Optimal. Leaf size=311 \[ -\text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^3-1\& ,\frac {\text {$\#$1}^6 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )-\text {$\#$1}^5 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )-\text {$\#$1}^2 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )+3 \text {$\#$1} \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )-4 \log \left (\sqrt {x-\sqrt {x^2+1}}-\text {$\#$1}\right )}{2 \text {$\#$1}^6+3 \text {$\#$1}}\& \right ]+\frac {1}{20} \left (x-\sqrt {x^2+1}\right )^{5/2}-\frac {3}{4} \sqrt {x-\sqrt {x^2+1}}+\frac {1}{4 \left (\sqrt {x^2+1}-x\right )^2}+\frac {4}{\sqrt {x-\sqrt {x^2+1}}}-\frac {1}{4 \left (x-\sqrt {x^2+1}\right )^{3/2}}+\frac {1}{28 \left (x-\sqrt {x^2+1}\right )^{7/2}}+\frac {3}{2} \log \left (\sqrt {x^2+1}-x\right ) \]
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Rubi [F] time = 8.48, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x^4}{1-x \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {x^4}{1-x \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}} \, dx &=\int \left (x+\sqrt {1+x^2}+\frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {2 x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {2 x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {x \left (-1+3 x^3+x^4+2 x^5+x^6\right )}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-2 \sqrt {x-\sqrt {1+x^2}}+x^3 \sqrt {x-\sqrt {1+x^2}}-\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}+x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}-\frac {2 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {5 x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {2 x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {4 x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {2 x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {2 x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {2 x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}\right ) \, dx\\ &=\frac {x^2}{2}-2 \int \frac {x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \sqrt {x-\sqrt {1+x^2}} \, dx-2 \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+4 \int \frac {x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+5 \int \frac {x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \sqrt {1+x^2} \, dx+\int \frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x \left (-1+3 x^3+x^4+2 x^5+x^6\right )}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int x^3 \sqrt {x-\sqrt {1+x^2}} \, dx-\int \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}} \, dx+\int x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx\\ &=\frac {x^2}{2}+\frac {1}{2} x \sqrt {1+x^2}-\frac {1}{8} \log \left (-1+2 x^3+x^4+2 x^5+2 x^6+x^8\right )+\frac {1}{16} \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^3 \left (1+x^2\right )}{x^{9/2}} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2 \left (1+x^2\right )^2}{x^{9/2}} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{8} \int \frac {-8 x-6 x^2-4 x^3+14 x^4-4 x^5+16 x^6}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^{5/2}} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \int \frac {1}{\sqrt {1+x^2}} \, dx-2 \int \frac {x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+4 \int \frac {x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+5 \int \frac {x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\operatorname {Subst}\left (\int \frac {1+x^2}{x^{3/2}} \, dx,x,x-\sqrt {1+x^2}\right )\\ &=\frac {x^2}{2}+\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{2} \sinh ^{-1}(x)-\frac {1}{8} \log \left (-1+2 x^3+x^4+2 x^5+2 x^6+x^8\right )+\frac {1}{16} \operatorname {Subst}\left (\int \left (-\frac {1}{x^{9/2}}+\frac {2}{x^{5/2}}-2 x^{3/2}+x^{7/2}\right ) \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{16} \operatorname {Subst}\left (\int \frac {\left (-1+x^4\right )^2}{x^{9/2}} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{8} \int \frac {2 x \left (4+3 x+2 x^2-7 x^3+2 x^4-8 x^5\right )}{1-2 x^3-x^4-2 x^5-2 x^6-x^8} \, dx+\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{x^{5/2}}+\frac {2}{\sqrt {x}}+x^{3/2}\right ) \, dx,x,x-\sqrt {1+x^2}\right )-2 \int \frac {x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+4 \int \frac {x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+5 \int \frac {x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\operatorname {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x-\sqrt {1+x^2}\right )\\ &=\frac {x^2}{2}+\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{56 \left (x-\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{4 \left (x-\sqrt {1+x^2}\right )^{3/2}}+\frac {2}{\sqrt {x-\sqrt {1+x^2}}}+\sqrt {x-\sqrt {1+x^2}}-\frac {2}{3} \left (x-\sqrt {1+x^2}\right )^{3/2}+\frac {1}{20} \left (x-\sqrt {1+x^2}\right )^{5/2}+\frac {1}{72} \left (x-\sqrt {1+x^2}\right )^{9/2}+\frac {1}{2} \sinh ^{-1}(x)-\frac {1}{8} \log \left (-1+2 x^3+x^4+2 x^5+2 x^6+x^8\right )-\frac {1}{16} \operatorname {Subst}\left (\int \left (\frac {1}{x^{9/2}}-\frac {2}{\sqrt {x}}+x^{7/2}\right ) \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{4} \int \frac {x \left (4+3 x+2 x^2-7 x^3+2 x^4-8 x^5\right )}{1-2 x^3-x^4-2 x^5-2 x^6-x^8} \, dx-2 \int \frac {x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+4 \int \frac {x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+5 \int \frac {x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx\\ &=\frac {x^2}{2}+\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{28 \left (x-\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{4 \left (x-\sqrt {1+x^2}\right )^{3/2}}+\frac {2}{\sqrt {x-\sqrt {1+x^2}}}+\frac {5}{4} \sqrt {x-\sqrt {1+x^2}}-\frac {2}{3} \left (x-\sqrt {1+x^2}\right )^{3/2}+\frac {1}{20} \left (x-\sqrt {1+x^2}\right )^{5/2}+\frac {1}{2} \sinh ^{-1}(x)-\frac {1}{8} \log \left (-1+2 x^3+x^4+2 x^5+2 x^6+x^8\right )-\frac {1}{4} \int \left (-\frac {4 x}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {3 x^2}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {2 x^3}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {7 x^4}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}-\frac {2 x^5}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}+\frac {8 x^6}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8}\right ) \, dx-2 \int \frac {x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+4 \int \frac {x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+5 \int \frac {x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx\\ &=\frac {x^2}{2}+\frac {1}{2} x \sqrt {1+x^2}+\frac {1}{28 \left (x-\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{4 \left (x-\sqrt {1+x^2}\right )^{3/2}}+\frac {2}{\sqrt {x-\sqrt {1+x^2}}}+\frac {5}{4} \sqrt {x-\sqrt {1+x^2}}-\frac {2}{3} \left (x-\sqrt {1+x^2}\right )^{3/2}+\frac {1}{20} \left (x-\sqrt {1+x^2}\right )^{5/2}+\frac {1}{2} \sinh ^{-1}(x)-\frac {1}{8} \log \left (-1+2 x^3+x^4+2 x^5+2 x^6+x^8\right )+\frac {1}{2} \int \frac {x^3}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\frac {1}{2} \int \frac {x^5}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\frac {3}{4} \int \frac {x^2}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\frac {7}{4} \int \frac {x^4}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^6}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^3 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^5 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^4 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^6 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+2 \int \frac {x^3 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-2 \int \frac {x^7 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+4 \int \frac {x^5 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+5 \int \frac {x^3 \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {\sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^4 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^6 \sqrt {1+x^2}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^4 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx-\int \frac {x^5 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx+\int \frac {x^6 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+2 x^3+x^4+2 x^5+2 x^6+x^8} \, dx\\ \end {align*}
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Mathematica [B] time = 17.59, size = 19036, normalized size = 61.21 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.43, size = 311, normalized size = 1.00 \begin {gather*} \frac {1}{28 \left (x-\sqrt {1+x^2}\right )^{7/2}}-\frac {1}{4 \left (x-\sqrt {1+x^2}\right )^{3/2}}+\frac {4}{\sqrt {x-\sqrt {1+x^2}}}-\frac {3}{4} \sqrt {x-\sqrt {1+x^2}}+\frac {1}{20} \left (x-\sqrt {1+x^2}\right )^{5/2}+\frac {1}{4 \left (-x+\sqrt {1+x^2}\right )^2}+\frac {3}{2} \log \left (-x+\sqrt {1+x^2}\right )-\text {RootSum}\left [-1+4 \text {$\#$1}^3+\text {$\#$1}^8\&,\frac {-4 \log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right )+3 \log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}-\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2-\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5+\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^6}{3 \text {$\#$1}+2 \text {$\#$1}^6}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int -\frac {x^{4}}{\sqrt {x^{2} + 1} \sqrt {x - \sqrt {x^{2} + 1}} x - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{1-x \sqrt {x^{2}+1}\, \sqrt {x -\sqrt {x^{2}+1}}}\, dx\]
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 \begin {gather*} -\int \frac {x^{4}}{\sqrt {x^{2} + 1} \sqrt {x - \sqrt {x^{2} + 1}} x - 1}\,{d x} \end {gather*}
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 \begin {gather*} -\int \frac {x^4}{x\,\sqrt {x^2+1}\,\sqrt {x-\sqrt {x^2+1}}-1} \,d x \end {gather*}
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 \begin {gather*} - \int \frac {x^{4}}{x \sqrt {x - \sqrt {x^{2} + 1}} \sqrt {x^{2} + 1} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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