Integrand size = 119, antiderivative size = 27 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-x-\frac {1}{3} x \log ^2\left (\frac {\log (3)}{\left (-\frac {1}{x}+x+x^2\right )^2}\right ) \]
[Out]
Time = 13.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19, number of steps used = 38, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.151, Rules used = {6820, 2608, 2603, 12, 6874, 2125, 2106, 2104, 814, 648, 632, 210, 642, 2092, 2090, 719, 31, 1642} \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (-x^3-x^2+1\right )^2}\right )-x \]
[In]
[Out]
Rule 12
Rule 31
Rule 210
Rule 632
Rule 642
Rule 648
Rule 719
Rule 814
Rule 1642
Rule 2090
Rule 2092
Rule 2104
Rule 2106
Rule 2125
Rule 2603
Rule 2608
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-1+\frac {4 \left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{3 \left (-1+x^2+x^3\right )}-\frac {1}{3} \log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )\right ) \, dx \\ & = -x-\frac {1}{3} \int \log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right ) \, dx+\frac {4}{3} \int \frac {\left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx \\ & = -x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {2}{3} \int \frac {2 \left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3} \, dx+\frac {4}{3} \int \left (2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )+\frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}\right ) \, dx \\ & = -x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {4}{3} \int \frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx+\frac {4}{3} \int \frac {\left (1+x^2+2 x^3\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3} \, dx+\frac {8}{3} \int \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right ) \, dx \\ & = -x+\frac {8}{3} x \log \left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {4}{3} \int \left (-2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )+\frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3}\right ) \, dx+\frac {4}{3} \int \left (\frac {3 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}-\frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}\right ) \, dx-\frac {8}{3} \int \frac {2 \left (1+x^2+2 x^3\right )}{1-x^2-x^3} \, dx \\ & = -x+\frac {8}{3} x \log \left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {4}{3} \int \frac {\left (3-x^2\right ) \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{1-x^2-x^3} \, dx-\frac {4}{3} \int \frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx-\frac {8}{3} \int \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right ) \, dx+4 \int \frac {\log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx-\frac {16}{3} \int \frac {1+x^2+2 x^3}{1-x^2-x^3} \, dx \\ & = -x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {4}{3} \int \frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx+\frac {4}{3} \int \left (-\frac {3 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}+\frac {x^2 \log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3}\right ) \, dx+\frac {8}{3} \int \frac {2 \left (1+x^2+2 x^3\right )}{1-x^2-x^3} \, dx+4 \int \frac {\log \left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )}{-1+x^2+x^3} \, dx-\frac {16}{3} \int \left (-2+\frac {3-x^2}{1-x^2-x^3}\right ) \, dx \\ & = \frac {29 x}{3}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{3} \int \frac {3-x^2}{1-x^2-x^3} \, dx+\frac {16}{3} \int \frac {1+x^2+2 x^3}{1-x^2-x^3} \, dx \\ & = \frac {29 x}{3}-\frac {16}{9} \log \left (1-x^2-x^3\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {16}{9} \int \frac {-9-2 x}{1-x^2-x^3} \, dx+\frac {16}{3} \int \left (-2+\frac {3-x^2}{1-x^2-x^3}\right ) \, dx \\ & = -x-\frac {16}{9} \log \left (1-x^2-x^3\right )-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {16}{9} \text {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\frac {25}{27}+\frac {x}{3}-x^3} \, dx,x,\frac {1}{3}+x\right )+\frac {16}{3} \int \frac {3-x^2}{1-x^2-x^3} \, dx \\ & = -x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \int \frac {-9-2 x}{1-x^2-x^3} \, dx+\frac {16}{9} \text {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\left (\frac {\frac {2}{\sqrt [3]{25+3 \sqrt {69}}}+\sqrt [3]{50+6 \sqrt {69}}}{3\ 2^{2/3}}-x\right ) \left (\frac {1}{18} \left (-2+2 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+\sqrt [3]{\frac {1}{2} \left (25+3 \sqrt {69}\right )}\right ) x+x^2\right )} \, dx,x,\frac {1}{3}+x\right ) \\ & = -x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \text {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\frac {25}{27}+\frac {x}{3}-x^3} \, dx,x,\frac {1}{3}+x\right )+\frac {16}{9} \text {Subst}\left (\int \left (\frac {36 \left (-2 \sqrt [3]{2}-25 \sqrt [3]{25+3 \sqrt {69}}-\left (50+6 \sqrt {69}\right )^{2/3}\right )}{\left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right ) \left (2 \sqrt [3]{2}+\left (2 \left (25+3 \sqrt {69}\right )\right )^{2/3}-6 \sqrt [3]{25+3 \sqrt {69}} x\right )}+\frac {36 \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )-3 \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right ) x\right )}{\left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right ) \left (2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2\right )}\right ) \, dx,x,\frac {1}{3}+x\right ) \\ & = -x+\frac {32 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \log \left (2 \sqrt [3]{2}+\left (50+6 \sqrt {69}\right )^{2/3}-2 \sqrt [3]{25+3 \sqrt {69}} (1+3 x)\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \text {Subst}\left (\int \frac {-\frac {25}{3}-2 x}{\left (\frac {\frac {2}{\sqrt [3]{25+3 \sqrt {69}}}+\sqrt [3]{50+6 \sqrt {69}}}{3\ 2^{2/3}}-x\right ) \left (\frac {1}{18} \left (-2+2 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+\sqrt [3]{\frac {1}{2} \left (25+3 \sqrt {69}\right )}\right ) x+x^2\right )} \, dx,x,\frac {1}{3}+x\right )+\frac {64 \text {Subst}\left (\int \frac {-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )-3 \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right ) x}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}} \\ & = -x+\frac {32 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \log \left (2 \sqrt [3]{2}+\left (50+6 \sqrt {69}\right )^{2/3}-2 \sqrt [3]{25+3 \sqrt {69}} (1+3 x)\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {16}{9} \text {Subst}\left (\int \left (\frac {36 \left (-2 \sqrt [3]{2}-25 \sqrt [3]{25+3 \sqrt {69}}-\left (50+6 \sqrt {69}\right )^{2/3}\right )}{\left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right ) \left (2 \sqrt [3]{2}+\left (2 \left (25+3 \sqrt {69}\right )\right )^{2/3}-6 \sqrt [3]{25+3 \sqrt {69}} x\right )}+\frac {36 \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )-3 \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right ) x\right )}{\left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right ) \left (2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2\right )}\right ) \, dx,x,\frac {1}{3}+x\right )-\frac {\left (16 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )\right ) \text {Subst}\left (\int \frac {3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )+36 \left (25+3 \sqrt {69}\right )^{2/3} x}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}+\frac {\left (16 \left (9\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right )+36 \left (25+3 \sqrt {69}\right )^{2/3} \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{9 \left (25+3 \sqrt {69}\right )^{2/3} \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )} \\ & = -x-\frac {16 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \log \left (\sqrt [3]{2} \left (9+\sqrt {69}\right ) \left (\sqrt [3]{2}+\sqrt [3]{25+3 \sqrt {69}}\right )+25\ 2^{2/3} x+3\ 2^{2/3} \sqrt {69} x+4 \left (25+3 \sqrt {69}\right )^{2/3} x+2 \sqrt [3]{2 \left (25+3 \sqrt {69}\right )} x+6 \left (25+3 \sqrt {69}\right )^{2/3} x^2\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )-\frac {64 \text {Subst}\left (\int \frac {-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )-3 \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right ) x}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}}-\frac {\left (32 \left (9\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right )+36 \left (25+3 \sqrt {69}\right )^{2/3} \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-108 \left (623 \sqrt [3]{2}+75 \sqrt [3]{2} \sqrt {69}-2 \left (25+3 \sqrt {69}\right )^{4/3}+\left (50+6 \sqrt {69}\right )^{2/3}\right )-x^2} \, dx,x,3 \left (2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )+4 \left (25+3 \sqrt {69}\right )^{2/3} (1+3 x)\right )\right )}{9 \left (25+3 \sqrt {69}\right )^{2/3} \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )} \\ & = -x+\frac {16\ 3^{5/6} \sqrt [3]{46 \left (207+25 \sqrt {69}\right )} \left (2-\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}\right ) \arctan \left (\frac {2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )+4 \left (25+3 \sqrt {69}\right )^{2/3} (1+3 x)}{2 \sqrt {3 \left (623 \sqrt [3]{2}+75 \sqrt [3]{2} \sqrt {69}-2 \left (25+3 \sqrt {69}\right )^{4/3}+\left (50+6 \sqrt {69}\right )^{2/3}\right )}}\right )}{\sqrt {623 \sqrt [3]{2}+75 \sqrt [3]{2} \sqrt {69}-2 \left (25+3 \sqrt {69}\right )^{4/3}+\left (50+6 \sqrt {69}\right )^{2/3}} \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {16 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \log \left (\sqrt [3]{2} \left (9+\sqrt {69}\right ) \left (\sqrt [3]{2}+\sqrt [3]{25+3 \sqrt {69}}\right )+25\ 2^{2/3} x+3\ 2^{2/3} \sqrt {69} x+4 \left (25+3 \sqrt {69}\right )^{2/3} x+2 \sqrt [3]{2 \left (25+3 \sqrt {69}\right )} x+6 \left (25+3 \sqrt {69}\right )^{2/3} x^2\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {\left (16 \left (25+2 \sqrt [3]{\frac {2}{25+3 \sqrt {69}}}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )\right ) \text {Subst}\left (\int \frac {3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )+36 \left (25+3 \sqrt {69}\right )^{2/3} x}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{3 \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {\left (16 \left (9\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right )+36 \left (25+3 \sqrt {69}\right )^{2/3} \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2\ 2^{2/3}-2 \left (25+3 \sqrt {69}\right )^{2/3}+\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{4/3}+3\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) x+18 \left (25+3 \sqrt {69}\right )^{2/3} x^2} \, dx,x,\frac {1}{3}+x\right )}{9 \left (25+3 \sqrt {69}\right )^{2/3} \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )} \\ & = -x+\frac {16\ 3^{5/6} \sqrt [3]{46 \left (207+25 \sqrt {69}\right )} \left (2-\sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}\right ) \arctan \left (\frac {2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )+4 \left (25+3 \sqrt {69}\right )^{2/3} (1+3 x)}{2 \sqrt {3 \left (623 \sqrt [3]{2}+75 \sqrt [3]{2} \sqrt {69}-2 \left (25+3 \sqrt {69}\right )^{4/3}+\left (50+6 \sqrt {69}\right )^{2/3}\right )}}\right )}{\sqrt {623 \sqrt [3]{2}+75 \sqrt [3]{2} \sqrt {69}-2 \left (25+3 \sqrt {69}\right )^{4/3}+\left (50+6 \sqrt {69}\right )^{2/3}} \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )}-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right )+\frac {\left (32 \left (9\ 2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right ) \left (25 \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \left (25+3 \sqrt {69}\right )+2 \sqrt [3]{50+6 \sqrt {69}}\right )+36 \left (25+3 \sqrt {69}\right )^{2/3} \left (-2 \left (25+3 \sqrt {69}\right )^{2/3}-\left (25-3 \sqrt {69}\right ) \sqrt [3]{50+6 \sqrt {69}}-2^{2/3} \left (623+75 \sqrt {69}\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-108 \left (623 \sqrt [3]{2}+75 \sqrt [3]{2} \sqrt {69}-2 \left (25+3 \sqrt {69}\right )^{4/3}+\left (50+6 \sqrt {69}\right )^{2/3}\right )-x^2} \, dx,x,3 \left (2^{2/3} \left (25+3 \sqrt {69}+2^{2/3} \sqrt [3]{25+3 \sqrt {69}}\right )+4 \left (25+3 \sqrt {69}\right )^{2/3} (1+3 x)\right )\right )}{9 \left (25+3 \sqrt {69}\right )^{2/3} \left (6+6 \left (\frac {2}{25+3 \sqrt {69}}\right )^{2/3}+2 \sqrt [3]{2} \left (25+3 \sqrt {69}\right )^{2/3}+2^{2/3} \sqrt [3]{623+75 \sqrt {69}}\right )} \\ & = -x-\frac {1}{3} x \log ^2\left (\frac {x^2 \log (3)}{\left (1-x^2-x^3\right )^2}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.85 (sec) , antiderivative size = 223, normalized size of antiderivative = 8.26 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=\frac {x \left (3+\log ^2\left (\frac {x^2 \log (3)}{\left (-1+x^2+x^3\right )^2}\right )\right ) \left (3+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]^2+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]^2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]+2 \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ] \text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]^2\right )}{3 \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,1\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]\right ) \left (\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,2\right ]-\text {Root}\left [\text {$\#$1}^3+\text {$\#$1}^2-1\&,3\right ]\right )} \]
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Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.56
method | result | size |
norman | \(-x -\frac {x \ln \left (\frac {x^{2} \ln \left (3\right )}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}\) | \(42\) |
risch | \(-x -\frac {x \ln \left (\frac {x^{2} \ln \left (3\right )}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}\) | \(42\) |
parallelrisch | \(-\frac {x \ln \left (\frac {x^{2} \ln \left (3\right )}{x^{6}+2 x^{5}+x^{4}-2 x^{3}-2 x^{2}+1}\right )^{2}}{3}+2-x\) | \(43\) |
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Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {1}{3} \, x \log \left (\frac {x^{2} \log \left (3\right )}{x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1}\right )^{2} - x \]
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Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=- \frac {x \log {\left (\frac {x^{2} \log {\left (3 \right )}}{x^{6} + 2 x^{5} + x^{4} - 2 x^{3} - 2 x^{2} + 1} \right )}^{2}}{3} - x \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.30 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {4}{3} \, x \log \left (x^{3} + x^{2} - 1\right )^{2} - \frac {4}{3} \, x \log \left (x\right )^{2} - \frac {4}{3} \, x \log \left (x\right ) \log \left (\log \left (3\right )\right ) - \frac {1}{3} \, {\left (\log \left (\log \left (3\right )\right )^{2} + 3\right )} x + \frac {4}{3} \, {\left (2 \, x \log \left (x\right ) + x \log \left (\log \left (3\right )\right )\right )} \log \left (x^{3} + x^{2} - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (25) = 50\).
Time = 0.91 (sec) , antiderivative size = 79, normalized size of antiderivative = 2.93 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {1}{3} \, x \log \left (x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1\right )^{2} + \frac {2}{3} \, x \log \left (x^{6} + 2 \, x^{5} + x^{4} - 2 \, x^{3} - 2 \, x^{2} + 1\right ) \log \left (x^{2} \log \left (3\right )\right ) - \frac {1}{3} \, x \log \left (x^{2} \log \left (3\right )\right )^{2} - x \]
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Time = 8.82 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {3-3 x^2-3 x^3+\left (4+4 x^2+8 x^3\right ) \log \left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )+\left (1-x^2-x^3\right ) \log ^2\left (\frac {x^2 \log (3)}{1-2 x^2-2 x^3+x^4+2 x^5+x^6}\right )}{-3+3 x^2+3 x^3} \, dx=-\frac {x\,\left ({\ln \left (\frac {x^2\,\ln \left (3\right )}{x^6+2\,x^5+x^4-2\,x^3-2\,x^2+1}\right )}^2+3\right )}{3} \]
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