Optimal. Leaf size=31 \[ \frac {10 \left (-2 x+\log \left (\frac {(x+4 (3+x))^2}{e^x+x}\right )\right )}{x \log (x)} \]
________________________________________________________________________________________
Rubi [F] time = 5.62, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {240 x^2+100 x^3+e^x \left (240 x+100 x^2\right )+\left (-120 x+50 x^2+e^x \left (-20 x-50 x^2\right )\right ) \log (x)+\left (e^x (-120-50 x)-120 x-50 x^2+\left (e^x (-120-50 x)-120 x-50 x^2\right ) \log (x)\right ) \log \left (\frac {144+120 x+25 x^2}{e^x+x}\right )}{\left (12 x^3+5 x^4+e^x \left (12 x^2+5 x^3\right )\right ) \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 \left (2 x+\log (x) \left (-\frac {x \left (12-5 x+e^x (2+5 x)\right )}{\left (e^x+x\right ) (12+5 x)}-\log \left (\frac {(12+5 x)^2}{e^x+x}\right )\right )-\log \left (\frac {(12+5 x)^2}{e^x+x}\right )\right )}{x^2 \log ^2(x)} \, dx\\ &=10 \int \frac {2 x+\log (x) \left (-\frac {x \left (12-5 x+e^x (2+5 x)\right )}{\left (e^x+x\right ) (12+5 x)}-\log \left (\frac {(12+5 x)^2}{e^x+x}\right )\right )-\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)} \, dx\\ &=10 \int \left (\frac {-1+x}{x \left (e^x+x\right ) \log (x)}+\frac {24 x+10 x^2-2 x \log (x)-5 x^2 \log (x)-12 \log \left (\frac {(12+5 x)^2}{e^x+x}\right )-5 x \log \left (\frac {(12+5 x)^2}{e^x+x}\right )-12 \log (x) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )-5 x \log (x) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 (12+5 x) \log ^2(x)}\right ) \, dx\\ &=10 \int \frac {-1+x}{x \left (e^x+x\right ) \log (x)} \, dx+10 \int \frac {24 x+10 x^2-2 x \log (x)-5 x^2 \log (x)-12 \log \left (\frac {(12+5 x)^2}{e^x+x}\right )-5 x \log \left (\frac {(12+5 x)^2}{e^x+x}\right )-12 \log (x) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )-5 x \log (x) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 (12+5 x) \log ^2(x)} \, dx\\ &=10 \int \left (\frac {1}{\left (e^x+x\right ) \log (x)}-\frac {1}{x \left (e^x+x\right ) \log (x)}\right ) \, dx+10 \int \frac {(12+5 x) \left (2 x-\log \left (\frac {(12+5 x)^2}{e^x+x}\right )\right )-\log (x) \left (x (2+5 x)+(12+5 x) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )\right )}{x^2 (12+5 x) \log ^2(x)} \, dx\\ &=10 \int \frac {1}{\left (e^x+x\right ) \log (x)} \, dx-10 \int \frac {1}{x \left (e^x+x\right ) \log (x)} \, dx+10 \int \left (\frac {24+10 x-2 \log (x)-5 x \log (x)}{x (12+5 x) \log ^2(x)}-\frac {(1+\log (x)) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)}\right ) \, dx\\ &=10 \int \frac {1}{\left (e^x+x\right ) \log (x)} \, dx-10 \int \frac {1}{x \left (e^x+x\right ) \log (x)} \, dx+10 \int \frac {24+10 x-2 \log (x)-5 x \log (x)}{x (12+5 x) \log ^2(x)} \, dx-10 \int \frac {(1+\log (x)) \log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)} \, dx\\ &=10 \int \left (\frac {2}{x \log ^2(x)}+\frac {-2-5 x}{x (12+5 x) \log (x)}\right ) \, dx+10 \int \frac {1}{\left (e^x+x\right ) \log (x)} \, dx-10 \int \frac {1}{x \left (e^x+x\right ) \log (x)} \, dx-10 \int \left (\frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)}+\frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log (x)}\right ) \, dx\\ &=10 \int \frac {1}{\left (e^x+x\right ) \log (x)} \, dx-10 \int \frac {1}{x \left (e^x+x\right ) \log (x)} \, dx+10 \int \frac {-2-5 x}{x (12+5 x) \log (x)} \, dx-10 \int \frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)} \, dx-10 \int \frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log (x)} \, dx+20 \int \frac {1}{x \log ^2(x)} \, dx\\ &=10 \int \frac {1}{\left (e^x+x\right ) \log (x)} \, dx-10 \int \frac {1}{x \left (e^x+x\right ) \log (x)} \, dx+10 \int \frac {-2-5 x}{x (12+5 x) \log (x)} \, dx-10 \int \frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)} \, dx-10 \int \frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log (x)} \, dx+20 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )\\ &=-\frac {20}{\log (x)}+10 \int \frac {1}{\left (e^x+x\right ) \log (x)} \, dx-10 \int \frac {1}{x \left (e^x+x\right ) \log (x)} \, dx+10 \int \frac {-2-5 x}{x (12+5 x) \log (x)} \, dx-10 \int \frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log ^2(x)} \, dx-10 \int \frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x^2 \log (x)} \, dx\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 33, normalized size = 1.06 \begin {gather*} 10 \left (-\frac {2}{\log (x)}+\frac {\log \left (\frac {(12+5 x)^2}{e^x+x}\right )}{x \log (x)}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.05, size = 33, normalized size = 1.06 \begin {gather*} -\frac {10 \, {\left (2 \, x - \log \left (\frac {25 \, x^{2} + 120 \, x + 144}{x + e^{x}}\right )\right )}}{x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.35, size = 31, normalized size = 1.00 \begin {gather*} -\frac {10 \, {\left (2 \, x - \log \left (25 \, x^{2} + 120 \, x + 144\right ) + \log \left (x + e^{x}\right )\right )}}{x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.18, size = 218, normalized size = 7.03
method | result | size |
risch | \(-\frac {10 \ln \left ({\mathrm e}^{x}+x \right )}{x \ln \relax (x )}+\frac {-5 i \pi \mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )\right )^{2} \mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )^{2}\right )+10 i \pi \,\mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )\right ) \mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )^{2}\right )^{2}-5 i \pi \mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )^{2}\right )^{3}-5 i \pi \,\mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )^{2}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}+x}\right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {12}{5}\right )^{2}}{{\mathrm e}^{x}+x}\right )+5 i \pi \,\mathrm {csgn}\left (i \left (x +\frac {12}{5}\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {12}{5}\right )^{2}}{{\mathrm e}^{x}+x}\right )^{2}+5 i \pi \,\mathrm {csgn}\left (\frac {i}{{\mathrm e}^{x}+x}\right ) \mathrm {csgn}\left (\frac {i \left (x +\frac {12}{5}\right )^{2}}{{\mathrm e}^{x}+x}\right )^{2}-5 i \pi \mathrm {csgn}\left (\frac {i \left (x +\frac {12}{5}\right )^{2}}{{\mathrm e}^{x}+x}\right )^{3}+20 \ln \relax (5)-20 x +20 \ln \left (x +\frac {12}{5}\right )}{x \ln \relax (x )}\) | \(218\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.40, size = 26, normalized size = 0.84 \begin {gather*} -\frac {10 \, {\left (2 \, x - 2 \, \log \left (5 \, x + 12\right ) + \log \left (x + e^{x}\right )\right )}}{x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.75, size = 33, normalized size = 1.06 \begin {gather*} -\frac {10\,\left (2\,x-\ln \left (\frac {25\,x^2+120\,x+144}{x+{\mathrm {e}}^x}\right )\right )}{x\,\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.78, size = 27, normalized size = 0.87 \begin {gather*} - \frac {20}{\log {\relax (x )}} + \frac {10 \log {\left (\frac {25 x^{2} + 120 x + 144}{x + e^{x}} \right )}}{x \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________