \text {Ei}(-5+x) \log \left (x^2\right )-2 \int \frac {2 e^5 \text {Ei}(-5+x)}{x} \, dx+2 \int \left (\frac {2 e^x \log (-5+x)}{x}+e^x \log (-5+x) \log \left (x^2\right )\right ) \, dx\\ &=2 x-2 x \log (x)+2 e^5 \text {Ei}(-5+x) \log \left (x^2\right )+2 \int e^x \log (-5+x) \log \left (x^2\right ) \, dx+4 \int \frac {e^x \log (-5+x)}{x} \, dx-\left (4 e^5\right ) \int \frac {\text {Ei}(-5+x)}{x} \, dx\\ &=2 x+4 \text {Ei}(x) \log (-5+x)-2 x \log (x)+2 e^5 \text {Ei}(-5+x) \log \left (x^2\right )+2 e^x \log (-5+x) \log \left (x^2\right )-2 \int \frac {2 e^x \log (-5+x)}{x} \, dx-2 \int \frac {e^x \log \left (x^2\right )}{-5+x} \, dx-4 \int \frac {\text {Ei}(x)}{-5+x} \, dx-\left (4 e^5\right ) \int \frac {\text {Ei}(-5+x)}{x} \, dx\\ &=2 x+4 \text {Ei}(x) \log (-5+x)-2 x \log (x)+2 e^x \log (-5+x) \log \left (x^2\right )+2 \int \frac {2 e^5 \text {Ei}(-5+x)}{x} \, dx-4 \int \frac {\text {Ei}(x)}{-5+x} \, dx-4 \int \frac {e^x \log (-5+x)}{x} \, dx-\left (4 e^5\right ) \int \frac {\text {Ei}(-5+x)}{x} \, dx\\ &=2 x-2 x \log (x)+2 e^x \log (-5+x) \log \left (x^2\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 21, normalized size = 0.88 \begin {gather*} 2 \left (x-x \log (x)+e^x \log (-5+x) \log \left (x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.83, size = 20, normalized size = 0.83 \begin {gather*} 2 \, {\left (2 \, e^{x} \log \left (x - 5\right ) - x\right )} \log \relax (x) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 21, normalized size = 0.88 \begin {gather*} 2 \, e^{x} \log \left (x^{2}\right ) \log \left (x - 5\right ) - 2 \, x \log \relax (x) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.32, size = 77, normalized size = 3.21
method | result | size |
risch | \(\left (-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x}+4 \,{\mathrm e}^{x} \ln \relax (x )\right ) \ln \left (x -5\right )-2 x \ln \relax (x )+2 x\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 19, normalized size = 0.79 \begin {gather*} 4 \, e^{x} \log \left (x - 5\right ) \log \relax (x) - 2 \, x \log \relax (x) + 2 \, x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.05, size = 21, normalized size = 0.88 \begin {gather*} 2\,x-2\,x\,\ln \relax (x)+2\,\ln \left (x-5\right )\,\ln \left (x^2\right )\,{\mathrm {e}}^x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.89, size = 22, normalized size = 0.92 \begin {gather*} - 2 x \log {\relax (x )} + 2 x + 4 e^{x} \log {\relax (x )} \log {\left (x - 5 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________