Integrand size = 44, antiderivative size = 136 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+3 x}{2 \sqrt {2}}\right )+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )+\frac {7}{27} \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x^2}}\right )-\frac {7}{54} \log \left (3+2 x+3 x^2\right ) \]
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Time = 1.18 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6874, 201, 221, 648, 632, 210, 642, 1034, 12, 1095, 1051, 1045, 212, 267} \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=-\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {x+1}{\sqrt {2} \sqrt {x^2+1}}\right )+\frac {4}{27} \sqrt {2} \arctan \left (\frac {3 x+1}{2 \sqrt {2}}\right )+\frac {7}{27} \text {arctanh}\left (\frac {1-x}{2 \sqrt {x^2+1}}\right )-\frac {x^2}{6}-\frac {1}{6} \sqrt {x^2+1} x+\frac {8 \sqrt {x^2+1}}{9}-\frac {7}{54} \log \left (3 x^2+2 x+3\right )+\frac {8 x}{9} \]
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Rule 12
Rule 201
Rule 210
Rule 212
Rule 221
Rule 267
Rule 632
Rule 642
Rule 648
Rule 1034
Rule 1045
Rule 1051
Rule 1095
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x^2}{1-x^3+\sqrt {1+x^2}+x^2 \sqrt {1+x^2}}-\frac {2 x^2}{\sqrt {1+x^2} \left (-1+x^3-\left (1+x^2\right )^{3/2}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {x^2}{\sqrt {1+x^2} \left (-1+x^3-\left (1+x^2\right )^{3/2}\right )} \, dx\right )-\int \frac {x^2}{1-x^3+\sqrt {1+x^2}+x^2 \sqrt {1+x^2}} \, dx \\ & = -\left (2 \int \left (-\frac {1}{3}+\frac {2}{9 \sqrt {1+x^2}}-\frac {x}{3 \sqrt {1+x^2}}+\frac {2 x}{3 \left (3+2 x+3 x^2\right )}+\frac {3+5 x}{9 \sqrt {1+x^2} \left (3+2 x+3 x^2\right )}\right ) \, dx\right )-\int \left (-\frac {2}{9}+\frac {x}{3}+\frac {\sqrt {1+x^2}}{3}+\frac {-3-5 x}{9 \left (3+2 x+3 x^2\right )}-\frac {2 x \sqrt {1+x^2}}{3 \left (3+2 x+3 x^2\right )}\right ) \, dx \\ & = \frac {8 x}{9}-\frac {x^2}{6}-\frac {1}{9} \int \frac {-3-5 x}{3+2 x+3 x^2} \, dx-\frac {2}{9} \int \frac {3+5 x}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx-\frac {1}{3} \int \sqrt {1+x^2} \, dx-\frac {4}{9} \int \frac {1}{\sqrt {1+x^2}} \, dx+\frac {2}{3} \int \frac {x}{\sqrt {1+x^2}} \, dx+\frac {2}{3} \int \frac {x \sqrt {1+x^2}}{3+2 x+3 x^2} \, dx-\frac {4}{3} \int \frac {x}{3+2 x+3 x^2} \, dx \\ & = \frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {4 \text {arcsinh}(x)}{9}+\frac {1}{18} \int \frac {4-4 x}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx-\frac {1}{18} \int \frac {16+16 x}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx+\frac {5}{54} \int \frac {2+6 x}{3+2 x+3 x^2} \, dx+\frac {4}{27} \int \frac {1}{3+2 x+3 x^2} \, dx-\frac {1}{6} \int \frac {1}{\sqrt {1+x^2}} \, dx-\frac {2}{9} \int \frac {2+6 x}{3+2 x+3 x^2} \, dx+\frac {2}{9} \int -\frac {2 x^2}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx+\frac {4}{9} \int \frac {1}{3+2 x+3 x^2} \, dx \\ & = \frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {11 \text {arcsinh}(x)}{18}-\frac {7}{54} \log \left (3+2 x+3 x^2\right )-\frac {8}{27} \text {Subst}\left (\int \frac {1}{-32-x^2} \, dx,x,2+6 x\right )-\frac {4}{9} \int \frac {x^2}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx-\frac {8}{9} \text {Subst}\left (\int \frac {1}{-32-x^2} \, dx,x,2+6 x\right )-\frac {128}{9} \text {Subst}\left (\int \frac {1}{-4096-2 x^2} \, dx,x,\frac {32+32 x}{\sqrt {1+x^2}}\right )-\frac {1024}{9} \text {Subst}\left (\int \frac {1}{32768-2 x^2} \, dx,x,\frac {-64+64 x}{\sqrt {1+x^2}}\right ) \\ & = \frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {11 \text {arcsinh}(x)}{18}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+3 x}{2 \sqrt {2}}\right )+\frac {1}{9} \sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )+\frac {4}{9} \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x^2}}\right )-\frac {7}{54} \log \left (3+2 x+3 x^2\right )-\frac {4}{27} \int \frac {1}{\sqrt {1+x^2}} \, dx-\frac {4}{27} \int \frac {-3-2 x}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx \\ & = \frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+3 x}{2 \sqrt {2}}\right )+\frac {1}{9} \sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )+\frac {4}{9} \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x^2}}\right )-\frac {7}{54} \log \left (3+2 x+3 x^2\right )-\frac {1}{27} \int \frac {-10-10 x}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx+\frac {1}{27} \int \frac {2-2 x}{\sqrt {1+x^2} \left (3+2 x+3 x^2\right )} \, dx \\ & = \frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+3 x}{2 \sqrt {2}}\right )+\frac {1}{9} \sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )+\frac {4}{9} \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x^2}}\right )-\frac {7}{54} \log \left (3+2 x+3 x^2\right )-\frac {64}{27} \text {Subst}\left (\int \frac {1}{-1024-2 x^2} \, dx,x,\frac {16+16 x}{\sqrt {1+x^2}}\right )-\frac {800}{27} \text {Subst}\left (\int \frac {1}{12800-2 x^2} \, dx,x,\frac {40-40 x}{\sqrt {1+x^2}}\right ) \\ & = \frac {8 x}{9}-\frac {x^2}{6}+\frac {8 \sqrt {1+x^2}}{9}-\frac {1}{6} x \sqrt {1+x^2}-\frac {41 \text {arcsinh}(x)}{54}+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+3 x}{2 \sqrt {2}}\right )+\frac {4}{27} \sqrt {2} \arctan \left (\frac {1+x}{\sqrt {2} \sqrt {1+x^2}}\right )+\frac {7}{27} \text {arctanh}\left (\frac {1-x}{2 \sqrt {1+x^2}}\right )-\frac {7}{54} \log \left (3+2 x+3 x^2\right ) \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {1}{54} \left (48 x-9 x^2+48 \sqrt {1+x^2}-9 x \sqrt {1+x^2}+16 \sqrt {2} \arctan \left (\frac {1+x-\sqrt {1+x^2}}{\sqrt {2}}\right )+55 \log \left (-x+\sqrt {1+x^2}\right )-14 \log \left (-2-x-x^2+(1+x) \sqrt {1+x^2}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(653\) vs. \(2(99)=198\).
