Optimal. Leaf size=402 \[ -\sqrt {3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \text {ArcTan}\left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )-\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\text {ArcTan}\left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \text {ArcTan}\left (\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \sqrt {3} \log \left (1-\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right ) \]
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Rubi [A]
time = 0.37, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 13, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {6306, 132,
65, 338, 301, 648, 632, 210, 642, 209, 95, 216, 212} \begin {gather*} -\sqrt {3} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}+1}{\sqrt {3}}\right )-\text {ArcTan}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )+\text {ArcTan}\left (\frac {2 \sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}+\sqrt {3}\right )+2 \text {ArcTan}\left (\frac {\sqrt [6]{\frac {x-1}{x}}}{\sqrt [6]{\frac {1}{x}+1}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \sqrt {3} \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 )-\frac {1}{2} \sqrt {3} \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 )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{\frac {x-1}{x}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 132
Rule 209
Rule 210
Rule 212
Rule 216
Rule 301
Rule 338
Rule 632
Rule 642
Rule 648
Rule 6306
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx &=-\text {Subst}\left (\int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x} x} \, dx,x,\frac {1}{x}\right )\\ &=-\text {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right )-\text {Subst}\left (\int \frac {1}{\sqrt [6]{1-x} x (1+x)^{5/6}} \, dx,x,\frac {1}{x}\right )\\ &=6 \text {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{\frac {-1+x}{x}}\right )-6 \text {Subst}\left (\int \frac {1}{-1+x^6} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )\\ &=2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+2 \text {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 )+2 \text {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 )+6 \text {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 )\\ &=2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \text {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 {1}{2} \text {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 {3}{2} \text {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 {3}{2} \text {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 )+2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \text {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 )+2 \text {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 )\\ &=2 \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \text {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}{2} \text {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 )-3 \text {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 )-3 \text {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 {1}{2} \sqrt {3} \text {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 )-\frac {1}{2} \sqrt {3} \text {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 )\\ &=-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )+2 \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \sqrt {3} \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 )-\frac {1}{2} \sqrt {3} \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 )-\text {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 )-\text {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 )\\ &=-\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}}{\sqrt {3}}\right )-\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tan ^{-1}\left (\frac {\sqrt [6]{-\frac {1-x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \tanh ^{-1}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{-\frac {1-x}{x}}}\right )-\frac {1}{2} \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 {1}{2} \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 {1}{2} \sqrt {3} \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 )-\frac {1}{2} \sqrt {3} \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 )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 26, normalized size = 0.06 \begin {gather*} \frac {12}{7} e^{\frac {7}{3} \coth ^{-1}(x)} \, _2F_1\left (\frac {7}{12},1;\frac {19}{12};e^{4 \coth ^{-1}(x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 7.82, size = 2714, normalized size = 6.75
method | result | size |
trager | \(\text {Expression too large to display}\) | \(2714\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 340, normalized size = 0.85 \begin {gather*} -\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 ) - \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 {1}{2} \, \sqrt {3} \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 ) + \frac {1}{2} \, \sqrt {3} \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 ) - 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 ) - 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 ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \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 {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 261, normalized size = 0.65 \begin {gather*} -\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \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 {1}{2} \, \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}{2} \, \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 ) + \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \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 {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.27, size = 167, normalized size = 0.42 \begin {gather*} 2\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )-\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}-\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}+1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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