Integrand size = 127, antiderivative size = 29 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=1-\left (e^3+\left (-x+\frac {4}{-5+3 x^3}\right )^2\right ) \log (3 x) \]
[Out]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.54 (sec) , antiderivative size = 772, normalized size of antiderivative = 26.62, number of steps used = 43, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.197, Rules used = {6820, 6874, 1843, 1848, 1889, 31, 648, 631, 210, 642, 2404, 2341, 2376, 272, 46, 205, 206, 2367, 2353, 2352, 2351, 2354, 2438, 27, 12} \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-\frac {8 \arctan \left (\frac {2 x}{\sqrt [6]{3} \sqrt [3]{5}}+\frac {1}{\sqrt {3}}\right )}{3^{5/6} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \operatorname {PolyLog}\left (2,(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {16}{15 \left (5-3 x^3\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {16}{75} \log \left (5-3 x^3\right )-x^2 \log (3 x)+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (3^{2/3} x^2+\sqrt [3]{15} x+5^{2/3}\right )-\frac {8 x \log (3 x)}{\sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{-1} 3^{2/3} x+\sqrt [3]{15}\right )}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)+\frac {16 \log (x)}{25}+\frac {8 \sqrt [3]{-1} 3^{2/3} \log \left (\sqrt [3]{-3} x+\sqrt [3]{5}\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^4}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (\sqrt [3]{-\frac {3}{5}} x+1\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {8 \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{3} x+\sqrt [3]{-5}\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )} \]
[In]
[Out]
Rule 12
Rule 27
Rule 31
Rule 46
Rule 205
Rule 206
Rule 210
Rule 272
Rule 631
Rule 642
Rule 648
Rule 1843
Rule 1848
Rule 1889
Rule 2341
Rule 2351
Rule 2352
Rule 2353
Rule 2354
Rule 2367
Rule 2376
Rule 2404
Rule 2438
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-5+3 x^3\right ) \left (e^3 \left (5-3 x^3\right )^2+\left (4+5 x-3 x^4\right )^2\right )-2 x \left (100+125 x+144 x^2+60 x^3-225 x^4-72 x^6+135 x^7-27 x^{10}\right ) \log (3 x)}{x \left (5-3 x^3\right )^3} \, dx \\ & = \int \left (\frac {-16 \left (1+\frac {25 e^3}{16}\right )-40 x-25 x^2+30 e^3 x^3+24 x^4+30 x^5-9 e^3 x^6-9 x^8}{x \left (5-3 x^3\right )^2}-\frac {2 \left (-4-5 x+3 x^4\right ) \left (25+36 x^2-30 x^3+9 x^6\right ) \log (3 x)}{\left (-5+3 x^3\right )^3}\right ) \, dx \\ & = -\left (2 \int \frac {\left (-4-5 x+3 x^4\right ) \left (25+36 x^2-30 x^3+9 x^6\right ) \log (3 x)}{\left (-5+3 x^3\right )^3} \, dx\right )+\int \frac {-16 \left (1+\frac {25 e^3}{16}\right )-40 x-25 x^2+30 e^3 x^3+24 x^4+30 x^5-9 e^3 x^6-9 x^8}{x \left (5-3 x^3\right )^2} \, dx \\ & = -\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {1}{405} \int \frac {-81 \left (16+25 e^3\right )-3240 x-2025 x^2+1215 e^3 x^3+1215 x^5}{x \left (5-3 x^3\right )} \, dx-2 \int \left (x \log (3 x)-\frac {144 x^2 \log (3 x)}{\left (-5+3 x^3\right )^3}+\frac {60 \log (3 x)}{\left (-5+3 x^3\right )^2}+\frac {8 \log (3 x)}{-5+3 x^3}\right ) \, dx \\ & = -\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {1}{405} \int \left (-\frac {81 \left (16+25 e^3\right )}{5 x}-405 x+\frac {648 \left (25+6 x^2\right )}{5 \left (-5+3 x^3\right )}\right ) \, dx-2 \int x \log (3 x) \, dx-16 \int \frac {\log (3 x)}{-5+3 x^3} \, dx-120 \int \frac {\log (3 x)}{\left (-5+3 x^3\right )^2} \, dx+288 \int \frac {x^2 \log (3 x)}{\left (-5+3 x^3\right )^3} \, dx \\ & = -\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}+\frac {8}{25} \int \frac {25+6 x^2}{-5+3 x^3} \, dx+16 \int \frac {1}{x \left (-5+3 x^3\right )^2} \, dx-16 \int \left (-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}+\sqrt [3]{-3} x\right )}-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}-\frac {\log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{5}-(-1)^{2/3} \sqrt [3]{3} x\right )}\right ) \, dx-120 \int \left (\frac {2 \log (3 x)}{15 \sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}+\frac {3 (-1)^{2/3} \sqrt [3]{\frac {3}{5}} \log (3 x)}{5 \left (1+\sqrt [3]{-1}\right )^4 \left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )^2}+\frac {6 (-1)^{5/6} \sqrt [6]{3} \log (3 x)}{5\ 5^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )}+\frac {3 \sqrt [3]{\frac {3}{5}} \log (3 x)}{5 \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x\right )^2}+\frac {4 \log (3 x)}{15 \sqrt [3]{3} 5^{2/3} \left (2\ 3^{2/3} \sqrt [3]{5}+3 \left (1-i \sqrt {3}\right ) x\right )}-\frac {\log (3 x)}{15\ 3^{2/3} \sqrt [3]{5} \left (-\sqrt [3]{3} 5^{2/3}+2\ 3^{2/3} \sqrt [3]{5} x-3 x^2\right )}\right ) \, dx \\ & = -\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}+\frac {16}{3} \text {Subst}\left (\int \frac {1}{x (-5+3 x)^2} \, dx,x,x^3\right )-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x\right )^2} \, dx-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x\right )^2} \, dx+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}+\sqrt [3]{-3} x} \, dx}{3\ 5^{2/3}}+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}-\sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}+\frac {16 \int \frac {\log (3 x)}{\sqrt [3]{5}-(-1)^{2/3} \sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}-\frac {8 \int \frac {\sqrt [3]{\frac {5}{3}} \left (50-2 \sqrt [3]{3} 5^{2/3}\right )+\left (25-4 \sqrt [3]{3} 5^{2/3}\right ) x}{\left (\frac {5}{3}\right )^{2/3}+\sqrt [3]{\frac {5}{3}} x+x^2} \, dx}{75 \sqrt [3]{3} 5^{2/3}}-\frac {16 \int \frac {\log (3 x)}{3^{2/3} \sqrt [3]{5}-3 x} \, dx}{\sqrt [3]{3} 5^{2/3}}-\frac {32 \int \frac {\log (3 x)}{2\ 3^{2/3} \sqrt [3]{5}+3 \left (1-i \sqrt {3}\right ) x} \, dx}{\sqrt [3]{3} 5^{2/3}}+\frac {8 \int \frac {\log (3 x)}{-\sqrt [3]{3} 5^{2/3}+2\ 3^{2/3} \sqrt [3]{5} x-3 x^2} \, dx}{3^{2/3} \sqrt [3]{5}}-\frac {\left (144 (-1)^{5/6} \sqrt [6]{3}\right ) \int \frac {\log (3 x)}{3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x} \, dx}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}-\frac {1}{225} \left (8 \left (6+5\ 3^{2/3} \sqrt [3]{5}\right )\right ) \int \frac {1}{\sqrt [3]{\frac {5}{3}}-x} \, dx \\ & = -\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{15}+\sqrt [3]{-1} 3^{2/3} x\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}+\frac {16}{3} \text {Subst}\left (\int \left (\frac {1}{25 x}+\frac {3}{5 (-5+3 x)^2}-\frac {3}{25 (-5+3 x)}\right ) \, dx,x,x^3\right )+\frac {16 \int \frac {\log \left (\sqrt [3]{\frac {3}{5}} x\right )}{\sqrt [3]{5}-\sqrt [3]{3} x} \, dx}{3\ 5^{2/3}}-\frac {\left (16 \sqrt [3]{-\frac {1}{3}}\right ) \int \frac {\log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{x} \, dx}{3\ 