Optimal. Leaf size=21 \[ \frac {2+e^4+\frac {2 x^2}{(16+\log (2))^4}}{\log (x)} \]
________________________________________________________________________________________
Rubi [A] time = 0.57, antiderivative size = 26, normalized size of antiderivative = 1.24, number of steps used = 17, number of rules used = 10, integrand size = 101, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.099, Rules used = {6, 12, 6688, 6742, 2353, 2306, 2309, 2178, 2302, 30} \begin {gather*} \frac {2 x^2}{(16+\log (2))^4 \log (x)}+\frac {2+e^4}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 6
Rule 12
Rule 30
Rule 2178
Rule 2302
Rule 2306
Rule 2309
Rule 2353
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{\left (1536 x \log ^2(2)+64 x \log ^3(2)+x \log ^4(2)+x (65536+16384 \log (2))\right ) \log ^2(x)} \, dx\\ &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{\left (x \log ^4(2)+x (65536+16384 \log (2))+x \left (1536 \log ^2(2)+64 \log ^3(2)\right )\right ) \log ^2(x)} \, dx\\ &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{\left (x \left (1536 \log ^2(2)+64 \log ^3(2)\right )+x \left (65536+16384 \log (2)+\log ^4(2)\right )\right ) \log ^2(x)} \, dx\\ &=\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{x \left (65536+16384 \log (2)+1536 \log ^2(2)+64 \log ^3(2)+\log ^4(2)\right ) \log ^2(x)} \, dx\\ &=\frac {\int \frac {-131072-65536 e^4-2 x^2+\left (-32768-16384 e^4\right ) \log (2)+\left (-3072-1536 e^4\right ) \log ^2(2)+\left (-128-64 e^4\right ) \log ^3(2)+\left (-2-e^4\right ) \log ^4(2)+4 x^2 \log (x)}{x \log ^2(x)} \, dx}{(16+\log (2))^4}\\ &=\frac {\int \frac {-e^4 (16+\log (2))^4-2 \left (x^2+(16+\log (2))^4\right )+4 x^2 \log (x)}{x \log ^2(x)} \, dx}{(16+\log (2))^4}\\ &=\frac {\int \left (\frac {-131072-65536 e^4-2 x^2-32768 \log (2)-16384 e^4 \log (2)-3072 \log ^2(2)-1536 e^4 \log ^2(2)-128 \log ^3(2)-64 e^4 \log ^3(2)-2 \log ^4(2)-e^4 \log ^4(2)}{x \log ^2(x)}+\frac {4 x}{\log (x)}\right ) \, dx}{(16+\log (2))^4}\\ &=\frac {\int \frac {-131072-65536 e^4-2 x^2-32768 \log (2)-16384 e^4 \log (2)-3072 \log ^2(2)-1536 e^4 \log ^2(2)-128 \log ^3(2)-64 e^4 \log ^3(2)-2 \log ^4(2)-e^4 \log ^4(2)}{x \log ^2(x)} \, dx}{(16+\log (2))^4}+\frac {4 \int \frac {x}{\log (x)} \, dx}{(16+\log (2))^4}\\ &=\frac {\int \left (-\frac {2 x}{\log ^2(x)}+\frac {-131072-65536 e^4-32768 \log (2)-16384 e^4 \log (2)-3072 \log ^2(2)-1536 e^4 \log ^2(2)-128 \log ^3(2)-64 e^4 \log ^3(2)-2 \log ^4(2)-e^4 \log ^4(2)}{x \log ^2(x)}\right ) \, dx}{(16+\log (2))^4}+\frac {4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{(16+\log (2))^4}\\ &=\frac {4 \text {Ei}(2 \log (x))}{(16+\log (2))^4}+\left (-2-e^4\right ) \int \frac {1}{x \log ^2(x)} \, dx-\frac {2 \int \frac {x}{\log ^2(x)} \, dx}{(16+\log (2))^4}\\ &=\frac {4 \text {Ei}(2 \log (x))}{(16+\log (2))^4}+\frac {2 x^2}{(16+\log (2))^4 \log (x)}+\left (-2-e^4\right ) \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (x)\right )-\frac {4 \int \frac {x}{\log (x)} \, dx}{(16+\log (2))^4}\\ &=\frac {4 \text {Ei}(2 \log (x))}{(16+\log (2))^4}+\frac {2+e^4}{\log (x)}+\frac {2 x^2}{(16+\log (2))^4 \log (x)}-\frac {4 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )}{(16+\log (2))^4}\\ &=\frac {2+e^4}{\log (x)}+\frac {2 x^2}{(16+\log (2))^4 \log (x)}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 34, normalized size = 1.62 \begin {gather*} \frac {e^4 (16+\log (2))^4+2 \left (x^2+(16+\log (2))^4\right )}{(16+\log (2))^4 \log (x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.04, size = 77, normalized size = 3.67 \begin {gather*} \frac {{\left (e^{4} + 2\right )} \log \relax (2)^{4} + 64 \, {\left (e^{4} + 2\right )} \log \relax (2)^{3} + 1536 \, {\left (e^{4} + 2\right )} \log \relax (2)^{2} + 2 \, x^{2} + 16384 \, {\left (e^{4} + 2\right )} \log \relax (2) + 65536 \, e^{4} + 131072}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.22, size = 99, normalized size = 4.71 \begin {gather*} \frac {e^{4} \log \relax (2)^{4} + 64 \, e^{4} \log \relax (2)^{3} + 2 \, \log \relax (2)^{4} + 1536 \, e^{4} \log \relax (2)^{2} + 128 \, \log \relax (2)^{3} + 2 \, x^{2} + 16384 \, e^{4} \log \relax (2) + 3072 \, \log \relax (2)^{2} + 65536 \, e^{4} + 32768 \, \log \relax (2) + 131072}{\log \relax (2)^{4} \log \relax (x) + 64 \, \log \relax (2)^{3} \log \relax (x) + 1536 \, \log \relax (2)^{2} \log \relax (x) + 16384 \, \log \relax (2) \log \relax (x) + 65536 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.11, size = 66, normalized size = 3.14
