Optimal. Leaf size=285 \[ \frac{1}{3} \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x^3+\frac{7}{18} \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x^2+\frac{11}{27} \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x-\frac{19}{324} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}-\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{19}{324} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}+\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )-\frac{19 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19 \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19}{81} \tanh ^{-1}\left (\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}\right ) \]
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Rubi [A] time = 0.249146, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6171, 99, 151, 12, 93, 210, 634, 618, 204, 628, 206} \[ \frac{1}{3} \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x^3+\frac{7}{18} \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x^2+\frac{11}{27} \sqrt [6]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{5/6} x-\frac{19}{324} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}-\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )+\frac{19}{324} \log \left (\frac{\sqrt [3]{\frac{1}{x}+1}}{\sqrt [3]{\frac{x-1}{x}}}+\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1\right )-\frac{19 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19 \tan ^{-1}\left (\frac{\frac{2 \sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}+1}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19}{81} \tanh ^{-1}\left (\frac{\sqrt [6]{\frac{1}{x}+1}}{\sqrt [6]{\frac{x-1}{x}}}\right ) \]
Antiderivative was successfully verified.
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Rule 6171
Rule 99
Rule 151
Rule 12
Rule 93
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int e^{\frac{1}{3} \coth ^{-1}(x)} x^2 \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x} x^4} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3-\frac{1}{3} \operatorname{Subst}\left (\int \frac{\frac{7}{3}+2 x}{\sqrt [6]{1-x} x^3 (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3+\frac{1}{6} \operatorname{Subst}\left (\int \frac{-\frac{22}{9}-\frac{7 x}{3}}{\sqrt [6]{1-x} x^2 (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3-\frac{1}{6} \operatorname{Subst}\left (\int \frac{19}{27 \sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3-\frac{19}{162} \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3-\frac{19}{27} \operatorname{Subst}\left (\int \frac{1}{-1+x^6} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3+\frac{19}{81} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{19}{81} \operatorname{Subst}\left (\int \frac{1-\frac{x}{2}}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{19}{81} \operatorname{Subst}\left (\int \frac{1+\frac{x}{2}}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3+\frac{19}{81} \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{19}{324} \operatorname{Subst}\left (\int \frac{-1+2 x}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{19}{324} \operatorname{Subst}\left (\int \frac{1+2 x}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{19}{108} \operatorname{Subst}\left (\int \frac{1}{1-x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )+\frac{19}{108} \operatorname{Subst}\left (\int \frac{1}{1+x+x^2} \, dx,x,\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3+\frac{19}{81} \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{19}{324} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}-\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{19}{324} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}+\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{19}{54} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )-\frac{19}{54} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{\frac{-1+x}{x}}}\right )\\ &=\frac{11}{27} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x+\frac{7}{18} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6} x^2+\frac{1}{3} \sqrt [6]{1+\frac{1}{x}} \left (\frac{-1+x}{x}\right )^{5/6} x^3-\frac{19 \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19 \tan ^{-1}\left (\frac{1+\frac{2 \sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}}{\sqrt{3}}\right )}{54 \sqrt{3}}+\frac{19}{81} \tanh ^{-1}\left (\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )-\frac{19}{324} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}-\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )+\frac{19}{324} \log \left (1+\frac{\sqrt [3]{1+\frac{1}{x}}}{\sqrt [3]{-\frac{1-x}{x}}}+\frac{\sqrt [6]{1+\frac{1}{x}}}{\sqrt [6]{-\frac{1-x}{x}}}\right )\\ \end{align*}
Mathematica [A] time = 5.249, size = 189, normalized size = 0.66 \[ \frac{1}{324} \left (\frac{732 e^{\frac{1}{3} \coth ^{-1}(x)}}{e^{2 \coth ^{-1}(x)}-1}+\frac{1368 e^{\frac{1}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^2}+\frac{864 e^{\frac{1}{3} \coth ^{-1}(x)}}{\left (e^{2 \coth ^{-1}(x)}-1\right )^3}-38 \log \left (1-e^{\frac{1}{3} \coth ^{-1}(x)}\right )+38 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}+1\right )-19 \log \left (-e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+19 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}+e^{\frac{2}{3} \coth ^{-1}(x)}+1\right )+38 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}-1}{\sqrt{3}}\right )+38 \sqrt{3} \tan ^{-1}\left (\frac{2 e^{\frac{1}{3} \coth ^{-1}(x)}+1}{\sqrt{3}}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\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 = 1.53666, size = 297, normalized size = 1.04 \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}{6}} + 1\right )}\right ) - \frac{19}{162} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1\right )}\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}{324} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{19}{324} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) + \frac{19}{162} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{19}{162} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64059, size = 581, normalized size = 2.04 \begin{align*} \frac{1}{54} \,{\left (18 \, x^{3} + 39 \, x^{2} + 43 \, x + 22\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}} - \frac{19}{162} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + \frac{1}{3} \, \sqrt{3}\right ) - \frac{19}{162} \, \sqrt{3} \arctan \left (\frac{2}{3} \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - \frac{1}{3} \, \sqrt{3}\right ) + \frac{19}{324} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{19}{324} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) + \frac{19}{162} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{19}{162} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1\right ) \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{x^{2}}{\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 = 1.16715, size = 290, normalized size = 1.02 \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}{6}} + 1\right )}\right ) - \frac{19}{162} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1\right )}\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}{324} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{19}{324} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) + \frac{19}{162} \, \log \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 1\right ) - \frac{19}{162} \, \log \left ({\left | \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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