5^{2/3}}+\frac {8 \int \frac {1}{3^{2/3} \sqrt [3]{5}+3 \sqrt [3]{-1} x} \, dx}{\sqrt [3]{3} 5^{2/3}}-\frac {8 \int \frac {1}{-3^{2/3} \sqrt [3]{5}+3 (-1)^{2/3} x} \, dx}{\sqrt [3]{3} 5^{2/3}}-\frac {16 \int \frac {\log \left (\sqrt [3]{\frac {3}{5}} x\right )}{3^{2/3} \sqrt [3]{5}-3 x} \, dx}{\sqrt [3]{3} 5^{2/3}}+\frac {\left (16 (-1)^{2/3}\right ) \int \frac {\log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{x} \, dx}{3 \sqrt [3]{3} 5^{2/3}}-\frac {4 \int \frac {1}{\left (\frac {5}{3}\right )^{2/3}+\sqrt [3]{\frac {5}{3}} x+x^2} \, dx}{3^{2/3} \sqrt [3]{5}}+\frac {8 \int -\frac {3 \log (3 x)}{\left (3^{2/3} \sqrt [3]{5}-3 x\right )^2} \, dx}{3^{2/3} \sqrt [3]{5}}+\frac {\left (48 i \sqrt [6]{3}\right ) \int \frac {\log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{x} \, dx}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {32 \int \frac {\log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{x} \, dx}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}+\frac {1}{225} \left (4 \left (12-5\ 3^{2/3} \sqrt [3]{5}\right )\right ) \int \frac {\sqrt [3]{\frac {5}{3}}+2 x}{\left (\frac {5}{3}\right )^{2/3}+\sqrt [3]{\frac {5}{3}} x+x^2} \, dx \\ & = \frac {16}{15 \left (5-3 x^3\right )}-\frac {16 x^3}{25 \left (5-3 x^3\right )}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {16 \log (x)}{25}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-x^2 \log (3 x)-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{15}+\sqrt [3]{-1} 3^{2/3} x\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {8 (-1)^{2/3} \log \left (\sqrt [3]{5}+\sqrt [3]{-3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (5^{2/3}+\sqrt [3]{15} x+3^{2/3} x^2\right )-\frac {16}{75} \log \left (5-3 x^3\right )-\frac {16 (-1)^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \operatorname {PolyLog}\left (2,(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}-\left (8 \sqrt [3]{\frac {3}{5}}\right ) \int \frac {\log (3 x)}{\left (3^{2/3} \sqrt [3]{5}-3 x\right )^2} \, dx+\frac {8 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{\frac {3}{5}} x\right )}{\sqrt [3]{3} 5^{2/3}} \\ & = \frac {16}{15 \left (5-3 x^3\right )}-\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {8 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt [6]{3} \sqrt [3]{5}}\right )}{3^{5/6} 5^{2/3}}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {16 \log (x)}{25}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-\frac {8 x \log (3 x)}{\sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}-x^2 \log (3 x)-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{15}+\sqrt [3]{-1} 3^{2/3} x\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {8 (-1)^{2/3} \log \left (\sqrt [3]{5}+\sqrt [3]{-3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (5^{2/3}+\sqrt [3]{15} x+3^{2/3} x^2\right )-\frac {16}{75} \log \left (5-3 x^3\right )-\frac {16 (-1)^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \operatorname {PolyLog}\left (2,(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}+\frac {8 \int \frac {1}{3^{2/3} \sqrt [3]{5}-3 x} \, dx}{\sqrt [3]{3} 5^{2/3}} \\ & = \frac {16}{15 \left (5-3 x^3\right )}-\frac {16 x^3}{25 \left (5-3 x^3\right )}-\frac {8 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt [6]{3} \sqrt [3]{5}}\right )}{3^{5/6} 5^{2/3}}+\frac {16 \log (45) \log \left (3^{2/3} \sqrt [3]{5}-3 x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {16 \log (x)}{25}-\frac {1}{25} \left (16+25 e^3\right ) \log (x)-\frac {8 x \log (3 x)}{\sqrt [3]{3} 5^{2/3} \left (3^{2/3} \sqrt [3]{5}-3 x\right )}-x^2 \log (3 x)-\frac {8 \sqrt [3]{-1} x \log (3 x)}{3\ 5^{2/3} \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}-\frac {8 x \log (3 x)}{15^{2/3} \left (\sqrt [3]{15}+\sqrt [3]{-1} 3^{2/3} x\right )}-\frac {16 \log (3 x)}{\left (5-3 x^3\right )^2}-\frac {8 (-1)^{2/3} \log \left (\sqrt [3]{5}+\sqrt [3]{-3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {16 (-1)^{2/3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \log (3 x) \log \left (1+\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \log (3 x) \log \left (1-(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {8 \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{3 \sqrt [3]{3} 5^{2/3}}+\frac {8}{225} \left (6+5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )-\frac {16 \log (45) \log \left (\sqrt [3]{5}-\sqrt [3]{3} x\right )}{9 \sqrt [3]{3} 5^{2/3}}+\frac {8 \sqrt [3]{-\frac {1}{3}} \log \left (\sqrt [3]{-5}+\sqrt [3]{3} x\right )}{3\ 5^{2/3}}-\frac {32 \log (3 x) \log \left (1+\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )}+\frac {4}{225} \left (12-5\ 3^{2/3} \sqrt [3]{5}\right ) \log \left (5^{2/3}+\sqrt [3]{15} x+3^{2/3} x^2\right )-\frac {16}{75} \log \left (5-3 x^3\right )-\frac {16 (-1)^{2/3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{3 \sqrt [3]{3} 5^{2/3}}-\frac {48 i \sqrt [6]{3} \operatorname {PolyLog}\left (2,-\sqrt [3]{-\frac {3}{5}} x\right )}{5^{2/3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {16 \sqrt [3]{-\frac {1}{3}} \operatorname {PolyLog}\left (2,(-1)^{2/3} \sqrt [3]{\frac {3}{5}} x\right )}{3\ 5^{2/3}}-\frac {32 \operatorname {PolyLog}\left (2,-\frac {1}{2} \sqrt [3]{\frac {3}{5}} \left (1-i \sqrt {3}\right ) x\right )}{3 \sqrt [3]{3} 5^{2/3} \left (1-i \sqrt {3}\right )} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-e^3 \log (x)-\frac {\left (-4-5 x+3 x^4\right )^2 \log (3 x)}{\left (-5+3 x^3\right )^2} \]
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Time = 0.47 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83
method | result | size |
risch | \(-\frac {\left (9 x^{8}-30 x^{5}-24 x^{4}+25 x^{2}+40 x +16\right ) \ln \left (3 x \right )}{9 x^{6}-30 x^{3}+25}-\ln \left (x \right ) {\mathrm e}^{3}\) | \(53\) |
derivativedivides | \(-\frac {\left (9 \,{\mathrm e}^{3}+\frac {144}{25}\right ) \ln \left (3 x \right )}{9}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(64\) |
default | \(-\frac {\left (9 \,{\mathrm e}^{3}+\frac {144}{25}\right ) \ln \left (3 x \right )}{9}-x^{2} \ln \left (3 x \right )+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(64\) |
norman | \(\frac {-16 \ln \left (3 x \right )-40 x \ln \left (3 x \right )-25 x^{2} \ln \left (3 x \right )+24 \ln \left (3 x \right ) x^{4}+30 \ln \left (3 x \right ) x^{5}-9 \ln \left (3 x \right ) x^{8}}{\left (3 x^{3}-5\right )^{2}}-\ln \left (x \right ) {\mathrm e}^{3}\) | \(68\) |
parallelrisch | \(-\frac {81 \ln \left (3 x \right ) x^{8}+81 \ln \left (x \right ) {\mathrm e}^{3} x^{6}-270 \ln \left (3 x \right ) x^{5}-270 \ln \left (x \right ) {\mathrm e}^{3} x^{3}-216 \ln \left (3 x \right ) x^{4}+225 x^{2} \ln \left (3 x \right )+225 \ln \left (x \right ) {\mathrm e}^{3}+360 x \ln \left (3 x \right )+144 \ln \left (3 x \right )}{9 \left (9 x^{6}-30 x^{3}+25\right )}\) | \(91\) |
