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 = 0.38, antiderivative size = 260, normalized size of antiderivative = 1.00, 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 50
Rule 63
Rule 80
Rule 203
Rule 204
Rule 295
Rule 331
Rule 618
Rule 628
Rule 634
Rule 6171
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 = 0.80, 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]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 240, normalized size = 0.92 \[ -\frac {\sqrt {3} x^{2} \log \left (16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16\right ) - \sqrt {3} x^{2} \log \left (-16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16\right ) + 4 \, x^{2} \arctan \left (\sqrt {3} + \frac {1}{2} \, \sqrt {-16 \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 16 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 16} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 4 \, x^{2} \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1} - 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 4 \, x^{2} \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - 6 \, {\left (4 \, x^{2} + 7 \, x + 3\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{36 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.16, size = 175, normalized size = 0.67 \[ -\frac {1}{36} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{36} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} + 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{2}} + \frac {1}{18} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{18} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{9} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 20.93, size = 2997, normalized size = 11.53 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 178, normalized size = 0.68 \[ -\frac {1}{36} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{36} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {\left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 7 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{3 \, {\left (\frac {2 \, {\left (x - 1\right )}}{x + 1} + \frac {{\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + 1\right )}} + \frac {1}{18} \, \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{18} \, \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{9} \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.11, size = 136, normalized size = 0.52 \[ \frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )}{9}+\frac {\frac {7\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{3}+\frac {{\left (\frac {x-1}{x+1}\right )}^{11/6}}{3}}{\frac {2\,\left (x-1\right )}{x+1}+\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+1}-\mathrm {atan}\left (\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{243\,\left (-\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\mathrm {atan}\left (\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{243\,\left (\frac {1}{243}+\frac {\sqrt {3}\,1{}\mathrm {i}}{243}\right )}\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________