Optimal. Leaf size=157 \[ \frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22 \text {ArcTan}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{27 \sqrt {3}}-\frac {11}{27} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {11 \log (x)}{81} \]
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Rubi [A]
time = 0.04, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6306, 101, 156,
12, 93} \begin {gather*} -\frac {22 \text {ArcTan}\left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )}{27 \sqrt {3}}+\frac {1}{3} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x^3+\frac {4}{9} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x^2+\frac {14}{27} \sqrt [3]{\frac {1}{x}+1} \left (\frac {x-1}{x}\right )^{2/3} x-\frac {11}{27} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {11 \log (x)}{81} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 101
Rule 156
Rule 6306
Rubi steps
\begin {align*} \int e^{\frac {2}{3} \coth ^{-1}(x)} x^2 \, dx &=-\text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {1}{3} \text {Subst}\left (\int \frac {\frac {8}{3}+2 x}{\sqrt [3]{1-x} x^3 (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3+\frac {1}{6} \text {Subst}\left (\int \frac {-\frac {28}{9}-\frac {8 x}{3}}{\sqrt [3]{1-x} x^2 (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {1}{6} \text {Subst}\left (\int \frac {44}{27 \sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22}{81} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {14}{27} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x+\frac {4}{9} \sqrt [3]{1+\frac {1}{x}} \left (-\frac {1-x}{x}\right )^{2/3} x^2+\frac {1}{3} \sqrt [3]{1+\frac {1}{x}} \left (\frac {-1+x}{x}\right )^{2/3} x^3-\frac {22 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-\frac {1-x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )}{27 \sqrt {3}}-\frac {11}{27} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{-\frac {1-x}{x}}\right )-\frac {11 \log (x)}{81}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 4.90, size = 340, normalized size = 2.17 \begin {gather*} -\frac {e^{-\frac {10}{3} \coth ^{-1}(x)} \left (-22750000-20915440 e^{2 \coth ^{-1}(x)}+7026175 e^{4 \coth ^{-1}(x)}+7394140 e^{6 \coth ^{-1}(x)}-433485 e^{8 \coth ^{-1}(x)}+22750000 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{2 \coth ^{-1}(x)}\right )+15227940 e^{2 \coth ^{-1}(x)} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{2 \coth ^{-1}(x)}\right )-14083160 e^{4 \coth ^{-1}(x)} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{2 \coth ^{-1}(x)}\right )-8250060 e^{6 \coth ^{-1}(x)} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{2 \coth ^{-1}(x)}\right )+1456000 e^{8 \coth ^{-1}(x)} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};e^{2 \coth ^{-1}(x)}\right )+54 e^{8 \coth ^{-1}(x)} \left (475+782 e^{2 \coth ^{-1}(x)}+325 e^{4 \coth ^{-1}(x)}\right ) \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {16}{3};e^{2 \coth ^{-1}(x)}\right )+162 e^{8 \coth ^{-1}(x)} \left (35+64 e^{2 \coth ^{-1}(x)}+29 e^{4 \coth ^{-1}(x)}\right ) \, _5F_4\left (2,2,2,2,\frac {7}{3};1,1,1,\frac {16}{3};e^{2 \coth ^{-1}(x)}\right )+486 e^{8 \coth ^{-1}(x)} \, _6F_5\left (2,2,2,2,2,\frac {7}{3};1,1,1,1,\frac {16}{3};e^{2 \coth ^{-1}(x)}\right )+972 e^{10 \coth ^{-1}(x)} \, _6F_5\left (2,2,2,2,2,\frac {7}{3};1,1,1,1,\frac {16}{3};e^{2 \coth ^{-1}(x)}\right )+486 e^{12 \coth ^{-1}(x)} \, _6F_5\left (2,2,2,2,2,\frac {7}{3};1,1,1,1,\frac {16}{3};e^{2 \coth ^{-1}(x)}\right )\right )}{49140} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.95, size = 613, normalized size = 3.90 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 149, normalized size = 0.95 \begin {gather*} -\frac {22}{81} \, \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 (11 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {8}{3}} - 10 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{3}} + 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{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 {11}{81} \, \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 {22}{81} \, \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 = 0.34, size = 100, normalized size = 0.64 \begin {gather*} \frac {1}{27} \, {\left (9 \, x^{3} + 21 \, x^{2} + 26 \, x + 14\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \frac {22}{81} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {11}{81} \, \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 {22}{81} \, \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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\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 = 144, normalized size = 0.92 \begin {gather*} -\frac {22}{81} \, \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 {10 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x + 1} - \frac {11 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{{\left (x + 1\right )}^{2}} - 35 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}\right )}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {11}{81} \, \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 {22}{81} \, \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 = 1.19, size = 171, normalized size = 1.09 \begin {gather*} -\frac {22\,\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-\frac {484}{729}\right )}{81}-\frac {\frac {70\,{\left (\frac {x-1}{x+1}\right )}^{2/3}}{27}-\frac {20\,{\left (\frac {x-1}{x+1}\right )}^{5/3}}{27}+\frac {22\,{\left (\frac {x-1}{x+1}\right )}^{8/3}}{27}}{\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}-\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (-\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )+\ln \left (\frac {484\,{\left (\frac {x-1}{x+1}\right )}^{1/3}}{729}-9\,{\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right )}^2\right )\,\left (\frac {11}{81}+\frac {\sqrt {3}\,11{}\mathrm {i}}{81}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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