parts | \(-\left ({\mathrm e}^{3}+\frac {16}{25}\right ) \ln \left (x \right )+\frac {16}{45 \left (x^{3}-\frac {5}{3}\right )}+\frac {8 \,5^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (x -\frac {5^{\frac {1}{3}} 3^{\frac {2}{3}}}{3}\right )}{45}-\frac {4 \,5^{\frac {1}{3}} 3^{\frac {2}{3}} \ln \left (x^{2}+\frac {5^{\frac {1}{3}} 3^{\frac {2}{3}} x}{3}+\frac {5^{\frac {2}{3}} 3^{\frac {1}{3}}}{3}\right )}{45}-\frac {8 \,5^{\frac {1}{3}} 3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,5^{\frac {2}{3}} 3^{\frac {1}{3}} x}{5}+1\right )}{3}\right )}{15}+\frac {16 \ln \left (3 x^{3}-5\right )}{75}-x^{2} \ln \left (3 x \right )-\frac {8 \,45^{\frac {1}{3}} \ln \left (3 x -45^{\frac {1}{3}}\right )}{45}+\frac {4 \,45^{\frac {1}{3}} \ln \left (9 x^{2}+3 \,45^{\frac {1}{3}} x +45^{\frac {2}{3}}\right )}{45}+\frac {8 \,45^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,45^{\frac {2}{3}} x}{15}+1\right )}{3}\right )}{45}+\frac {72 \ln \left (3 x \right ) x}{27 x^{3}-45}-\frac {48}{5 \left (27 x^{3}-45\right )}-\frac {16 \ln \left (27 x^{3}-45\right )}{75}+\frac {432 \ln \left (3 x \right ) x^{3} \left (27 x^{3}-90\right )}{25 \left (27 x^{3}-45\right )^{2}}\) | \(232\) |
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Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-\frac {{\left (9 \, x^{8} - 30 \, x^{5} - 24 \, x^{4} + 25 \, x^{2} + {\left (9 \, x^{6} - 30 \, x^{3} + 25\right )} e^{3} + 40 \, x + 16\right )} \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \]
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Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
Time = 0.16 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=- e^{3} \log {\left (x \right )} + \frac {\left (- 9 x^{8} + 30 x^{5} + 24 x^{4} - 25 x^{2} - 40 x - 16\right ) \log {\left (3 x \right )}}{9 x^{6} - 30 x^{3} + 25} \]
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Exception generated. \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=\text {Exception raised: RuntimeError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.10 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-\frac {9 \, x^{8} \log \left (3 \, x\right ) + 9 \, x^{6} e^{3} \log \left (x\right ) - 30 \, x^{5} \log \left (3 \, x\right ) - 24 \, x^{4} \log \left (3 \, x\right ) - 30 \, x^{3} e^{3} \log \left (x\right ) + 25 \, x^{2} \log \left (3 \, x\right ) + 40 \, x \log \left (3 \, x\right ) + 25 \, e^{3} \log \left (x\right ) + 16 \, \log \left (3 \, x\right )}{9 \, x^{6} - 30 \, x^{3} + 25} \]
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Time = 13.90 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {80+200 x+125 x^2-48 x^3-240 x^4-225 x^5+72 x^7+135 x^8-27 x^{11}+e^3 \left (125-225 x^3+135 x^6-27 x^9\right )+\left (200 x+250 x^2+288 x^3+120 x^4-450 x^5-144 x^7+270 x^8-54 x^{11}\right ) \log (3 x)}{-125 x+225 x^4-135 x^7+27 x^{10}} \, dx=-{\mathrm {e}}^3\,\ln \left (x\right )-\frac {\ln \left (3\,x\right )\,\left (x^8-\frac {10\,x^5}{3}-\frac {8\,x^4}{3}+\frac {25\,x^2}{9}+\frac {40\,x}{9}+\frac {16}{9}\right )}{x^6-\frac {10\,x^3}{3}+\frac {25}{9}} \]
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