Optimal. Leaf size=157 \[ \frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^3+\frac{4}{9} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{14}{27} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{11}{27} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{11 \log (x)}{81}-\frac{22 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3}} \]
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Rubi [A] time = 0.0701047, antiderivative size = 157, normalized size of antiderivative = 1., 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 = {6171, 99, 151, 12, 91} \[ \frac{1}{3} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^3+\frac{4}{9} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x^2+\frac{14}{27} \sqrt [3]{\frac{1}{x}+1} \left (\frac{x-1}{x}\right )^{2/3} x-\frac{11}{27} \log \left (\sqrt [3]{\frac{1}{x}+1}-\sqrt [3]{\frac{x-1}{x}}\right )-\frac{11 \log (x)}{81}-\frac{22 \tan ^{-1}\left (\frac{2 \sqrt [3]{\frac{x-1}{x}}}{\sqrt{3} \sqrt [3]{\frac{1}{x}+1}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 6171
Rule 99
Rule 151
Rule 12
Rule 91
Rubi steps
\begin{align*} \int e^{\frac{2}{3} \coth ^{-1}(x)} x^2 \, dx &=-\operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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} \operatorname{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*}
Mathematica [C] time = 7.16166, size = 340, normalized size = 2.17 \[ -\frac{e^{-\frac{10}{3} \coth ^{-1}(x)} \left (54 e^{8 \coth ^{-1}(x)} \left (782 e^{2 \coth ^{-1}(x)}+325 e^{4 \coth ^{-1}(x)}+475\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,\frac{7}{3}\right \},\left \{1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+162 e^{8 \coth ^{-1}(x)} \left (64 e^{2 \coth ^{-1}(x)}+29 e^{4 \coth ^{-1}(x)}+35\right ) \text{HypergeometricPFQ}\left (\left \{2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+486 e^{8 \coth ^{-1}(x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+972 e^{10 \coth ^{-1}(x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+486 e^{12 \coth ^{-1}(x)} \text{HypergeometricPFQ}\left (\left \{2,2,2,2,2,\frac{7}{3}\right \},\left \{1,1,1,1,\frac{16}{3}\right \},e^{2 \coth ^{-1}(x)}\right )+15227940 e^{2 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )-14083160 e^{4 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )-8250060 e^{6 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )+1456000 e^{8 \coth ^{-1}(x)} \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )+22750000 \text{Hypergeometric2F1}\left (\frac{1}{3},1,\frac{4}{3},e^{2 \coth ^{-1}(x)}\right )-20915440 e^{2 \coth ^{-1}(x)}+7026175 e^{4 \coth ^{-1}(x)}+7394140 e^{6 \coth ^{-1}(x)}-433485 e^{8 \coth ^{-1}(x)}-22750000\right )}{49140} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}{\frac{1}{\sqrt [3]{{\frac{-1+x}{1+x}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54007, size = 201, normalized size = 1.28 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46335, size = 325, normalized size = 2.07 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt [3]{\frac{x - 1}{x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18165, size = 194, normalized size = 1.24 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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