3.29.84 \(\int \frac {x^4}{1-x \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}} \, dx\)

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]

Int[x^4/(1 - x*Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]]),x]

[Out]

x^2/2 + (x*Sqrt[1 + x^2])/2 + 1/(28*(x - Sqrt[1 + x^2])^(7/2)) - 1/(4*(x - Sqrt[1 + x^2])^(3/2)) + 2/Sqrt[x -
Sqrt[1 + x^2]] + (5*Sqrt[x - Sqrt[1 + x^2]])/4 - (2*(x - Sqrt[1 + x^2])^(3/2))/3 + (x - Sqrt[1 + x^2])^(5/2)/2
0 + ArcSinh[x]/2 - Log[-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8]/8 + Defer[Int][x/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*
x^6 + x^8), x] + (3*Defer[Int][x^2/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x])/4 + Defer[Int][x^3/(-1 + 2*x^
3 + x^4 + 2*x^5 + 2*x^6 + x^8), x]/2 - (7*Defer[Int][x^4/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x])/4 + Def
er[Int][x^5/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x]/2 - 2*Defer[Int][x^6/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^
6 + x^8), x] + Defer[Int][Sqrt[1 + x^2]/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] - 2*Defer[Int][(x^3*Sqrt[
1 + x^2])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] - Defer[Int][(x^4*Sqrt[1 + x^2])/(-1 + 2*x^3 + x^4 + 2*
x^5 + 2*x^6 + x^8), x] - 2*Defer[Int][(x^5*Sqrt[1 + x^2])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] - Defer
[Int][(x^6*Sqrt[1 + x^2])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] - 2*Defer[Int][Sqrt[x - Sqrt[1 + x^2]]/
(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] + 5*Defer[Int][(x^3*Sqrt[x - Sqrt[1 + x^2]])/(-1 + 2*x^3 + x^4 +
2*x^5 + 2*x^6 + x^8), x] + 2*Defer[Int][(x^4*Sqrt[x - Sqrt[1 + x^2]])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8)
, x] + 4*Defer[Int][(x^5*Sqrt[x - Sqrt[1 + x^2]])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] + 2*Defer[Int][
(x^6*Sqrt[x - Sqrt[1 + x^2]])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] - Defer[Int][(Sqrt[1 + x^2]*Sqrt[x
- Sqrt[1 + x^2]])/(-1 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + x^8), x] + Defer[Int][(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), x] + 2*Defer[Int][(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), x] + Defer[Int][(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), x] - Defer[Int][(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), x] + Defer[Int][(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), x] - 2*Defer[Int][(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), x]

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.

[In]

Integrate[x^4/(1 - x*Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]]),x]

[Out]

Result too large to show

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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.

[In]

IntegrateAlgebraic[x^4/(1 - x*Sqrt[1 + x^2]*Sqrt[x - Sqrt[1 + x^2]]),x]

[Out]

1/(28*(x - Sqrt[1 + x^2])^(7/2)) - 1/(4*(x - Sqrt[1 + x^2])^(3/2)) + 4/Sqrt[x - Sqrt[1 + x^2]] - (3*Sqrt[x - S
qrt[1 + x^2]])/4 + (x - Sqrt[1 + x^2])^(5/2)/20 + 1/(4*(-x + Sqrt[1 + x^2])^2) + (3*Log[-x + Sqrt[1 + x^2]])/2
- RootSum[-1 + 4*#1^3 + #1^8 & , (-4*Log[Sqrt[x - Sqrt[1 + x^2]] - #1] + 3*Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*
#1 - Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1^2 - Log[Sqrt[x - Sqrt[1 + x^2]] - #1]*#1^5 + Log[Sqrt[x - Sqrt[1 + x
^2]] - #1]*#1^6)/(3*#1 + 2*#1^6) & ]

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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.

[In]

integrate(x^4/(1-x*(x^2+1)^(1/2)*(x-(x^2+1)^(1/2))^(1/2)),x, algorithm="fricas")

[Out]

Timed out

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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.

[In]

integrate(x^4/(1-x*(x^2+1)^(1/2)*(x-(x^2+1)^(1/2))^(1/2)),x, algorithm="giac")

[Out]

integrate(-x^4/(sqrt(x^2 + 1)*sqrt(x - sqrt(x^2 + 1))*x - 1), x)

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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.

[In]

int(x^4/(1-x*(x^2+1)^(1/2)*(x-(x^2+1)^(1/2))^(1/2)),x)

[Out]

int(x^4/(1-x*(x^2+1)^(1/2)*(x-(x^2+1)^(1/2))^(1/2)),x)

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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.

[In]

integrate(x^4/(1-x*(x^2+1)^(1/2)*(x-(x^2+1)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

-integrate(x^4/(sqrt(x^2 + 1)*sqrt(x - sqrt(x^2 + 1))*x - 1), x)

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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.

[In]

int(-x^4/(x*(x^2 + 1)^(1/2)*(x - (x^2 + 1)^(1/2))^(1/2) - 1),x)

[Out]

-int(x^4/(x*(x^2 + 1)^(1/2)*(x - (x^2 + 1)^(1/2))^(1/2) - 1), x)

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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.

[In]

integrate(x**4/(1-x*(x**2+1)**(1/2)*(x-(x**2+1)**(1/2))**(1/2)),x)

[Out]

-Integral(x**4/(x*sqrt(x - sqrt(x**2 + 1))*sqrt(x**2 + 1) - 1), x)

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