Optimal. Leaf size=130 \[ \frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{9} \log \left (1+\frac {1}{x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6306, 81, 52,
62} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )}{3 \sqrt {3}}+\frac {1}{2} \left (\frac {x-1}{x}\right )^{2/3} \left (\frac {1}{x}+1\right )^{4/3}+\frac {1}{3} \left (\frac {x-1}{x}\right )^{2/3} \sqrt [3]{\frac {1}{x}+1}-\frac {1}{3} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {1}{9} \log \left (\frac {1}{x}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 62
Rule 81
Rule 6306
Rubi steps
\begin {align*} \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x^3} \, dx &=-\text {Subst}\left (\int \frac {x \sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {1}{3} \text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {2}{9} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3}+\frac {1}{2} \left (1+\frac {1}{x}\right )^{4/3} \left (\frac {-1+x}{x}\right )^{2/3}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-\frac {1-x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{-\frac {1-x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{9} \log \left (1+\frac {1}{x}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 0.17, size = 134, normalized size = 1.03 \begin {gather*} -\frac {2}{27} \left (\frac {27 e^{\frac {2}{3} \coth ^{-1}(x)}}{\left (1+e^{2 \coth ^{-1}(x)}\right )^2}-\frac {36 e^{\frac {2}{3} \coth ^{-1}(x)}}{1+e^{2 \coth ^{-1}(x)}}-2 \coth ^{-1}(x)+3 \log \left (1+e^{\frac {2}{3} \coth ^{-1}(x)}\right )-\text {RootSum}\left [1-\text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\coth ^{-1}(x)-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right )+\coth ^{-1}(x) \text {$\#$1}^2-3 \log \left (e^{\frac {1}{3} \coth ^{-1}(x)}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+\text {$\#$1}^2}\&\right ]\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 = 1.16, size = 507, normalized size = 3.90
method | result | size |
risch | \(\frac {\left (-1+x \right ) \left (5 x +3\right )}{6 x^{2} \left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}}}+\frac {\left (-\frac {2 \ln \left (\frac {8 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-45 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -8 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -30 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}-216 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}-45 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x -16 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}-54 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -27 x^{2}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-24 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-36 x -9}{x \left (1+x \right )}\right )}{9}+\frac {2 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x^{2}+27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+72 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +2 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2} x -27 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x^{2}+135 \left (x^{3}+x^{2}-x -1\right )^{\frac {2}{3}}+72 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}} x +4 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )^{2}+6 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right ) x -36 x^{2}-81 \left (x^{3}+x^{2}-x -1\right )^{\frac {1}{3}}+33 \RootOf \left (\textit {\_Z}^{2}-3 \textit {\_Z} +9\right )-216 x -180}{x \left (1+x \right )}\right )}{27}\right ) \left (\left (1+x \right )^{2} \left (-1+x \right )\right )^{\frac {1}{3}}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}} \left (1+x \right )}\) | \(507\) |
trager | \(\frac {\left (1+x \right ) \left (5 x +3\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}}{6 x^{2}}-\frac {2 \ln \left (\frac {9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}-3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -3 \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-36 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}+6 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +12 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x -1}{x}\right )}{9}+\frac {2 \ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x +1}{x}\right )}{9}-\frac {2 \ln \left (-\frac {9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}} x +9 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) \left (-\frac {1-x}{1+x}\right )^{\frac {2}{3}}+3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}} x -18 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )^{2}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right ) x +3 \left (-\frac {1-x}{1+x}\right )^{\frac {1}{3}}-3 \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )-x +1}{x}\right ) \RootOf \left (9 \textit {\_Z}^{2}-3 \textit {\_Z} +1\right )}{3}\) | \(523\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 124, normalized size = 0.95 \begin {gather*} \frac {2}{9} \, \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 (\left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{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}{9} \, \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}{9} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 111, normalized size = 0.85 \begin {gather*} \frac {4 \, \sqrt {3} x^{2} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 2 \, x^{2} \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - 4 \, x^{2} \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{2} + 8 \, x + 3\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{18 \, x^{2}} \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^{3} \sqrt [3]{\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 = 0.42, size = 122, normalized size = 0.94 \begin {gather*} \frac {2}{9} \, \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 {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{3 \, {\left (\frac {x - 1}{x + 1} + 1\right )}^{2}} + \frac {1}{9} \, \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}{9} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 145, normalized size = 1.12 \begin {gather*} \frac {\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{3}+\frac {2\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{3}}{\frac {2\,\left (x-1\right )}{x+1}+\frac {{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+1}-\frac {2\,\ln \left (\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}+\frac {4}{9}\right )}{9}-\ln \left (9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{9}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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