Optimal. Leaf size=30 \[ \frac {3+x}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \]
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Rubi [F] time = 37.51, 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 {\left (-18 x+2 x^3\right ) \log (5)+\left (-18-6 x+\left (-6 x^2-2 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )+\left (54-18 x+\left (18 x^2-6 x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (9-3 x+\left (3 x^2-x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )}{\left (-81+81 x-27 x^2+3 x^3+\left (-27 x^2+27 x^3-9 x^4+x^5\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )+\left (-54+36 x-6 x^2+\left (-18 x^2+12 x^3-2 x^4\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log \left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )+\left (-9+3 x+\left (-3 x^2+x^3\right ) \log (5)\right ) \log \left (\frac {3+x^2 \log (5)}{\log (5)}\right ) \log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right ) \log ^2\left (\log \left (\frac {\log \left (\frac {3+x^2 \log (5)}{\log (5)}\right )}{9-6 x+x^2}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18 x \log (5)-x^3 \log (25)+\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2 (3+x)+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (6+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx\\ &=\int \left (\frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}+\frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2}\right ) \, dx\\ &=\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx+\int \frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx\\ &=\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx+\int \frac {18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{(3-x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx\\ &=\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx+\int \left (\frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{3 (3-x) (1+\log (125)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2}+\frac {(3+x) \log (5) \left (18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{3 \left (3+x^2 \log (5)\right ) (1+\log (125)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2}\right ) \, dx\\ &=\frac {\int \frac {18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )}{(3-x) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\frac {\log (5) \int \frac {(3+x) \left (18 x \log (5)-x^3 \log (25)+18 \log \left (x^2+\frac {3}{\log (5)}\right )+6 x \log \left (x^2+\frac {3}{\log (5)}\right )+6 x^2 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )+2 x^3 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right )-27 \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+3 x^2 (1-3 \log (5)) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )+x^4 \log (5) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx\\ &=\frac {\int \frac {18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{(3-x) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\frac {\log (5) \int \frac {(3+x) \left (18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )}{\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right ) \left (3-x+\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )^2} \, dx}{3 (1+\log (125))}+\int \frac {1}{-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [B] time = 0.22, size = 142, normalized size = 4.73 \begin {gather*} \frac {18 x \log (5)-x^3 \log (25)+(3+x) \left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )}{\left (-2 (-3+x) x \log (5)+\left (3+x^2 \log (5)\right ) \log \left (x^2+\frac {3}{\log (5)}\right ) \left (2+(-3+x) \log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right ) \left (-3+x-\log \left (\log \left (\frac {\log \left (x^2+\frac {3}{\log (5)}\right )}{(-3+x)^2}\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 38, normalized size = 1.27 \begin {gather*} \frac {x + 3}{x - \log \left (\log \left (\frac {\log \left (\frac {x^{2} \log \relax (5) + 3}{\log \relax (5)}\right )}{x^{2} - 6 \, x + 9}\right )\right ) - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.26, size = 175, normalized size = 5.83
method | result | size |
risch | \(\frac {3+x}{x -\ln \left (\ln \left (\ln \left (\frac {x^{2} \ln \relax (5)+3}{\ln \relax (5)}\right )\right )-2 \ln \left (x -3\right )+\frac {i \pi \,\mathrm {csgn}\left (i \left (x -3\right )^{2}\right ) \left (-\mathrm {csgn}\left (i \left (x -3\right )^{2}\right )+\mathrm {csgn}\left (i \left (x -3\right )\right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\frac {i \ln \left (\frac {x^{2} \ln \relax (5)+3}{\ln \relax (5)}\right )}{\left (x -3\right )^{2}}\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \left (\frac {x^{2} \ln \relax (5)+3}{\ln \relax (5)}\right )}{\left (x -3\right )^{2}}\right )+\mathrm {csgn}\left (i \ln \left (\frac {x^{2} \ln \relax (5)+3}{\ln \relax (5)}\right )\right )\right ) \left (-\mathrm {csgn}\left (\frac {i \ln \left (\frac {x^{2} \ln \relax (5)+3}{\ln \relax (5)}\right )}{\left (x -3\right )^{2}}\right )+\mathrm {csgn}\left (\frac {i}{\left (x -3\right )^{2}}\right )\right )}{2}\right )-3}\) | \(175\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 35, normalized size = 1.17 \begin {gather*} \frac {x + 3}{x - \log \left (-2 \, \log \left (x - 3\right ) + \log \left (\log \left (x^{2} \log \relax (5) + 3\right ) - \log \left (\log \relax (5)\right )\right )\right ) - 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \relax (5)\,\left (18\,x-2\,x^3\right )+\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )\,\left (6\,x+\ln \relax (5)\,\left (2\,x^3+6\,x^2\right )+18\right )-\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\,\left (\ln \relax (5)\,\left (18\,x^2-6\,x^3\right )-18\,x+54\right )-\ln \left (\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\right )\,\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\,\left (\ln \relax (5)\,\left (3\,x^2-x^3\right )-3\,x+9\right )}{\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\,\left (\ln \relax (5)\,\left (3\,x^2-x^3\right )-3\,x+9\right )\,{\ln \left (\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\right )}^2+\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\,\left (\ln \relax (5)\,\left (2\,x^4-12\,x^3+18\,x^2\right )-36\,x+6\,x^2+54\right )\,\ln \left (\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\right )+\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )\,\ln \left (\frac {\ln \left (\frac {\ln \relax (5)\,x^2+3}{\ln \relax (5)}\right )}{x^2-6\,x+9}\right )\,\left (\ln \relax (5)\,\left (-x^5+9\,x^4-27\,x^3+27\,x^2\right )-81\,x+27\,x^2-3\,x^3+81\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.01, size = 32, normalized size = 1.07 \begin {gather*} \frac {- x - 3}{- x + \log {\left (\log {\left (\frac {\log {\left (\frac {x^{2} \log {\relax (5 )} + 3}{\log {\relax (5 )}} \right )}}{x^{2} - 6 x + 9} \right )} \right )} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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