Time = 0.04 (sec) , antiderivative size = 654, normalized size of antiderivative = 4.81
\[-\frac {x^{2}}{6}+\frac {8 x}{9}-\frac {7 \ln \left (3 x^{2}+2 x +3\right )}{54}+\frac {4 \sqrt {2}\, \arctan \left (\frac {\left (6 x +2\right ) \sqrt {2}}{8}\right )}{27}-\frac {41 \,\operatorname {arcsinh}\left (x \right )}{54}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+5 \,\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{12 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}+\frac {3 \sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{8 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}-\frac {x \sqrt {x^{2}+1}}{6}+\frac {8 \sqrt {x^{2}+1}}{9}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (13 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+43 \,\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{216 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}-\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (-11 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\, \left (1+x \right )}{2 \left (\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1\right ) \left (1-x \right )}\right )+\operatorname {arctanh}\left (\sqrt {\frac {2 \left (1+x \right )^{2}}{\left (1-x \right )^{2}}+2}\right )\right )}{36 \sqrt {\frac {\frac {\left (1+x \right )^{2}}{\left (1-x \right )^{2}}+1}{\left (\frac {1+x}{1-x}+1\right )^{2}}}\, \left (\frac {1+x}{1-x}+1\right )}\]
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Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=-\frac {1}{6} \, x^{2} - \frac {1}{18} \, \sqrt {x^{2} + 1} {\left (3 \, x - 16\right )} + \frac {4}{27} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) + \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 1\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 1}\right ) - \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x + 1\right )} + \frac {1}{2} \, \sqrt {2} \sqrt {x^{2} + 1}\right ) + \frac {8}{9} \, x + \frac {7}{54} \, \log \left (3 \, x^{2} - \sqrt {x^{2} + 1} {\left (3 \, x - 1\right )} - x + 2\right ) - \frac {7}{54} \, \log \left (3 \, x^{2} + 2 \, x + 3\right ) - \frac {7}{54} \, \log \left (x^{2} - \sqrt {x^{2} + 1} {\left (x + 1\right )} + x + 2\right ) + \frac {41}{54} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
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Timed out. \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\text {Timed out} \]
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\[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\int { \frac {x^{2} {\left (\sqrt {x^{2} + 1} - 2\right )}}{{\left (x^{3} - {\left (x^{2} + 1\right )}^{\frac {3}{2}} - 1\right )} \sqrt {x^{2} + 1}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.29 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=-\frac {1}{6} \, x^{2} - \frac {1}{18} \, \sqrt {x^{2} + 1} {\left (3 \, x - 16\right )} + \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 3 \, \sqrt {x^{2} + 1} - 1\right )}\right ) + \frac {4}{27} \, \sqrt {2} \arctan \left (\frac {1}{4} \, \sqrt {2} {\left (3 \, x + 1\right )}\right ) - \frac {4}{27} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (x - \sqrt {x^{2} + 1} + 1\right )}\right ) + \frac {8}{9} \, x + \frac {7}{54} \, \log \left (3 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 2 \, x + 2 \, \sqrt {x^{2} + 1} + 1\right ) - \frac {7}{54} \, \log \left ({\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, x - 2 \, \sqrt {x^{2} + 1} + 3\right ) - \frac {7}{54} \, \log \left (3 \, x^{2} + 2 \, x + 3\right ) + \frac {41}{54} \, \log \left (-x + \sqrt {x^{2} + 1}\right ) \]
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Time = 0.64 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.59 \[ \int \frac {x^2 \left (2-\sqrt {1+x^2}\right )}{\sqrt {1+x^2} \left (1-x^3+\left (1+x^2\right )^{3/2}\right )} \, dx=\frac {8\,x}{9}-\frac {41\,\mathrm {asinh}\left (x\right )}{54}-\left (\frac {x}{6}-\frac {8}{9}\right )\,\sqrt {x^2+1}-\frac {x^2}{6}+\frac {\sqrt {2}\,\ln \left (x+\frac {1}{3}-\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )\,\left (-\frac {16}{27}+\frac {\sqrt {2}\,14{}\mathrm {i}}{27}\right )\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\ln \left (x+\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )\,\left (\frac {16}{27}+\frac {\sqrt {2}\,14{}\mathrm {i}}{27}\right )\,1{}\mathrm {i}}{8}+\frac {\sqrt {2}\,\left (\frac {4}{81}+\frac {\sqrt {2}\,44{}\mathrm {i}}{81}\right )\,\left (\ln \left (x+\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )-\ln \left (1+\left (\frac {2}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}\right )\,\sqrt {x^2+1}-\frac {x}{3}-\frac {\sqrt {2}\,x\,2{}\mathrm {i}}{3}\right )\right )\,1{}\mathrm {i}}{8\,\sqrt {{\left (\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )}^2+1}}+\frac {\sqrt {2}\,\left (-\frac {4}{81}+\frac {\sqrt {2}\,44{}\mathrm {i}}{81}\right )\,\left (\ln \left (x+\frac {1}{3}-\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )-\ln \left (1-\left (-\frac {2}{3}+\frac {\sqrt {2}\,1{}\mathrm {i}}{3}\right )\,\sqrt {x^2+1}-\frac {x}{3}+\frac {\sqrt {2}\,x\,2{}\mathrm {i}}{3}\right )\right )\,1{}\mathrm {i}}{8\,\sqrt {{\left (-\frac {1}{3}+\frac {\sqrt {2}\,2{}\mathrm {i}}{3}\right )}^2+1}} \]
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