method | result | size |
norman | \(\frac {\frac {2 x^{2}}{16+\ln \relax (2)}+8192+\ln \relax (2)^{3} {\mathrm e}^{4}+48 \ln \relax (2)^{2} {\mathrm e}^{4}+2 \ln \relax (2)^{3}+768 \,{\mathrm e}^{4} \ln \relax (2)+96 \ln \relax (2)^{2}+4096 \,{\mathrm e}^{4}+1536 \ln \relax (2)}{\left (16+\ln \relax (2)\right )^{3} \ln \relax (x )}\) | \(66\) |
risch | \(\frac {\ln \relax (2)^{4} {\mathrm e}^{4}+64 \ln \relax (2)^{3} {\mathrm e}^{4}+2 \ln \relax (2)^{4}+1536 \ln \relax (2)^{2} {\mathrm e}^{4}+128 \ln \relax (2)^{3}+16384 \,{\mathrm e}^{4} \ln \relax (2)+3072 \ln \relax (2)^{2}+2 x^{2}+65536 \,{\mathrm e}^{4}+32768 \ln \relax (2)+131072}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}\) | \(92\) |
default | \(\frac {\ln \relax (2)^{4} {\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {64 \ln \relax (2)^{3} {\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {2 \ln \relax (2)^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}-\frac {4 \expIntegralEi \left (1, -2 \ln \relax (x )\right )}{\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536}+\frac {1536 \ln \relax (2)^{2} {\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {128 \ln \relax (2)^{3}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {16384 \,{\mathrm e}^{4} \ln \relax (2)}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {3072 \ln \relax (2)^{2}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}-\frac {2 \left (-\frac {x^{2}}{\ln \relax (x )}-2 \expIntegralEi \left (1, -2 \ln \relax (x )\right )\right )}{\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536}+\frac {65536 \,{\mathrm e}^{4}}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {32768 \ln \relax (2)}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}+\frac {131072}{\left (\ln \relax (2)^{4}+64 \ln \relax (2)^{3}+1536 \ln \relax (2)^{2}+16384 \ln \relax (2)+65536\right ) \ln \relax (x )}\) | \(415\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.61, size = 371, normalized size = 17.67 \begin {gather*} \frac {e^{4} \log \relax (2)^{4}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {64 \, e^{4} \log \relax (2)^{3}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {2 \, \log \relax (2)^{4}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {1536 \, e^{4} \log \relax (2)^{2}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {128 \, \log \relax (2)^{3}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {2 \, x^{2}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {16384 \, e^{4} \log \relax (2)}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {3072 \, \log \relax (2)^{2}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {65536 \, e^{4}}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {32768 \, \log \relax (2)}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} + \frac {131072}{{\left (\log \relax (2)^{4} + 64 \, \log \relax (2)^{3} + 1536 \, \log \relax (2)^{2} + 16384 \, \log \relax (2) + 65536\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.85, size = 73, normalized size = 3.48 \begin {gather*} \frac {2\,x^2+65536\,{\mathrm {e}}^4+32768\,\ln \relax (2)+16384\,{\mathrm {e}}^4\,\ln \relax (2)+1536\,{\mathrm {e}}^4\,{\ln \relax (2)}^2+64\,{\mathrm {e}}^4\,{\ln \relax (2)}^3+{\mathrm {e}}^4\,{\ln \relax (2)}^4+3072\,{\ln \relax (2)}^2+128\,{\ln \relax (2)}^3+2\,{\ln \relax (2)}^4+131072}{\ln \relax (x)\,{\left (\ln \relax (2)+16\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.15, size = 102, normalized size = 4.86 \begin {gather*} \frac {2 x^{2} + 2 \log {\relax (2 )}^{4} + e^{4} \log {\relax (2 )}^{4} + 128 \log {\relax (2 )}^{3} + 64 e^{4} \log {\relax (2 )}^{3} + 3072 \log {\relax (2 )}^{2} + 32768 \log {\relax (2 )} + 1536 e^{4} \log {\relax (2 )}^{2} + 131072 + 16384 e^{4} \log {\relax (2 )} + 65536 e^{4}}{\left (\log {\relax (2 )}^{4} + 64 \log {\relax (2 )}^{3} + 1536 \log {\relax (2 )}^{2} + 16384 \log {\relax (2 )} + 65536\right ) \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________