Optimal. Leaf size=287 \[ \frac{1}{18} \left (\frac{x-1}{x}\right )^{5/6} \left (\frac{1}{x}+1\right )^{7/6}+\frac{\left (\frac{x-1}{x}\right )^{5/6} \left (\frac{1}{x}+1\right )^{7/6}}{3 x}+\frac{19}{54} \left (\frac{x-1}{x}\right )^{5/6} \sqrt [6]{\frac{1}{x}+1}+\frac{19 \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{108 \sqrt{3}}-\frac{19 \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{108 \sqrt{3}}-\frac{19}{162} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\frac{19}{162} \tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+\frac{19}{81} \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right ) \]
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Rubi [A] time = 0.398351, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {6171, 90, 80, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ \frac{1}{18} \left (\frac{x-1}{x}\right )^{5/6} \left (\frac{1}{x}+1\right )^{7/6}+\frac{\left (\frac{x-1}{x}\right )^{5/6} \left (\frac{1}{x}+1\right )^{7/6}}{3 x}+\frac{19}{54} \left (\frac{x-1}{x}\right )^{5/6} \sqrt [6]{\frac{1}{x}+1}+\frac{19 \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{108 \sqrt{3}}-\frac{19 \log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{108 \sqrt{3}}-\frac{19}{162} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\frac{19}{162} \tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+\frac{19}{81} \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 6171
Rule 90
Rule 80
Rule 50
Rule 63
Rule 331
Rule 295
Rule 634
Rule 618
Rule 204
Rule 628
Rule 203
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{3} \coth ^{-1}(x)}}{x^4} \, dx &=-\operatorname{Subst}\left (\int \frac{x^2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{\left (-1-\frac{x}{3}\right ) \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}-\frac{19}{54} \operatorname{Subst}\left (\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}-\frac{19}{162} \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}+\frac{19}{27} \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac{-1+x}{x}}\right )\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}+\frac{19}{27} \operatorname{Subst}\left (\int \frac{x^4}{1+x^6} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}+\frac{19}{81} \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19}{81} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}+\frac{\sqrt{3} x}{2}}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19}{81} \operatorname{Subst}\left (\int \frac{-\frac{1}{2}-\frac{\sqrt{3} x}{2}}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}+\frac{19}{81} \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19}{324} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19}{324} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19 \operatorname{Subst}\left (\int \frac{-\sqrt{3}+2 x}{1-\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )}{108 \sqrt{3}}-\frac{19 \operatorname{Subst}\left (\int \frac{\sqrt{3}+2 x}{1+\sqrt{3} x+x^2} \, dx,x,\frac{\sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )}{108 \sqrt{3}}\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}+\frac{19}{81} \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19 \log \left (1-\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )}{108 \sqrt{3}}-\frac{19 \log \left (1+\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )}{108 \sqrt{3}}-\frac{19}{162} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,-\sqrt{3}+\frac{2 \sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )-\frac{19}{162} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{3}+\frac{2 \sqrt [6]{\frac{-1+x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )\\ &=\frac{19}{54} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{18} \left (1+\frac{1}{x}\right )^{7/6} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{\left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}}{3 x}-\frac{19}{162} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19}{162} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19}{81} \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{19 \log \left (1-\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )}{108 \sqrt{3}}-\frac{19 \log \left (1+\frac{\sqrt{3} \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}+\frac{\sqrt [3]{-\frac{1-x}{x}}}{\sqrt [3]{1+\frac{1}{x}}}\right )}{108 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.206159, size = 133, normalized size = 0.46 \[ \frac{1}{486} \left (-19 \text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\text{$\#$1}^2 \coth ^{-1}(x)-3 \text{$\#$1}^2 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right )+6 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right )-2 \coth ^{-1}(x)}{2 \text{$\#$1}^3-\text{$\#$1}}\& \right ]+\frac{18 e^{\frac{1}{3} \coth ^{-1}(x)} \left (8 e^{2 \coth ^{-1}(x)}+61 e^{4 \coth ^{-1}(x)}+19\right )}{\left (e^{2 \coth ^{-1}(x)}+1\right )^3}-114 \tan ^{-1}\left (e^{\frac{1}{3} \coth ^{-1}(x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}}{\frac{1}{\sqrt [6]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56902, size = 277, normalized size = 0.97 \begin{align*} -\frac{19}{324} \, \sqrt{3} \log \left (\sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + \frac{19}{324} \, \sqrt{3} \log \left (-\sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + \frac{19 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{17}{6}} + 8 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{11}{6}} + 61 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{27 \,{\left (\frac{3 \,{\left (x - 1\right )}}{x + 1} + \frac{3 \,{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac{{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} + 1\right )}} + \frac{19}{162} \, \arctan \left (\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{19}{162} \, \arctan \left (-\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{19}{81} \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74442, size = 775, normalized size = 2.7 \begin{align*} -\frac{19 \, \sqrt{3} x^{3} \log \left (5776 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 5776 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 5776\right ) - 19 \, \sqrt{3} x^{3} \log \left (-5776 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 5776 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 5776\right ) + 76 \, x^{3} \arctan \left (\sqrt{3} + \frac{1}{38} \, \sqrt{-5776 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 5776 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 5776} - 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + 76 \, x^{3} \arctan \left (-\sqrt{3} + 2 \, \sqrt{\sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1} - 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) - 76 \, x^{3} \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) - 6 \,{\left (22 \, x^{3} + 43 \, x^{2} + 39 \, x + 18\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{324 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{4} \sqrt [6]{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37161, size = 381, normalized size = 1.33 \begin{align*} \frac{19}{162} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}\right ) - \frac{19}{54} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right )}\right ) - \frac{19 \,{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right )}}{81 \,{\left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right )}{\left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1\right )}} + \frac{\frac{8 \,{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{x + 1} + \frac{19 \,{\left (x - 1\right )}^{2} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{{\left (x + 1\right )}^{2}} + 61 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{27 \,{\left (\frac{x - 1}{x + 1} + 1\right )}^{3}} + \frac{19}{81} \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{95}{972} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) - \frac{19}{108} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 1\right ) + \frac{38}{243} \, \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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