Optimal. Leaf size=263 \[ \frac{1}{4} \sqrt{\frac{3}{13}} \tan ^{-1}\left (\frac{-5 x^2+12 x+8}{\sqrt{39} x^2}\right )-\frac{1}{8} \sqrt{\frac{235 \sqrt{517}-5167}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2-\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )+\frac{1}{8} \sqrt{\frac{235 \sqrt{517}-5167}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2+\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )-\frac{1}{4} \sqrt{\frac{5167+235 \sqrt{517}}{40326}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )-\frac{1}{4} \sqrt{\frac{5167+235 \sqrt{517}}{40326}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right ) \]
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Rubi [A] time = 0.492824, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {2069, 12, 1673, 1169, 634, 618, 204, 628, 1107} \[ \frac{1}{4} \sqrt{\frac{3}{13}} \tan ^{-1}\left (\frac{-5 x^2+12 x+8}{\sqrt{39} x^2}\right )-\frac{1}{8} \sqrt{\frac{235 \sqrt{517}-5167}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2-\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )+\frac{1}{8} \sqrt{\frac{235 \sqrt{517}-5167}{40326}} \log \left (\left (\frac{4}{x}+3\right )^2+\sqrt{2 \left (19+\sqrt{517}\right )} \left (\frac{4}{x}+3\right )+\sqrt{517}\right )-\frac{1}{4} \sqrt{\frac{5167+235 \sqrt{517}}{40326}} \tan ^{-1}\left (\frac{\frac{8}{x}-\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right )-\frac{1}{4} \sqrt{\frac{5167+235 \sqrt{517}}{40326}} \tan ^{-1}\left (\frac{\frac{8}{x}+\sqrt{2 \left (19+\sqrt{517}\right )}+6}{\sqrt{2 \left (\sqrt{517}-19\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 2069
Rule 12
Rule 1673
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rule 1107
Rubi steps
\begin{align*} \int \frac{1}{8+24 x+8 x^2-15 x^3+8 x^4} \, dx &=-\left (1024 \operatorname{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 \operatorname{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 \operatorname{Subst}\left (\int -\frac{1536 x}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )\right )-128 \operatorname{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 \operatorname{Subst}\left (\int \frac{x}{2117632-2490368 x^2+1048576 x^4} \, dx,x,\frac{3}{4}+\frac{1}{x}\right )-\frac{\operatorname{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{\operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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}} \operatorname{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}} \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.0096563, size = 55, normalized size = 0.21 \[ \text{RootSum}\left [8 \text{$\#$1}^4-15 \text{$\#$1}^3+8 \text{$\#$1}^2+24 \text{$\#$1}+8\& ,\frac{\log (x-\text{$\#$1})}{32 \text{$\#$1}^3-45 \text{$\#$1}^2+16 \text{$\#$1}+24}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.005, size = 49, normalized size = 0.2 \begin{align*} \sum _{{\it \_R}={\it RootOf} \left ( 8\,{{\it \_Z}}^{4}-15\,{{\it \_Z}}^{3}+8\,{{\it \_Z}}^{2}+24\,{\it \_Z}+8 \right ) }{\frac{\ln \left ( x-{\it \_R} \right ) }{32\,{{\it \_R}}^{3}-45\,{{\it \_R}}^{2}+16\,{\it \_R}+24}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.851635, size = 41, normalized size = 0.16 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{8 \, x^{4} - 15 \, x^{3} + 8 \, x^{2} + 24 \, x + 8}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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