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

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

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Rubi [F]  time = 2.34, 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}{\sqrt {1+x} \left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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) \sqrt {1+x}}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx\\ &=2 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 \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 [A]  time = 0.62, size = 191, normalized size = 1.12 \begin {gather*} \frac {8}{3} \left (\sqrt {\sqrt {x+1}+1}-2\right ) \sqrt {\sqrt {\sqrt {x+1}+1}+1}-\text {RootSum}\left [2 \text {$\#$1}^{16}-8 \text {$\#$1}^{12}+8 \text {$\#$1}^{10}+14 \text {$\#$1}^8-32 \text {$\#$1}^6+24 \text {$\#$1}^4-8 \text {$\#$1}^2+1\&,\frac {\text {$\#$1}^{13} \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )-\text {$\#$1}^{11} \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )}{2 \text {$\#$1}^{14}-6 \text {$\#$1}^{10}+5 \text {$\#$1}^8+7 \text {$\#$1}^6-12 \text {$\#$1}^4+6 \text {$\#$1}^2-1}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[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[1 - 8*#1^2 + 24*#1^4 - 32*#1^6 +
14*#1^8 + 8*#1^10 - 8*#1^12 + 2*#1^16 & , (-(Log[1/Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]] - #1]*#1^11) + Log[1/Sqrt[1
+ Sqrt[1 + Sqrt[1 + x]]] - #1]*#1^13)/(-1 + 6*#1^2 - 12*#1^4 + 7*#1^6 + 5*#1^8 - 6*#1^10 + 2*#1^14) & ]

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IntegrateAlgebraic [A]  time = 0.31, size = 170, normalized size = 1.00 \begin {gather*} -\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 + (8*Sqrt[1 + Sqrt[1 + x]]*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/3 - RootSu
m[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]/(2*#1^3 - #1^5 - 8*#1^7 + 12*#1^9 - 6*#1^11 + #1^13) & ]

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

[Out]

sage0*x

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maple [B]  time = 0.20, size = 133, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

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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 {x + 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)/(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)

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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 {x+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)),x)

[Out]

int((x^2 - 1)/((x^2 + 1)*(((x + 1)^(1/2) + 1)^(1/2) + 1)^(1/2)*(x + 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 ) \sqrt {x + 1}}{\left (x^{2} + 1\right ) \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)/(1+x)**(1/2)/(x**2+1)/(1+(1+(1+x)**(1/2))**(1/2))**(1/2),x)

[Out]

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

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