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

Optimal. Leaf size=196 \[ -\text {RootSum}\left [\text {$\#$1}^{16}-8 \text {$\#$1}^{14}+24 \text {$\#$1}^{12}-32 \text {$\#$1}^{10}+14 \text {$\#$1}^8+8 \text {$\#$1}^6-8 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x+1}+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^{11}-5 \text {$\#$1}^9+8 \text {$\#$1}^7-4 \text {$\#$1}^5-\text {$\#$1}^3+\text {$\#$1}}\& \right ]+\frac {8}{5} \sqrt {x+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}-\frac {32}{15} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}-\frac {32}{15} \sqrt {\sqrt {\sqrt {x+1}+1}+1} \]

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Rubi [F]  time = 2.72, 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 {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-16*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 + (8*(1 + Sqrt[1 + Sqrt[1 + x]])^(5/2))/5 + 32*Defer[Subst][Defer[In
t][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]]
]] - 16*Defer[Subst][Defer[Int][x^4/(2 - 8*x^4 + 8*x^6 + 14*x^8 - 32*x^10 + 24*x^12 - 8*x^14 + x^16), x], x, S
qrt[1 + Sqrt[1 + Sqrt[1 + x]]]]

Rubi steps

\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3 \left (-2+x^2\right )}{\sqrt {1+x} \left (2-2 x^2+x^4\right ) \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^3 \left (-1-2 x^2+x^4\right )}{\sqrt {1+x} \left (1+4 x^4-4 x^6+x^8\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {(-1+x)^3 (1+x)^{5/2} \left (-1-2 x^2+x^4\right )}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x^6 \left (-2+x^2\right )^3 \left (-2+4 x^4-4 x^6+x^8\right )}{1+4 \left (-1+x^2\right )^4-4 \left (-1+x^2\right )^6+\left (-1+x^2\right )^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (-2 x^2+x^4-\frac {2 x^2 \left (-2+x^2\right )}{1+4 \left (-1+x^2\right )^4-4 \left (-1+x^2\right )^6+\left (-1+x^2\right )^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=-\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+\frac {8}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}-16 \operatorname {Subst}\left (\int \frac {x^2 \left (-2+x^2\right )}{1+4 \left (-1+x^2\right )^4-4 \left (-1+x^2\right )^6+\left (-1+x^2\right )^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=-\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+\frac {8}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}-16 \operatorname {Subst}\left (\int \left (-\frac {2 x^2}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}+\frac {x^4}{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 )\\ &=-\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}+\frac {8}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}-16 \operatorname {Subst}\left (\int \frac {x^4}{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 )+32 \operatorname {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*}

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Mathematica [F]  time = 3.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.30, size = 177, normalized size = 0.90 \begin {gather*} -\frac {32}{15} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{15} \left (-4+3 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}-\text {$\#$1}^3-4 \text {$\#$1}^5+8 \text {$\#$1}^7-5 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-32*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/15 + (8*(-4 + 3*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqr
t[1 + x]]])/15 - RootSum[2 - 8*#1^4 + 8*#1^6 + 14*#1^8 - 32*#1^10 + 24*#1^12 - 8*#1^14 + #1^16 & , Log[Sqrt[1
+ Sqrt[1 + Sqrt[1 + x]]] - #1]/(#1 - #1^3 - 4*#1^5 + 8*#1^7 - 5*#1^9 + #1^11) & ]

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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^2-1)/(x^2+1)/(1+(1+x)^(1/2))^(1/2)/(1+(1+(1+x)^(1/2))^(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*} \mathit {sage}_{0} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x

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maple [A]  time = 0.19, size = 137, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}-\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\)	\(137\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}-\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)-16/3*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)-sum((_R^4-2*_R^2)/(_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+14*_Z^8+8*_Z^6-8*_Z^4+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{2} + 1\right )} \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\sqrt {\sqrt {x+1}+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {\sqrt {x + 1} + 1} \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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