Optimal. Leaf size=245 \[ -\frac {1}{3} (1-x)^{5/6} x (x+1)^{7/6}-\frac {1}{18} (1-x)^{5/6} (x+1)^{7/6}-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{x+1}-\frac {19 \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{108 \sqrt {3}}+\frac {19 \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{108 \sqrt {3}}-\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {19}{162} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {19}{162} \tan ^{-1}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right ) \]
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Rubi [A] time = 0.39, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6126, 90, 80, 50, 63, 331, 295, 634, 618, 204, 628, 203} \[ -\frac {1}{3} (1-x)^{5/6} x (x+1)^{7/6}-\frac {1}{18} (1-x)^{5/6} (x+1)^{7/6}-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{x+1}-\frac {19 \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{108 \sqrt {3}}+\frac {19 \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{108 \sqrt {3}}-\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {19}{162} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {19}{162} \tan ^{-1}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right ) \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 203
Rule 204
Rule 295
Rule 331
Rule 618
Rule 628
Rule 634
Rule 6126
Rubi steps
\begin {align*} \int e^{\frac {1}{3} \tanh ^{-1}(x)} x^2 \, dx &=\int \frac {x^2 \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {1}{3} \int \frac {\left (-1-\frac {x}{3}\right ) \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}+\frac {19}{54} \int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}+\frac {19}{162} \int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {19}{27} \operatorname {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {19}{27} \operatorname {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {19}{81} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19}{81} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}+\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19}{81} \operatorname {Subst}\left (\int \frac {-\frac {1}{2}-\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19}{324} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19}{324} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19 \operatorname {Subst}\left (\int \frac {-\sqrt {3}+2 x}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{108 \sqrt {3}}+\frac {19 \operatorname {Subst}\left (\int \frac {\sqrt {3}+2 x}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{108 \sqrt {3}}\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19 \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{108 \sqrt {3}}+\frac {19 \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{108 \sqrt {3}}+\frac {19}{162} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,-\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {19}{162} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=-\frac {19}{54} (1-x)^{5/6} \sqrt [6]{1+x}-\frac {1}{18} (1-x)^{5/6} (1+x)^{7/6}-\frac {1}{3} (1-x)^{5/6} x (1+x)^{7/6}-\frac {19}{81} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {19}{162} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19}{162} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {19 \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{108 \sqrt {3}}+\frac {19 \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{108 \sqrt {3}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 59, normalized size = 0.24 \[ -\frac {1}{90} (1-x)^{5/6} \left (38 \sqrt [6]{2} \, _2F_1\left (-\frac {1}{6},\frac {5}{6};\frac {11}{6};\frac {1-x}{2}\right )+5 \sqrt [6]{x+1} \left (6 x^2+7 x+1\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 308, normalized size = 1.26 \[ \frac {19}{324} \, \sqrt {3} \log \left (1444 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1444 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1444\right ) - \frac {19}{324} \, \sqrt {3} \log \left (-1444 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1444 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1444\right ) + \frac {1}{54} \, {\left (18 \, x^{3} + 3 \, x^{2} + x - 22\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} - \frac {19}{81} \, \arctan \left (\sqrt {3} + \frac {1}{19} \, \sqrt {-1444 \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1444 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1444} - 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) - \frac {19}{81} \, \arctan \left (-\sqrt {3} + 2 \, \sqrt {\sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1} - 2 \, \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) + \frac {19}{81} \, \arctan \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )^{\frac {1}{3}} x^{2}\, dx \]
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 \[ \int x^{2} \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {1}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{1/3} \,d x \]
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 \[ \int x^{2} \sqrt [3]{\frac {x + 1}{\sqrt {1 - x^{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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