Integrand size = 22, antiderivative size = 263 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \arctan \left (\frac {6-\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \arctan \left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )+\frac {1}{4} \sqrt {\frac {3}{13}} \arctan \left (\frac {8+12 x-5 x^2}{\sqrt {39} x^2}\right )-\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )+\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right ) \]
[Out]
Time = 0.39 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2094, 12, 1687, 1183, 648, 632, 210, 642, 1121} \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\frac {1}{4} \sqrt {\frac {3}{13}} \arctan \left (\frac {-5 x^2+12 x+8}{\sqrt {39} x^2}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \arctan \left (\frac {\frac {8}{x}-\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \arctan \left (\frac {\frac {8}{x}+\sqrt {2 \left (19+\sqrt {517}\right )}+6}{\sqrt {2 \left (\sqrt {517}-19\right )}}\right )-\frac {1}{8} \sqrt {\frac {235 \sqrt {517}-5167}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2-\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right )+\frac {1}{8} \sqrt {\frac {235 \sqrt {517}-5167}{40326}} \log \left (\left (\frac {4}{x}+3\right )^2+\sqrt {2 \left (19+\sqrt {517}\right )} \left (\frac {4}{x}+3\right )+\sqrt {517}\right ) \]
[In]
[Out]
Rule 12
Rule 210
Rule 632
Rule 642
Rule 648
Rule 1121
Rule 1183
Rule 1687
Rule 2094
Rubi steps \begin{align*} \text {integral}& = -\left (1024 \text {Subst}\left (\int \frac {(24-32 x)^2}{8 \left (2117632-2490368 x^2+1048576 x^4\right )} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right ) \\ & = -\left (128 \text {Subst}\left (\int \frac {(24-32 x)^2}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right ) \\ & = -\left (128 \text {Subst}\left (\int -\frac {1536 x}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )\right )-128 \text {Subst}\left (\int \frac {576+1024 x^2}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right ) \\ & = 196608 \text {Subst}\left (\int \frac {x}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )-\frac {\text {Subst}\left (\int \frac {144 \sqrt {2 \left (19+\sqrt {517}\right )}-\left (576-64 \sqrt {517}\right ) x}{\frac {\sqrt {517}}{16}-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{256 \sqrt {1034 \left (19+\sqrt {517}\right )}}-\frac {\text {Subst}\left (\int \frac {144 \sqrt {2 \left (19+\sqrt {517}\right )}+\left (576-64 \sqrt {517}\right ) x}{\frac {\sqrt {517}}{16}+\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{256 \sqrt {1034 \left (19+\sqrt {517}\right )}} \\ & = 98304 \text {Subst}\left (\int \frac {1}{2117632-2490368 x+1048576 x^2} \, dx,x,\left (\frac {3}{4}+\frac {1}{x}\right )^2\right )-\frac {\left (517+9 \sqrt {517}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {517}}{16}-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{16544}-\frac {\left (517+9 \sqrt {517}\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {517}}{16}+\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )}{16544}-\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \text {Subst}\left (\int \frac {-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}+2 x}{\frac {\sqrt {517}}{16}-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right )+\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \text {Subst}\left (\int \frac {\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}+2 x}{\frac {\sqrt {517}}{16}+\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )} x+x^2} \, dx,x,\frac {3}{4}+\frac {1}{x}\right ) \\ & = -\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )+\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )-196608 \text {Subst}\left (\int \frac {1}{-2680059592704-x^2} \, dx,x,-2490368+2097152 \left (\frac {3}{4}+\frac {1}{x}\right )^2\right )+\frac {\left (517+9 \sqrt {517}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{8} \left (19-\sqrt {517}\right )-x^2} \, dx,x,-\frac {1}{2} \sqrt {\frac {1}{2} \left (19+\sqrt {517}\right )}+2 \left (\frac {3}{4}+\frac {1}{x}\right )\right )}{8272}+\frac {\left (517+9 \sqrt {517}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{8} \left (19-\sqrt {517}\right )-x^2} \, dx,x,\frac {1}{4} \left (6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}\right )\right )}{8272} \\ & = -\frac {1}{4} \sqrt {\frac {3}{13}} \tan ^{-1}\left (\frac {19-\left (3+\frac {4}{x}\right )^2}{2 \sqrt {39}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \tan ^{-1}\left (\frac {6+\sqrt {2 \left (19+\sqrt {517}\right )}+\frac {8}{x}}{\sqrt {2 \left (-19+\sqrt {517}\right )}}\right )-\frac {1}{4} \sqrt {\frac {5167+235 \sqrt {517}}{40326}} \tan ^{-1}\left (\frac {8+\left (6-\sqrt {2 \left (19+\sqrt {517}\right )}\right ) x}{\sqrt {2 \left (-19+\sqrt {517}\right )} x}\right )-\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}-\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right )+\frac {1}{8} \sqrt {\frac {-5167+235 \sqrt {517}}{40326}} \log \left (\sqrt {517}+\sqrt {2 \left (19+\sqrt {517}\right )} \left (3+\frac {4}{x}\right )+\left (3+\frac {4}{x}\right )^2\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.21 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\text {RootSum}\left [8+24 \text {$\#$1}+8 \text {$\#$1}^2-15 \text {$\#$1}^3+8 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{24+16 \text {$\#$1}-45 \text {$\#$1}^2+32 \text {$\#$1}^3}\&\right ] \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.19
method | result | size |
default | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\) | \(49\) |
risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{4}-15 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}+24 \textit {\_Z} +8\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{32 \textit {\_R}^{3}-45 \textit {\_R}^{2}+16 \textit {\_R} +24}\) | \(49\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.98 (sec) , antiderivative size = 1297, normalized size of antiderivative = 4.93 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 1.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.16 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\operatorname {RootSum} {\left (50326848 t^{4} + 765960 t^{2} + 12753 t + 64, \left ( t \mapsto t \log {\left (\frac {100785893208 t^{3}}{4758335} - \frac {1430512512 t^{2}}{4758335} + \frac {72982352521 t}{223641745} + x + \frac {2270349121}{1789133960} \right )} \right )\right )} \]
[In]
[Out]
\[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\int { \frac {1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8} \,d x } \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.47 \[ \int \frac {1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx=\sum _{k=1}^4\ln \left (-\frac {\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )\,\left (2184\,\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )+256\,x+\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )\,x\,38259+{\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )}^2\,x\,1531920+805896\,{\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right )}^2-120\right )}{4096}\right )\,\mathrm {root}\left (z^4+\frac {2455\,z^2}{161304}+\frac {109\,z}{430144}+\frac {1}{786357},z,k\right ) \]
[In]
[Out]