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

Optimal. Leaf size=224 \[ -\text {RootSum}\left [\text {$\#$1}^{16}-8 \text {$\#$1}^{14}+24 \text {$\#$1}^{12}-32 \text {$\#$1}^{10}+14 \text {$\#$1}^8+8 \text {$\#$1}^6-8 \text {$\#$1}^4+2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x+1}+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^9-4 \text {$\#$1}^7+4 \text {$\#$1}^5-\text {$\#$1}}\& \right ]+\sqrt {x+1} \left (\frac {8}{7} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}-\frac {48}{35} \sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )+\frac {32}{105} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}+\frac {32}{105} \sqrt {\sqrt {\sqrt {x+1}+1}+1} \]

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Rubi [F]  time = 2.70, 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+\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 + Sqrt[1 + x]]]),x]

[Out]

(16*(1 + Sqrt[1 + Sqrt[1 + x]])^(3/2))/3 - (24*(1 + Sqrt[1 + Sqrt[1 + x]])^(5/2))/5 + (8*(1 + Sqrt[1 + Sqrt[1
+ x]])^(7/2))/7 - 32*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]]]] + 48*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, Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]]] - 16*Defer[Subst][Defer[Int][x^6/(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}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3 \left (-2+x^2\right )}{\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 {x \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 x (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 (-1+x^2\right ) \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^2-3 x^4+x^6-\frac {2 x^2 \left (2-3 x^2+x^4\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 {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-16 \operatorname {Subst}\left (\int \frac {x^2 \left (2-3 x^2+x^4\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 {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-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 {3 x^4}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}+\frac {x^6}{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 {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-16 \operatorname {Subst}\left (\int \frac {x^6}{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 )+48 \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 )\\ \end {align*}

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Mathematica [A]  time = 0.71, size = 257, normalized size = 1.15 \begin {gather*} 8 \left (\frac {1}{8} \text {RootSum}\left [2 \text {$\#$1}^{16}-8 \text {$\#$1}^{12}+8 \text {$\#$1}^{10}+14 \text {$\#$1}^8-32 \text {$\#$1}^6+24 \text {$\#$1}^4-8 \text {$\#$1}^2+1\&,\frac {2 \text {$\#$1}^{11} \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )-3 \text {$\#$1}^9 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )+\text {$\#$1}^7 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )}{2 \text {$\#$1}^{14}-6 \text {$\#$1}^{10}+5 \text {$\#$1}^8+7 \text {$\#$1}^6-12 \text {$\#$1}^4+6 \text {$\#$1}^2-1}\&\right ]+\frac {1}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {3}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {2}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.28, size = 178, normalized size = 0.79 \begin {gather*} -\frac {16}{105} \left (-2+9 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{105} \sqrt {1+\sqrt {1+x}} \left (4+15 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{-\text {$\#$1}+4 \text {$\#$1}^5-4 \text {$\#$1}^7+\text {$\#$1}^9}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(-2 + 9*Sqrt[1 + x])*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 + (8*Sqrt[1 + Sqrt[1 + x]]*(4 + 15*Sqrt[1 + x])
*Sqrt[1 + Sqrt[1 + Sqrt[1 + x]]])/105 - 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 + 4*#1^5 - 4*#1^7 + #1^9) & ]

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

[Out]

sage0*x

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maple [A]  time = 0.25, size = 157, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\)	\(157\)
default	\(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\)	\(157\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/7*(1+(1+(1+x)^(1/2))^(1/2))^(7/2)-24/5*(1+(1+(1+x)^(1/2))^(1/2))^(5/2)+16/3*(1+(1+(1+x)^(1/2))^(1/2))^(3/2)-
sum((_R^6-3*_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 {\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+(1+x)^(1/2))^(1/2))^(1/2),x, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

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

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