Optimal. Leaf size=20 \[ \frac {x \left (1+e^x+x\right ) \log (x \log (4))}{8 \log (x)} \]
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Rubi [B] time = 0.77, antiderivative size = 48, normalized size of antiderivative = 2.40, number of steps used = 29, number of rules used = 15, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {12, 6742, 2288, 6688, 2330, 2298, 2309, 2178, 2320, 2297, 2361, 6496, 2306, 2366, 6482} \begin {gather*} \frac {x^2 \log (x \log (4))}{8 \log (x)}+\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {x \log (x \log (4))}{8 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2288
Rule 2297
Rule 2298
Rule 2306
Rule 2309
Rule 2320
Rule 2330
Rule 2361
Rule 2366
Rule 6482
Rule 6496
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {\left (1+e^x+x\right ) \log (x)+\left (-1-e^x-x+\left (1+2 x+e^x (1+x)\right ) \log (x)\right ) \log (x \log (4))}{\log ^2(x)} \, dx\\ &=\frac {1}{8} \int \left (\frac {e^x (\log (x)-\log (x \log (4))+\log (x) \log (x \log (4))+x \log (x) \log (x \log (4)))}{\log ^2(x)}+\frac {\log (x)+x \log (x)-\log (x \log (4))-x \log (x \log (4))+\log (x) \log (x \log (4))+2 x \log (x) \log (x \log (4))}{\log ^2(x)}\right ) \, dx\\ &=\frac {1}{8} \int \frac {e^x (\log (x)-\log (x \log (4))+\log (x) \log (x \log (4))+x \log (x) \log (x \log (4)))}{\log ^2(x)} \, dx+\frac {1}{8} \int \frac {\log (x)+x \log (x)-\log (x \log (4))-x \log (x \log (4))+\log (x) \log (x \log (4))+2 x \log (x) \log (x \log (4))}{\log ^2(x)} \, dx\\ &=\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {1}{8} \int \frac {-((1+x) \log (x \log (4)))+\log (x) (1+x+(1+2 x) \log (x \log (4)))}{\log ^2(x)} \, dx\\ &=\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {1}{8} \int \left (\frac {1+x}{\log (x)}+\frac {(-1-x+\log (x)+2 x \log (x)) \log (x \log (4))}{\log ^2(x)}\right ) \, dx\\ &=\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {1}{8} \int \frac {1+x}{\log (x)} \, dx+\frac {1}{8} \int \frac {(-1-x+\log (x)+2 x \log (x)) \log (x \log (4))}{\log ^2(x)} \, dx\\ &=\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {1}{8} \int \left (\frac {1}{\log (x)}+\frac {x}{\log (x)}\right ) \, dx+\frac {1}{8} \int \left (-\frac {\log (x \log (4))}{\log ^2(x)}-\frac {x \log (x \log (4))}{\log ^2(x)}+\frac {\log (x \log (4))}{\log (x)}+\frac {2 x \log (x \log (4))}{\log (x)}\right ) \, dx\\ &=\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {1}{8} \int \frac {1}{\log (x)} \, dx+\frac {1}{8} \int \frac {x}{\log (x)} \, dx-\frac {1}{8} \int \frac {\log (x \log (4))}{\log ^2(x)} \, dx-\frac {1}{8} \int \frac {x \log (x \log (4))}{\log ^2(x)} \, dx+\frac {1}{8} \int \frac {\log (x \log (4))}{\log (x)} \, dx+\frac {1}{4} \int \frac {x \log (x \log (4))}{\log (x)} \, dx\\ &=\frac {x \log (x \log (4))}{8 \log (x)}+\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {x^2 \log (x \log (4))}{8 \log (x)}+\frac {\text {li}(x)}{8}+\frac {1}{8} \int \left (\frac {2 \text {Ei}(2 \log (x))}{x}-\frac {x}{\log (x)}\right ) \, dx-\frac {1}{8} \int \frac {\text {li}(x)}{x} \, dx+\frac {1}{8} \int \left (-\frac {1}{\log (x)}+\frac {\text {li}(x)}{x}\right ) \, dx+\frac {1}{8} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-\frac {1}{4} \int \frac {\text {Ei}(2 \log (x))}{x} \, dx\\ &=\frac {x}{8}+\frac {1}{8} \text {Ei}(2 \log (x))+\frac {x \log (x \log (4))}{8 \log (x)}+\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {x^2 \log (x \log (4))}{8 \log (x)}+\frac {\text {li}(x)}{8}-\frac {1}{8} \log (x) \text {li}(x)-\frac {1}{8} \int \frac {1}{\log (x)} \, dx-\frac {1}{8} \int \frac {x}{\log (x)} \, dx+\frac {1}{8} \int \frac {\text {li}(x)}{x} \, dx+\frac {1}{4} \int \frac {\text {Ei}(2 \log (x))}{x} \, dx-\frac {1}{4} \operatorname {Subst}(\int \text {Ei}(2 x) \, dx,x,\log (x))\\ &=\frac {x^2}{8}+\frac {1}{8} \text {Ei}(2 \log (x))-\frac {1}{4} \text {Ei}(2 \log (x)) \log (x)+\frac {x \log (x \log (4))}{8 \log (x)}+\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {x^2 \log (x \log (4))}{8 \log (x)}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+\frac {1}{4} \operatorname {Subst}(\int \text {Ei}(2 x) \, dx,x,\log (x))\\ &=\frac {x \log (x \log (4))}{8 \log (x)}+\frac {e^x x \log (x \log (4))}{8 \log (x)}+\frac {x^2 \log (x \log (4))}{8 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.23, size = 20, normalized size = 1.00 \begin {gather*} \frac {x \left (1+e^x+x\right ) \log (x \log (4))}{8 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 34, normalized size = 1.70 \begin {gather*} \frac {{\left (x^{2} + x e^{x} + x\right )} \log \relax (x) + {\left (x^{2} + x e^{x} + x\right )} \log \left (2 \, \log \relax (2)\right )}{8 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 58, normalized size = 2.90 \begin {gather*} \frac {x^{2} \log \relax (2) + x e^{x} \log \relax (2) + x^{2} \log \relax (x) + x e^{x} \log \relax (x) + x^{2} \log \left (\log \relax (2)\right ) + x e^{x} \log \left (\log \relax (2)\right ) + x \log \relax (2) + x \log \relax (x) + x \log \left (\log \relax (2)\right )}{8 \, \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 56, normalized size = 2.80
method | result | size |
risch | \(\frac {x^{2}}{8}+\frac {{\mathrm e}^{x} x}{8}+\frac {x}{8}+\frac {x \left (2 x \ln \relax (2)+2 \,{\mathrm e}^{x} \ln \relax (2)+2 x \ln \left (\ln \relax (2)\right )+2 \,{\mathrm e}^{x} \ln \left (\ln \relax (2)\right )+2 \ln \relax (2)+2 \ln \left (\ln \relax (2)\right )\right )}{16 \ln \relax (x )}\) | \(56\) |
default | \(\frac {x}{8}+\frac {\left (\ln \relax (2)+\ln \left (\ln \relax (2)\right )\right ) x \,{\mathrm e}^{x}+x \,{\mathrm e}^{x} \ln \relax (x )}{8 \ln \relax (x )}+\frac {x^{2}}{8}+\frac {\ln \relax (2) x^{2}}{8 \ln \relax (x )}+\frac {\ln \relax (2) x}{8 \ln \relax (x )}+\frac {\ln \left (\ln \relax (2)\right ) x^{2}}{8 \ln \relax (x )}+\frac {\ln \left (\ln \relax (2)\right ) x}{8 \ln \relax (x )}\) | \(75\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {x^{2} {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + x {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + {\left (x {\left (\log \relax (2) + \log \left (\log \relax (2)\right )\right )} + x \log \relax (x)\right )} e^{x} + {\left (x^{2} + x\right )} \log \relax (x)}{8 \, \log \relax (x)} + \frac {1}{8} \, {\rm Ei}\left (2 \, \log \relax (x)\right ) + \frac {1}{8} \, {\rm Ei}\left (\log \relax (x)\right ) - \frac {1}{8} \, \int \frac {x + 1}{\log \relax (x)}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.38, size = 20, normalized size = 1.00 \begin {gather*} \frac {x\,\left (\ln \left (2\,\ln \relax (2)\right )+\ln \relax (x)\right )\,\left (x+{\mathrm {e}}^x+1\right )}{8\,\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.36, size = 65, normalized size = 3.25 \begin {gather*} \frac {x^{2}}{8} + \frac {x}{8} + \frac {\left (x \log {\relax (x )} + x \log {\left (\log {\relax (2 )} \right )} + x \log {\relax (2 )}\right ) e^{x}}{8 \log {\relax (x )}} + \frac {x^{2} \log {\left (\log {\relax (2 )} \right )} + x^{2} \log {\relax (2 )} + x \log {\left (\log {\relax (2 )} \right )} + x \log {\relax (2 )}}{8 \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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