Optimal. Leaf size=292 \[ -\frac {1}{12} (1-x)^{2/3} \sqrt [3]{x+1} (1-3 x)-\frac {1}{4} (1-x) (x+3)+\frac {1}{12} \sqrt [3]{1-x} (x+1)^{2/3} (3 x+1)+\frac {1}{12} \sqrt [6]{1-x} (x+1)^{5/6} (3 x+2)-\frac {1}{12} (1-x)^{5/6} \sqrt [6]{x+1} (3 x+10)+\frac {1}{4} \sqrt {1-x} x \sqrt {x+1}+\frac {1}{6} \tan ^{-1}\left (\frac {\sqrt [6]{x+1}}{\sqrt [6]{1-x}}\right )-\frac {4 \tan ^{-1}\left (\frac {\sqrt [3]{1-x}-2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {5}{6} \tan ^{-1}\left (\frac {\sqrt [3]{1-x}-\sqrt [3]{x+1}}{\sqrt [6]{1-x} \sqrt [6]{x+1}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{1-x} \sqrt [6]{x+1}}{\sqrt [3]{1-x}+\sqrt [3]{x+1}}\right )}{6 \sqrt {3}} \]
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Rubi [A] time = 1.85, antiderivative size = 522, normalized size of antiderivative = 1.79, number of steps used = 46, number of rules used = 21, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6688, 6742, 50, 60, 517, 195, 216, 675, 890, 63, 240, 209, 634, 618, 204, 628, 203, 26, 21, 331, 295} \[ \frac {x^2}{4}+\frac {1}{4} \sqrt {1-x^2} x+\frac {x}{2}-\frac {1}{4} (1-x)^{5/6} (x+1)^{7/6}-\frac {1}{4} (1-x)^{7/6} (x+1)^{5/6}+\frac {5}{12} \sqrt [6]{1-x} (x+1)^{5/6}-\frac {1}{4} (1-x)^{4/3} (x+1)^{2/3}+\frac {1}{3} \sqrt [3]{1-x} (x+1)^{2/3}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{x+1}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{x+1}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{x+1}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (x+1)+\frac {1}{3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )-\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt {3}}+\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt {3}}-\frac {1}{3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+1\right )+\frac {1}{4} \sin ^{-1}(x)-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )}{3 \sqrt {3}}+\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {1}{3} \tan ^{-1}\left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 21
Rule 26
Rule 50
Rule 60
Rule 63
Rule 195
Rule 203
Rule 204
Rule 209
Rule 216
Rule 240
Rule 295
Rule 331
Rule 517
Rule 618
Rule 628
Rule 634
Rule 675
Rule 890
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx &=\int \frac {x \sqrt [3]{1+x}}{-\sqrt [3]{1-x}+\sqrt [6]{1-x^2}} \, dx\\ &=\int \left (\frac {1}{2} (1-x)^{2/3} \sqrt [3]{1+x}+\frac {1}{2} \sqrt [3]{1-x} \sqrt [3]{1+x} \sqrt [6]{1-x^2}+\frac {1}{2} \sqrt [3]{1+x} \sqrt [3]{1-x^2}+\frac {\sqrt [3]{1+x} \sqrt {1-x^2}}{2 \sqrt [3]{1-x}}+\frac {\sqrt [3]{1+x} \left (1-x^2\right )^{2/3}}{2 (1-x)^{2/3}}-\frac {\sqrt [3]{1+x} \left (1-x^2\right )^{5/6}}{2 (-1+x)}\right ) \, dx\\ &=\frac {1}{2} \int (1-x)^{2/3} \sqrt [3]{1+x} \, dx+\frac {1}{2} \int \sqrt [3]{1-x} \sqrt [3]{1+x} \sqrt [6]{1-x^2} \, dx+\frac {1}{2} \int \sqrt [3]{1+x} \sqrt [3]{1-x^2} \, dx+\frac {1}{2} \int \frac {\sqrt [3]{1+x} \sqrt {1-x^2}}{\sqrt [3]{1-x}} \, dx+\frac {1}{2} \int \frac {\sqrt [3]{1+x} \left (1-x^2\right )^{2/3}}{(1-x)^{2/3}} \, dx-\frac {1}{2} \int \frac {\sqrt [3]{1+x} \left (1-x^2\right )^{5/6}}{-1+x} \, dx\\ &=-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{6} \int \frac {(1-x)^{2/3}}{(1+x)^{2/3}} \, dx+\frac {1}{2} \int \sqrt [3]{1-x} (1+x)^{2/3} \, dx+\frac {1}{2} \int \sqrt [6]{1-x} (1+x)^{5/6} \, dx+\frac {1}{2} \int (1+x) \, dx-\frac {1}{2} \int \frac {(1-x)^{5/6} (1+x)^{7/6}}{-1+x} \, dx+\frac {1}{2} \int \sqrt {1-x^2} \, dx\\ &=\frac {x}{2}+\frac {x^2}{4}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {2}{9} \int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx+\frac {1}{4} \int \frac {1}{\sqrt {1-x^2}} \, dx+\frac {1}{3} \int \frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}} \, dx+\frac {5}{12} \int \frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}} \, dx+\frac {1}{2} \int \frac {(1+x)^{7/6}}{\sqrt [6]{1-x}} \, dx\\ &=\frac {x}{2}+\frac {x^2}{4}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )+\frac {5}{36} \int \frac {1}{(1-x)^{5/6} \sqrt [6]{1+x}} \, dx+\frac {2}{9} \int \frac {1}{(1-x)^{2/3} \sqrt [3]{1+x}} \, dx+\frac {7}{12} \int \frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=\frac {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\frac {7}{36} \int \frac {1}{\sqrt [6]{1-x} (1+x)^{5/6}} \, dx-\frac {5}{6} \operatorname {Subst}\left (\int \frac {1}{\sqrt [6]{2-x^6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=\frac {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {5}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{6} \operatorname {Subst}\left (\int \frac {x^4}{\left (2-x^6\right )^{5/6}} \, dx,x,\sqrt [6]{1-x}\right )\\ &=\frac {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {5}{18} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {5}{18} \operatorname {Subst}\left (\int \frac {1-\frac {\sqrt {3} x}{2}}{1-\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {5}{18} \operatorname {Subst}\left (\int \frac {1+\frac {\sqrt {3} x}{2}}{1+\sqrt {3} x+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{6} \operatorname {Subst}\left (\int \frac {x^4}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )\\ &=\frac {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)-\frac {5}{18} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {5}{72} \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 {5}{72} \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 {7}{18} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{18} \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 {7}{18} \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 {5 \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 )}{24 \sqrt {3}}-\frac {5 \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 )}{24 \sqrt {3}}\\ &=\frac {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )+\frac {5 \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt {3}}-\frac {5 \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{24 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )-\frac {7}{72} \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 {7}{72} \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 {5}{36} \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 {5}{36} \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 {7 \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 )}{24 \sqrt {3}}+\frac {7 \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 )}{24 \sqrt {3}}\\ &=\frac {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {5}{36} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {5}{36} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt {3}}+\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )+\frac {7}{36} \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 {7}{36} \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 {x}{2}+\frac {x^2}{4}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{1+x}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{1+x}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{1+x}+\frac {1}{3} \sqrt [3]{1-x} (1+x)^{2/3}-\frac {1}{4} (1-x)^{4/3} (1+x)^{2/3}+\frac {5}{12} \sqrt [6]{1-x} (1+x)^{5/6}-\frac {1}{4} (1-x)^{7/6} (1+x)^{5/6}-\frac {1}{4} (1-x)^{5/6} (1+x)^{7/6}+\frac {1}{4} x \sqrt {1-x^2}+\frac {1}{4} \sin ^{-1}(x)-\frac {2}{3} \tan ^{-1}\left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {1}{3} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {1}{3} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {2 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (1+x)+\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right )-\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt {3}}+\frac {\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )}{12 \sqrt {3}}-\frac {1}{3} \log \left (1+\frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.78, size = 348, normalized size = 1.19 \[ -\frac {5 \sqrt {1-x^2} \, _2F_1\left (\frac {1}{6},\frac {1}{6};\frac {7}{6};\frac {1-x}{2}\right )}{6 \sqrt [6]{2} \sqrt [3]{1-x} \sqrt {x+1}}-\frac {2^{2/3} \sqrt [3]{1-x^2} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {1-x}{2}\right )}{3 \sqrt [3]{x+1}}-\frac {1}{12} \sqrt [3]{x+1} \left (-4\ 2^{2/3} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {x+1}{2}\right )+\frac {(3 x+10) \left (1-x^2\right )^{5/6}}{x+1}-\frac {(3 x+2) \sqrt {1-x^2}}{\sqrt [3]{1-x}}-(3 x+1) \sqrt [3]{1-x^2}-3 \sqrt [3]{1-x} x \sqrt [6]{1-x^2}-\frac {3 \sqrt [3]{1-x} x (x+2)}{\sqrt [3]{1-x^2}}+(1-x)^{2/3} (1-3 x)\right )-\frac {7 \left (1-x^2\right )^{5/6} \, _2F_1\left (\frac {5}{6},\frac {5}{6};\frac {11}{6};\frac {1-x}{2}\right )}{30\ 2^{5/6} (x+1)^{5/6}}+\frac {(1-x)^{5/6} (x+1)^{5/6} \sin ^{-1}(x)}{4 \left (1-x^2\right )^{5/6}} \]
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [F] time = 117.70, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.28, size = 865, normalized size = 2.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (1+x \right )^{\frac {2}{3}} \sqrt {1-x}}{-\left (1-x \right )^{\frac {5}{6}} \left (1+x \right )^{\frac {1}{3}}+\left (1-x \right )^{\frac {2}{3}} \sqrt {1+x}}\, dx\]
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 \[ \int \frac {{\left (x + 1\right )}^{\frac {2}{3}} x \sqrt {-x + 1}}{\sqrt {x + 1} {\left (-x + 1\right )}^{\frac {2}{3}} - {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {5}{6}}}\,{d x} \]
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 \[ \int \frac {x\,\sqrt {1-x}\,{\left (x+1\right )}^{2/3}}{{\left (1-x\right )}^{2/3}\,\sqrt {x+1}-{\left (1-x\right )}^{5/6}\,{\left (x+1\right )}^{1/3}} \,d x \]
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 \[ \int \frac {x \sqrt {1 - x} \left (x + 1\right )^{\frac {2}{3}}}{- \left (1 - x\right )^{\frac {5}{6}} \sqrt [3]{x + 1} + \left (1 - x\right )^{\frac {2}{3}} \sqrt {x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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