Optimal. Leaf size=260 \[ \frac{1}{2} \left (\frac{x-1}{x}\right )^{5/6} \left (\frac{1}{x}+1\right )^{7/6}+\frac{1}{6} \left (\frac{x-1}{x}\right )^{5/6} \sqrt [6]{\frac{1}{x}+1}+\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{12 \sqrt{3}}-\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{12 \sqrt{3}}-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.379102, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6171, 80, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ \frac{1}{2} \left (\frac{x-1}{x}\right )^{5/6} \left (\frac{1}{x}+1\right )^{7/6}+\frac{1}{6} \left (\frac{x-1}{x}\right )^{5/6} \sqrt [6]{\frac{1}{x}+1}+\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}-\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{12 \sqrt{3}}-\frac{\log \left (\frac{\sqrt [3]{\frac{x-1}{x}}}{\sqrt [3]{\frac{1}{x}+1}}+\frac{\sqrt{3} \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+1\right )}{12 \sqrt{3}}-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right )+\frac{1}{18} \tan ^{-1}\left (\frac{2 \sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}+\sqrt{3}\right )+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{\frac{x-1}{x}}}{\sqrt [6]{\frac{1}{x}+1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6171
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^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}-\frac{1}{18} \operatorname{Subst}\left (\int \frac{1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}+\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac{-1+x}{x}}\right )\\ &=\frac{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}+\frac{1}{3} \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{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}+\frac{1}{9} \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{1}{9} \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{1}{9} \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{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{1}{36} \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{1}{36} \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{\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 )}{12 \sqrt{3}}-\frac{\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 )}{12 \sqrt{3}}\\ &=\frac{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{\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 )}{12 \sqrt{3}}-\frac{\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 )}{12 \sqrt{3}}-\frac{1}{18} \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{1}{18} \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{1}{6} \sqrt [6]{1+\frac{1}{x}} \left (-\frac{1-x}{x}\right )^{5/6}+\frac{1}{2} \left (1+\frac{1}{x}\right )^{7/6} \left (\frac{-1+x}{x}\right )^{5/6}-\frac{1}{18} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{1}{18} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{1}{9} \tan ^{-1}\left (\frac{\sqrt [6]{-\frac{1-x}{x}}}{\sqrt [6]{1+\frac{1}{x}}}\right )+\frac{\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 )}{12 \sqrt{3}}-\frac{\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 )}{12 \sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.7009, size = 124, normalized size = 0.48 \[ \frac{1}{54} \left (\text{RootSum}\left [\text{$\#$1}^4-\text{$\#$1}^2+1\& ,\frac{\text{$\#$1}^2 \left (-\coth ^{-1}(x)\right )+3 \text{$\#$1}^2 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right )-6 \log \left (e^{\frac{1}{3} \coth ^{-1}(x)}-\text{$\#$1}\right )+2 \coth ^{-1}(x)}{2 \text{$\#$1}^3-\text{$\#$1}}\& \right ]+\frac{18 e^{\frac{1}{3} \coth ^{-1}(x)} \left (7 e^{2 \coth ^{-1}(x)}+1\right )}{\left (e^{2 \coth ^{-1}(x)}+1\right )^2}-6 \tan ^{-1}\left (e^{\frac{1}{3} \coth ^{-1}(x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}{\frac{1}{\sqrt [6]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.59628, size = 240, normalized size = 0.92 \begin{align*} -\frac{1}{36} \, \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{1}{36} \, \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{\left (\frac{x - 1}{x + 1}\right )^{\frac{11}{6}} + 7 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{3 \,{\left (\frac{2 \,{\left (x - 1\right )}}{x + 1} + \frac{{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + 1\right )}} + \frac{1}{18} \, \arctan \left (\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{1}{18} \, \arctan \left (-\sqrt{3} + 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{1}{9} \, \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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.76004, size = 720, normalized size = 2.77 \begin{align*} -\frac{\sqrt{3} x^{2} \log \left (16 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 16 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 16\right ) - \sqrt{3} x^{2} \log \left (-16 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 16 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 16\right ) + 4 \, x^{2} \arctan \left (\sqrt{3} + \frac{1}{2} \, \sqrt{-16 \, \sqrt{3} \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}} + 16 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} + 16} - 2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + 4 \, x^{2} \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 ) - 4 \, x^{2} \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) - 6 \,{\left (4 \, x^{2} + 7 \, x + 3\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{36 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \sqrt [6]{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.41753, size = 348, normalized size = 1.34 \begin{align*} \frac{1}{18} \, \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{1}{6} \, \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{2 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{2}{3}} + \left (\frac{x - 1}{x + 1}\right )^{\frac{1}{3}} - 1}{9 \,{\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{{\left (x - 1\right )} \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{x + 1} + 7 \, \left (\frac{x - 1}{x + 1}\right )^{\frac{5}{6}}}{3 \,{\left (\frac{x - 1}{x + 1} + 1\right )}^{2}} + \frac{1}{9} \, \arctan \left (\left (\frac{x - 1}{x + 1}\right )^{\frac{1}{6}}\right ) + \frac{5}{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{1}{12} \, \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{2}{27} \, \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.
[In]
[Out]