\(\int \frac {-1+x^2}{\sqrt {1+x} (1+x^2) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\) [2256]

Optimal result

Integrand size = 39, antiderivative size = 170 \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=-\frac {16}{3} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{3} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3-\text {$\#$1}^5-8 \text {$\#$1}^7+12 \text {$\#$1}^9-6 \text {$\#$1}^{11}+\text {$\#$1}^{13}}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \]

[In]

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

[Out]

-8*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] + (8*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 + 16*Defer[Subst][Defer[Int][(2 -
8*x^4 + 8*x^6 + 14*x^8 - 32*x^10 + 24*x^12 - 8*x^14 + x^16)^(-1), x], x, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] - 1
6*Defer[Subst][Defer[Int][x^2/(2 - 8*x^4 + 8*x^6 + 14*x^8 - 32*x^10 + 24*x^12 - 8*x^14 + x^16), x], x, Sqrt[1
+ Sqrt[1 + Sqrt[1 + x]]]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(-1+x) \sqrt {1+x}}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \\ & = 2 \text {Subst}\left (\int \frac {x^2 \left (-2+x^2\right )}{\sqrt {1+\sqrt {1+x}} \left (1+\left (-1+x^2\right )^2\right )} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )^2 \left (-2+\left (-1+x^2\right )^2\right )}{\sqrt {1+x} \left (1+x^4 \left (-2+x^2\right )^2\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \frac {(-1+x)^2 x (1+x)^{3/2} \left (-2+\left (-1+x^2\right )^2\right )}{1+x^4 \left (-2+x^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 8 \text {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )^2 \left (-1+x^2\right ) \left (-2+x^4 \left (-2+x^2\right )^2\right )}{1+\left (-1+x^2\right )^4 \left (-2+\left (-1+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = 8 \text {Subst}\left (\int \left (-1+x^2+\frac {2 \left (1-x^2\right )}{1+\left (-1+x^2\right )^4 \left (-2+\left (-1+x^2\right )^2\right )^2}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = -8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+16 \text {Subst}\left (\int \frac {1-x^2}{1+\left (-1+x^2\right )^4 \left (-2+\left (-1+x^2\right )^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = -8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+16 \text {Subst}\left (\int \left (\frac {1}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}-\frac {x^2}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ & = -8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+16 \text {Subst}\left (\int \frac {1}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-16 \text {Subst}\left (\int \frac {x^2}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.88 \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\frac {8}{3} \left (-2+\sqrt {1+\sqrt {1+x}}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{2 \text {$\#$1}^3-\text {$\#$1}^5-8 \text {$\#$1}^7+12 \text {$\#$1}^9-6 \text {$\#$1}^{11}+\text {$\#$1}^{13}}\&\right ] \]

[In]

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

[Out]

(8*(-2 + Sqrt[1 + Sqrt[1 + x]])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 3
2*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]/(2*#1^3 - #1^5 - 8*#1^7 + 1
2*#1^9 - 6*#1^11 + #1^13) & ]

Maple [N/A] (verified)

Time = 0.18 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\)	\(133\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-8 \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+1\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\)	\(133\)

[In]

int((x^2-1)/(1+x)^(1/2)/(x^2+1)/(1+(1+(1+x)^(1/2))^(1/2))^(1/2),x,method=_RETURNVERBOSE)

[Out]

8/3*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)-8*(1+(1+(1+x)^(1/2))^(1/2))^(1/2)+sum((-_R^2+1)/(_R^15-7*_R^13+18*_R^11-20
*_R^9+7*_R^7+3*_R^5-2*_R^3)*ln((1+(1+(1+x)^(1/2))^(1/2))^(1/2)-_R),_R=RootOf(_Z^16-8*_Z^14+24*_Z^12-32*_Z^10+1
4*_Z^8+8*_Z^6-8*_Z^4+2))

Fricas [F(-1)]

Timed out. \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.19 \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{2} + 1\right )} \sqrt {x + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.19 \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{2} + 1\right )} \sqrt {x + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.95 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.19 \[ \int \frac {-1+x^2}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\sqrt {x+1}} \,d x \]

[In]

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

[Out]

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