Integrand size = 56, antiderivative size = 292 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=-\frac {1}{12} (1-3 x) (1-x)^{2/3} \sqrt [3]{1+x}+\frac {1}{4} \sqrt {1-x} x \sqrt {1+x}-\frac {1}{4} (1-x) (3+x)+\frac {1}{12} \sqrt [3]{1-x} (1+x)^{2/3} (1+3 x)+\frac {1}{12} \sqrt [6]{1-x} (1+x)^{5/6} (2+3 x)-\frac {1}{12} (1-x)^{5/6} \sqrt [6]{1+x} (10+3 x)+\frac {1}{6} \arctan \left (\frac {\sqrt [6]{1+x}}{\sqrt [6]{1-x}}\right )-\frac {4 \arctan \left (\frac {\sqrt [3]{1-x}-2 \sqrt [3]{1+x}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}-\frac {5}{6} \arctan \left (\frac {\sqrt [3]{1-x}-\sqrt [3]{1+x}}{\sqrt [6]{1-x} \sqrt [6]{1+x}}\right )+\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{1-x} \sqrt [6]{1+x}}{\sqrt [3]{1-x}+\sqrt [3]{1+x}}\right )}{6 \sqrt {3}} \]
[Out]
Time = 1.52 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.79, number of steps used = 46, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6820, 6874, 52, 62, 531, 201, 222, 689, 904, 65, 246, 215, 648, 632, 210, 642, 209, 26, 21, 338, 301} \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\frac {\arcsin (x)}{4}-\frac {2}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )}{3 \sqrt {3}}+\frac {1}{3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}\right )-\frac {1}{3} \arctan \left (\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+\sqrt {3}\right )-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x+1}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{3 \sqrt {3}}+\frac {x^2}{4}+\frac {1}{4} \sqrt {1-x^2} x+\frac {x}{2}-\frac {1}{4} (1-x)^{5/6} (x+1)^{7/6}-\frac {1}{4} (1-x)^{7/6} (x+1)^{5/6}+\frac {5}{12} \sqrt [6]{1-x} (x+1)^{5/6}-\frac {1}{4} (1-x)^{4/3} (x+1)^{2/3}+\frac {1}{3} \sqrt [3]{1-x} (x+1)^{2/3}-\frac {1}{4} (1-x)^{5/3} \sqrt [3]{x+1}+\frac {1}{6} (1-x)^{2/3} \sqrt [3]{x+1}-\frac {7}{12} (1-x)^{5/6} \sqrt [6]{x+1}-\frac {1}{9} \log (1-x)+\frac {1}{9} \log (x+1)+\frac {1}{3} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )-\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}-\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt {3}}+\frac {\log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+\frac {\sqrt {3} \sqrt [6]{1-x}}{\sqrt [6]{x+1}}+1\right )}{12 \sqrt {3}}-\frac {1}{3} \log \left (\frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}+1\right ) \]
[In]
[Out]
Rule 21
Rule 26
Rule 52
Rule 62
Rule 65
Rule 201
Rule 209
Rule 210
Rule 215
Rule 222
Rule 246
Rule 301
Rule 338
Rule 531
Rule 632
Rule 642
Rule 648
Rule 689
Rule 904
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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 {\arcsin (x)}{4}+\frac {2 \arctan \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 {\arcsin (x)}{4}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \arctan \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} \text {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 {\arcsin (x)}{4}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \arctan \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} \text {Subst}\left (\int \frac {1}{1+x^6} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{6} \text {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 {\arcsin (x)}{4}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \arctan \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} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {5}{18} \text {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} \text {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} \text {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 {\arcsin (x)}{4}-\frac {5}{18} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \arctan \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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {7}{18} \text {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} \text {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 \text {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 \text {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 {\arcsin (x)}{4}-\frac {2}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}-\frac {2 \arctan \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} \text {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} \text {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} \text {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} \text {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 \text {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 \text {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 {\arcsin (x)}{4}-\frac {2}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {5}{36} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {5}{36} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {2 \arctan \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} \text {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} \text {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 {\arcsin (x)}{4}-\frac {2}{3} \arctan \left (\frac {\sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{3 \sqrt {3}}+\frac {1}{3} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {1}{3} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{1-x}}{\sqrt [6]{1+x}}\right )-\frac {2 \arctan \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*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 36.73 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=-\frac {1}{12} \sqrt [3]{1+x} \left ((1-3 x) (1-x)^{2/3}-\frac {3 \sqrt [3]{1-x} x (2+x)}{\sqrt [3]{1-x^2}}-3 \sqrt [3]{1-x} x \sqrt [6]{1-x^2}-(1+3 x) \sqrt [3]{1-x^2}-\frac {(2+3 x) \sqrt {1-x^2}}{\sqrt [3]{1-x}}+\frac {(10+3 x) \left (1-x^2\right )^{5/6}}{1+x}-4\ 2^{2/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},\frac {1+x}{2}\right )\right )+\frac {1}{36} \left (9 \arcsin (x)+14 \arctan \left (\frac {\sqrt [3]{1+x}}{\sqrt [6]{1-x^2}}\right )+7 \left (1+i \sqrt {3}\right ) \arctan \left (\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{1+x}}{2 \sqrt [6]{1-x^2}}\right )+7 \left (1-i \sqrt {3}\right ) \arctan \left (\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{1+x}}{2 \sqrt [6]{1-x^2}}\right )+8 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{1-x^2}}{(1+x)^{2/3}}}{\sqrt {3}}\right )-\frac {15\ 2^{5/6} \sqrt {1-x^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{6},\frac {1}{6},\frac {7}{6},\frac {1-x}{2}\right )}{\sqrt [3]{1-x} \sqrt {1+x}}-8 \log \left ((1+x)^{2/3}+\sqrt [3]{1-x^2}\right )+4 \log \left (\sqrt [3]{1+x}+x \sqrt [3]{1+x}-(1+x)^{2/3} \sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}\right )\right ) \]
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\[\int \frac {x \left (1+x \right )^{\frac {2}{3}} \sqrt {1-x}}{-\left (1-x \right )^{\frac {5}{6}} \left (1+x \right )^{\frac {1}{3}}+\left (1-x \right )^{\frac {2}{3}} \sqrt {1+x}}d x\]
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Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 785, normalized size of antiderivative = 2.69 \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\int { \frac {{\left (x + 1\right )}^{\frac {2}{3}} x \sqrt {-x + 1}}{\sqrt {x + 1} {\left (-x + 1\right )}^{\frac {2}{3}} - {\left (x + 1\right )}^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {5}{6}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\sqrt {1-x} x (1+x)^{2/3}}{-(1-x)^{5/6} \sqrt [3]{1+x}+(1-x)^{2/3} \sqrt {1+x}} \, dx=\int \frac {x\,\sqrt {1-x}\,{\left (x+1\right )}^{2/3}}{{\left (1-x\right )}^{2/3}\,\sqrt {x+1}-{\left (1-x\right )}^{5/6}\,{\left (x+1\right )}^{1/3}} \,d x \]
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